Kate bakes a $20$-inch by $18$-inch pan of cornbread. The cornbread is cut into pieces that measure $2$ inches by $2$ inches. How many pieces of cornbread does the pan contain?

1. **Calculate the area of the pan**:
   The pan has dimensions $20$ inches by $18$ inches. Therefore, the area of the pan is calculated by multiplying these dimensions:
   \[
   \text{Area of the pan} = 20 \times 18 = 360 \text{ square inches}
   \]

2. **Calculate the area of each piece of cornbread**:
   Each piece of cornbread measures $2$ inches by $2$ inches. Thus, the area of each piece is:
   \[
   \text{Area of each piece} = 2 \times 2 = 4 \text{ square inches}
   \]

3. **Determine the number of pieces**:
   To find out how many $2$-inch by $2$-inch pieces can be cut from the pan, divide the total area of the pan by the area of one piece:
   \[
   \text{Number of pieces} = \frac{\text{Area of the pan}}{\text{Area of each piece}} = \frac{360}{4} = 90
   \]

Thus, the pan contains $\boxed{\textbf{(A) } 90}$ pieces of cornbread.

Answer: 90


In triangle $\triangle ABC$, $a$, $b$, and $c$ are the opposite sides of angles $A$, $B$, and $C$ respectively, and $\dfrac{\cos B}{\cos C}=-\dfrac{b}{2a+c}$.
(1) Find the measure of angle $B$;
(2) If $b=\sqrt {13}$ and $a+c=4$, find the area of $\triangle ABC$.

Answer: \dfrac{3\sqrt{3}}{4}


Let $d(n)$ denote the number of positive divisors of $n$. For positive integer $n$ we define $f(n)$ as $$f(n) = d\left(k_1\right) + d\left(k_2\right)+ \cdots + d\left(k_m\right),$$ where $1 = k_1 < k_2 < \cdots < k_m = n$ are all divisors of the number $n$. We call an integer $n > 1$ [i]almost perfect[/i] if $f(n) = n$. Find all almost perfect numbers.

To find all almost perfect numbers, we first consider the function \( f(n) \). For a given positive integer \( n \), we define \( f(n) \) as:

\[
f(n) = d(k_1) + d(k_2) + \cdots + d(k_m),
\]

where \( 1 = k_1 < k_2 < \cdots < k_m = n \) are all the divisors of the number \( n \). Here, \( d(k) \) denotes the number of positive divisors of \( k \).

An integer \( n > 1 \) is called almost perfect if \( f(n) = n \).

We aim to identify all integers \( n \) for which this condition holds.

### Step-by-step Analysis

For small values of \( n \), we calculate \( f(n) \) directly and check if it equals \( n \).

1. **\( n = 1 \):**
   - Divisors of 1: \(\{1\}\)
   - \( f(1) = d(1) = 1 \)
   - \( n = 1 \) is not valid as \( n > 1 \).

2. **\( n = 3 \):**
   - Divisors of 3: \(\{1, 3\}\)
   - \( f(3) = d(1) + d(3) = 1 + 2 = 3 \)
   - Thus, \( 3 \) is almost perfect.

3. **\( n = 18 \):**
   - Divisors of 18: \(\{1, 2, 3, 6, 9, 18\}\)
   - \( f(18) = d(1) + d(2) + d(3) + d(6) + d(9) + d(18) = 1 + 2 + 2 + 4 + 3 + 6 = 18 \)
   - Thus, \( 18 \) is almost perfect.

4. **\( n = 36 \):**
   - Divisors of 36: \(\{1, 2, 3, 4, 6, 9, 12, 18, 36\}\)
   - \( f(36) = d(1) + d(2) + d(3) + d(4) + d(6) + d(9) + d(12) + d(18) + d(36) \)
   - \(\phantom{f(36)}= 1 + 2 + 2 + 3 + 4 + 3 + 6 + 6 + 9 = 36 \)
   - Thus, \( 36 \) is almost perfect.

### Conclusion

After manually checking these cases and realizing the specific structure of these numbers, we conclude that the set of almost perfect numbers is:

\[
\boxed{3, 18, 36}
\]

These solutions can be further supported by observing the structure of the divisors and the counting of divisors function, \( d(n) \), which leads to equality with \( n \) only in these specific cases.

Answer: 1, 3, 18, 36


Two distinct positive integers from 1 to 60 inclusive are chosen. Let the sum of the integers equal $S$ and the product equal $P$. What is the probability that $P+S$ is one less than a multiple of 7?

Answer: \frac{148}{590}


Given $(a+i)i=b+ai$, solve for $|a+bi|$.

Answer: \sqrt{2}


Given that \(AD\), \(BE\), and \(CF\) are the altitudes of the acute triangle \(\triangle ABC\). If \(AB = 26\) and \(\frac{EF}{BC} = \frac{5}{13}\), what is the length of \(BE\)?

Answer: 24


Find, with proof, all positive integers $n$ for which $2^n + 12^n + 2011^n$ is a perfect square.

The answer is $n=1$ , which is easily verified to be a valid integer $n$ .
Notice that \[2^n+12^n+2011^n\equiv 2^n+7^n \pmod{12}.\] Then for $n\geq 2$ , we have $2^n+7^n\equiv 3,5 \pmod{12}$ depending on the parity of $n$ . But perfect squares can only be $0,1,4,9\pmod{12}$ , contradiction. Therefore, we are done. $\blacksquare$

Answer: \[ n = 1 \]


In a sequence of positive integers each term after the first is $\frac{1}{3}$ of the sum of the term that precedes it and the term that follows it in the sequence. What is the 5th term of this sequence if the 1st term is 2 and the 4th term is 34?

Answer: 89


The vertices of an equilateral triangle lie on the hyperbola $xy = 4$, and a vertex of this hyperbola is the centroid of the triangle. What is the square of the area of the triangle?

Answer: 1728


Answer: \frac{\sqrt{15}-2}{6}


Let  $ABCD$  be a rectangle. Circles with diameters  $AB$  and  $CD$  meet at points  $P$  and  $Q$  inside the rectangle such that  $P$  is closer to segment  $BC$  than  $Q$ . Let  $M$  and  $N$  be the midpoints of segments  $AB$  and  $CD$ . If  $\angle MPN = 40^\circ$ , find the degree measure of  $\angle BPC$ .

*Ray Li.*

Answer: 80


Elizabetta wants to write the integers 1 to 9 in the regions of the shape shown, with one integer in each region. She wants the product of the integers in any two regions that have a common edge to be not more than 15. In how many ways can she do this?

Answer: 16


If $x=11$, $y=-8$, and $2x-3z=5y$, what is the value of $z$?

Answer: \frac{62}{3}


Let $p,$ $q,$ $r,$ $x,$ $y,$ and $z$ be positive real numbers such that $p + q + r = 2$ and $x + y + z = 1$. Find the maximum value of:
\[\frac{1}{p + q} + \frac{1}{p + r} + \frac{1}{q + r} + \frac{1}{x + y} + \frac{1}{x + z} + \frac{1}{y + z}.\]

Answer: \frac{27}{4}


Bill buys a stock for $100. On the first day, the stock decreases by $25\%$, on the second day it increases by $35\%$ from its value at the end of the first day, and on the third day, it decreases again by $15\%$. What is the overall percentage change in the stock's value over the three days?

Answer: -13.9375\%


Consider the line $15x + 6y = 90$ which forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle?
A) $21$
B) $35$
C) $41$
D) $21 + 10\sqrt{\frac{1}{29}}$

Answer: 21 + 10\sqrt{\frac{1}{29}}


In a box, there are 4 cards each with a function defined on the domain \\( R \\): \\( f_{1}(x)=x \\), \\( f_{2}(x)=|x| \\), \\( f_{3}(x)=\sin x \\), \\( f_{4}(x)=\cos x \\). Now, two cards are randomly selected from the box, and the functions written on the cards are multiplied together to form a new function. The probability that the resulting function is an odd function is __________.

Answer: \frac{2}{3}


If the legs of a right triangle are in the ratio $3:4$, find the ratio of the areas of the two triangles created by dropping an altitude from the right-angle vertex to the hypotenuse.

Answer: \frac{9}{16}


Caroline starts with the number 1, and every second she flips a fair coin; if it lands heads, she adds 1 to her number, and if it lands tails she multiplies her number by 2. Compute the expected number of seconds it takes for her number to become a multiple of 2021.

Consider this as a Markov chain on $\mathbb{Z} / 2021 \mathbb{Z}$. This Markov chain is aperiodic (since 0 can go to 0) and any number can be reached from any other number (by adding 1), so it has a unique stationary distribution $\pi$, which is uniform (since the uniform distribution is stationary). It is a well-known theorem on Markov chains that the expected return time from a state $i$ back to $i$ is equal to the inverse of the probability $\pi_{i}$ of $i$ in the stationary distribution. (One way to see this is to take a length $n \rightarrow \infty$ random walk on this chain, and note that $i$ occurs roughly $\pi_{i}$ of the time.) Since the probability of 0 is $\frac{1}{2021}$, the expected return time from 0 to 0 is 2021. After the first step (from 0), we are at 1 with probability $1 / 2$ and 0 with probability $1 / 2$, so the number of turns it takes to get from 1 to 0 on expectation is $2 \cdot 2021-2=4040$.

Answer: 4040


Determine the number of increasing sequences of positive integers $a_1 \le a_2 \le a_3 \le \cdots \le a_8 \le 1023$ such that $a_i-i$ is even for $1\le i \le 8$. The answer can be expressed as $\binom{m}{n}$ for some $m > n$. Compute the remainder when $m$ is divided by 1000.

Answer: 515


Let \(a_{1}, a_{2}, \ldots\) be an infinite sequence of integers such that \(a_{i}\) divides \(a_{i+1}\) for all \(i \geq 1\), and let \(b_{i}\) be the remainder when \(a_{i}\) is divided by 210. What is the maximal number of distinct terms in the sequence \(b_{1}, b_{2}, \ldots\)?

It is clear that the sequence \(\{a_{i}\}\) will be a concatenation of sequences of the form \(\{v_{i}\}_{i=1}^{N_{0}},\{w_{i} \cdot p_{1}\}_{i=1}^{N_{1}},\{x_{i} \cdot p_{1} p_{2}\}_{i=1}^{N_{2}},\{y_{i} \cdot p_{1} p_{2} p_{3}\}_{i=1}^{N_{3}}\), and \(\{z_{i} \cdot p_{1} p_{2} p_{3} p_{4}\}_{i=1}^{N_{4}}\), for some permutation \((p_{1}, p_{2}, p_{3}, p_{4})\) of \((2,3,5,7)\) and some sequences of integers \(\{v_{i}\} \cdot\{w_{i}\} \cdot\{x_{i}\} \cdot\{y_{i}\} \cdot\{z_{i}\}\), each coprime with 210. In \(\{v_{i}\}_{i=1}^{N_{0}}\), there are a maximum of \(\phi(210)\) distinct terms \(\bmod 210\). In \(\{w_{i} \cdot p_{1}\}_{i=1}^{N_{1}}\), there are a maximum of \(\phi\left(\frac{210}{p_{1}}\right)\) distinct terms mod 210. In \(\{x_{i} \cdot p_{1} p_{2}\}_{i=1}^{N_{2}}\), there are a maximum of \(\phi\left(\frac{210}{p_{1} p_{2}}\right)\) distinct terms \(\bmod 210\). In \(\{y_{i} \cdot p_{1} p_{2} p_{3}\}_{i=1}^{N_{3}}\), there are a maximum of \(\phi\left(\frac{210}{p_{1} p_{2} p_{3}}\right)\) distinct terms \(\bmod 210\). In \(\{z_{i} \cdot p_{1} p_{2} p_{3} p_{4}\}_{i=1}^{N_{4}}\), there can only be one distinct term \(\bmod 210\). Therefore we wish to maximize \(\phi(210)+\phi\left(\frac{210}{p_{1}}\right)+\phi\left(\frac{210}{p_{1} p_{2}}\right)+\phi\left(\frac{210}{p_{1} p_{2} p_{3}}\right)+1\) over all permutations \((p_{1}, p_{2}, p_{3}, p_{4})\) of \((2,3,5,7)\). It's easy to see that the maximum occurs when we take \(p_{1}=2, p_{2}=3, p_{3}=5, p_{4}=7\) for an answer of \(\phi(210)+\phi(105)+\phi(35)+\phi(7)+1=127\). This upper bound is clearly attainable by having the \(v_{i}\)'s cycle through the \(\phi(210)\) integers less than 210 coprime to 210, the \(w_{i}\)'s cycle through the \(\phi\left(\frac{210}{p_{1}}\right)\) integers less than \(\frac{210}{p_{1}}\) coprime to \(\frac{210}{p_{1}}\), etc.

Answer: 127


Which number is closest to \(-3.4\) on a number line?

On a number line, \(-3.4\) is between \(-4\) and \(-3\). This means that \(-3.4\) is closer to \(-3\) than to \(-4\), and so the answer is \(-3\).

Answer: -3


In a tournament, any two players play against each other. Each player earns one point for a victory, 1/2 for a draw, and 0 points for a loss. Let \( S \) be the set of the 10 lowest scores. We know that each player obtained half of their score by playing against players in \( S \).

a) What is the sum of the scores of the players in \( S \)?

b) Determine how many participants are in the tournament.

Note: Each player plays only once with each opponent.

Answer: 25


In an arithmetic sequence \(\{a_{n}\}\), where \(a_{1} > 0\) and \(5a_{8} = 8a_{13}\), what is the value of \(n\) when the sum of the first \(n\) terms \(S_{n}\) is maximized?

Answer: 21


In a survey, $82.5\%$ of respondents believed that mice were dangerous. Of these, $52.4\%$ incorrectly thought that mice commonly caused electrical fires. Given that these 27 respondents were mistaken, determine the total number of people surveyed.

Answer: 63


Given $f(x)=2x^3-6x^2+a$ (where $a$ is a constant), the function has a maximum value of 3 on the interval $[-2, 2]$. Find the minimum value of $f(x)$ on the interval $[-2, 2]$.

Answer: -29


Given a cube with an unknown volume, two of its dimensions are increased by $1$ and the third is decreased by $2$, and the volume of the resulting rectangular solid is $27$ less than that of the cube. Determine the volume of the original cube.

Answer: 125


Let \( M = \{1, 2, 3, \cdots, 1995\} \) and \( A \subseteq M \), with the constraint that if \( x \in A \), then \( 19x \notin A \). Find the maximum value of \( |A| \).

Answer: 1890


Find the product of all positive integral values of $x$ such that $x^2 - 40x + 399 = q$ for some prime number $q$. Note that there must be at least one such $x$.

Answer: 396


Find the number of subsets $S$ of $\{1,2, \ldots, 48\}$ satisfying both of the following properties: - For each integer $1 \leq k \leq 24$, exactly one of $2 k-1$ and $2 k$ is in $S$. - There are exactly nine integers $1 \leq m \leq 47$ so that both $m$ and $m+1$ are in $S$.

This problem can be thought of as laying down a series of $1 \times 2$ dominoes, with each one having either the left or right square marked. The second condition states that exactly 9 pairs of consecutive dominoes will have the leftmost one with the right square marked and the rightmost one with the left square marked. Therefore, this problem can be thought of as laying down a series of dominoes with the left square marked, followed by a series with the right square marked, followed by left square and so on and so forth, with the pattern LRLRLRL...LR. However, the left end is not guaranteed to be left marked dominoes and the right end is not guaranteed to be right marked dominoes. However, we can add a left marked domino to the left end and a right marked domino to the right end without changing the number of right-left combinations in the sequence. Further, there will be 10 of each left and right blocks, and a total of 26 dominoes, such that each block has at least 1 domino. If there are $a_{1}, a_{2}, \ldots, a_{20}$ dominoes in each block, then $a_{1}+a_{2}+\ldots+a_{20}=26$ and $a_{i}>0$ for all $1 \leq i \leq 20$. Therefore, from stars and bars, we find that there are \binom{25}{6} ways to select the dominoes and thus the subset $S$. Surprisingly, \binom{25}{6} is not too hard to compute and is just 177100.

Answer: 177100


Let $x,$ $y,$ and $z$ be positive real numbers such that
\[\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 6.\]Find the minimum value of $x^3 y^2 z.$

Answer: \frac{1}{108}


If $x=3$, $y=2x$, and $z=3y$, what is the average of $x$, $y$, and $z$?

Since $x=3$ and $y=2x$, then $y=2 \times 3=6$. Since $y=6$ and $z=3y$, then $z=3 \times 6=18$. Therefore, the average of $x, y$ and $z$ is $\frac{x+y+z}{3}=\frac{3+6+18}{3}=9$.

Answer: 9


In $\triangle ABC, AB = 10, BC = 8, CA = 7$ and side $BC$ is extended to a point $P$ such that $\triangle PAB$ is similar to $\triangle PCA$. Calculate the length of $PC$.

Answer: \frac{56}{3}


What is the value of $x$ if a cube's volume is $5x$ cubic units and its surface area is $x$ square units?

Answer: 5400


Two linear functions \( f(x) \) and \( g(x) \) satisfy the properties that for all \( x \),
- \( f(x) + g(x) = 2 \)
- \( f(f(x)) = g(g(x)) \)
and \( f(0) = 2022 \). Compute \( f(1) \).

Answer: 2021


Given that point $P$ is an intersection point of the ellipse $\frac{x^{2}}{a_{1}^{2}} + \frac{y^{2}}{b_{1}^{2}} = 1 (a_{1} > b_{1} > 0)$ and the hyperbola $\frac{x^{2}}{a_{2}^{2}} - \frac{y^{2}}{b_{2}^{2}} = 1 (a_{2} > 0, b_{2} > 0)$, $F_{1}$, $F_{2}$ are the common foci of the ellipse and hyperbola, $e_{1}$, $e_{2}$ are the eccentricities of the ellipse and hyperbola respectively, and $\angle F_{1}PF_{2} = \frac{2\pi}{3}$, find the maximum value of $\frac{1}{e_{1}} + \frac{1}{e_{2}}$.

Answer: \frac{4 \sqrt{3}}{3}


Answer: 25


The functions $f(x) = x^2-2x + m$ and $g(x) = x^2-2x + 4m$ are evaluated when $x = 4$. What is the value of $m$ if $2f(4) = g(4)$?

Answer: 4


A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is preparing to offer a discount on the selling price, but the profit margin must not be less than $10\%$. Find the maximum discount rate that can be offered.

Answer: 12\%


Determine the number $ABCC$ (written in decimal system) given that

$$
ABCC = (DD - E) \cdot 100 + DD \cdot E
$$

where $A, B, C, D,$ and $E$ are distinct digits.

Answer: 1966


Bob chooses a  $4$ -digit binary string uniformly at random, and examines an infinite sequence of uniformly and independently random binary bits. If  $N$  is the least number of bits Bob has to examine in order to find his chosen string, then find the expected value of  $N$ . For example, if Bob’s string is  $0000$  and the stream of bits begins  $101000001 \dots$ , then  $N = 7$ .

Answer: 30


What is the smallest positive integer with exactly 12 positive integer divisors?

Answer: 108


What is the least five-digit positive integer which is congruent to 5 (mod 15)?

Answer: 10,\!010


The area of a square inscribed in a semicircle compared to the area of a square inscribed in a full circle.

Answer: 2:5


In the country of Taxonia, each person pays as many thousandths of their salary in taxes as the number of tugriks that constitutes their salary. What salary is most advantageous to have?

(Salary is measured in a positive number of tugriks, not necessarily an integer.)

Answer: 500


Answer: 6


The volume of the parallelepiped determined by vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ is 6. Find the volume of the parallelepiped determined by $\mathbf{a} + 2\mathbf{b},$ $\mathbf{b} + \mathbf{c},$ and $2\mathbf{c} - 5\mathbf{a}.$

Answer: 48


Find the smallest positive integer divisible by $10$, $11$, and $12$.

Answer: 660


Two circles, one of radius 5 inches, the other of radius 2 inches, are tangent at point P. Two bugs start crawling at the same time from point P, one crawling along the larger circle at $3\pi$ inches per minute, the other crawling along the smaller circle at $2.5\pi$ inches per minute. How many minutes is it before their next meeting at point P?

Answer: 40


Given that the sum of the first $n$ terms of a geometric sequence ${a_{n}}$ is $S_{n}$, if $a_{3}=4$, $S_{3}=12$, find the common ratio.

Answer: -\frac{1}{2}


John borrows $2000$ from Mary, who charges an interest rate of $6\%$ per month (which compounds monthly). What is the least integer number of months after which John will owe more than triple what he borrowed?

Answer: 19


In an exam every question is solved by exactly four students, every pair of questions is solved by exactly one student, and none of the students solved all of the questions. Find the maximum possible number of questions in this exam.

Answer: 13


Call a number prime-looking if it is composite but not divisible by $2, 3,$ or $5.$ The three smallest prime-looking numbers are $49, 77$, and $91$. There are $168$ prime numbers less than $1000$. How many prime-looking numbers are there less than $1000$?

1. **Identify the total number of integers less than 1000**:
   There are 999 integers from 1 to 999.

2. **Calculate the number of prime numbers less than 1000**:
   It is given that there are 168 prime numbers less than 1000.

3. **Exclude the primes 2, 3, and 5 from the count**:
   Since 2, 3, and 5 are primes, we need to exclude them from the prime count when considering prime-looking numbers. This leaves us with $168 - 3 = 165$ primes.

4. **Identify the sets of numbers divisible by 2, 3, and 5**:
   - $|S_2|$: Numbers divisible by 2 are 2, 4, 6, ..., 998. This forms an arithmetic sequence with the first term 2, common difference 2, and last term 998. The number of terms is $\frac{998 - 2}{2} + 1 = 499$.
   - $|S_3|$: Numbers divisible by 3 are 3, 6, 9, ..., 999. This forms an arithmetic sequence with the first term 3, common difference 3, and last term 999. The number of terms is $\frac{999 - 3}{3} + 1 = 333$.
   - $|S_5|$: Numbers divisible by 5 are 5, 10, 15, ..., 995. This forms an arithmetic sequence with the first term 5, common difference 5, and last term 995. The number of terms is $\frac{995 - 5}{5} + 1 = 199$.

5. **Apply the Principle of Inclusion-Exclusion to find $|S_2 \cup S_3 \cup S_5|$**:
   - $|S_2 \cap S_3|$: Numbers divisible by both 2 and 3 are divisible by 6. The sequence is 6, 12, ..., 996. The number of terms is $\frac{996 - 6}{6} + 1 = 166$.
   - $|S_2 \cap S_5|$: Numbers divisible by both 2 and 5 are divisible by 10. The sequence is 10, 20, ..., 990. The number of terms is $\frac{990 - 10}{10} + 1 = 99$.
   - $|S_3 \cap S_5|$: Numbers divisible by both 3 and 5 are divisible by 15. The sequence is 15, 30, ..., 990. The number of terms is $\frac{990 - 15}{15} + 1 = 66$.
   - $|S_2 \cap S_3 \cap S_5|$: Numbers divisible by 2, 3, and 5 are divisible by 30. The sequence is 30, 60, ..., 990. The number of terms is $\frac{990 - 30}{30} + 1 = 33$.

   Using inclusion-exclusion, we calculate:
   \[
   |S_2 \cup S_3 \cup S_5| = 499 + 333 + 199 - 166 - 99 - 66 + 33 = 733
   \]

6. **Calculate the number of prime-looking numbers**:
   Prime-looking numbers are those which are not prime and not divisible by 2, 3, or 5. Using complementary counting, the number of prime-looking numbers is:
   \[
   999 - 165 - 733 = 101
   \]

   However, we must also exclude the number 1, which is neither prime nor composite, thus:
   \[
   101 - 1 = 100
   \]

   Therefore, the number of prime-looking numbers less than 1000 is $\boxed{100}$.

Answer: 100


Answer: \frac{2 \sqrt{5}}{5}


Given a function $f(x) = (m^2 - m - 1)x^{m^2 - 2m - 1}$ which is a power function and is increasing on the interval $(0, \infty)$, find the value of the real number $m$.

Answer: -1


Eric is taking a biology class. His problem sets are worth 100 points in total, his three midterms are worth 100 points each, and his final is worth 300 points. If he gets a perfect score on his problem sets and scores $60 \%, 70 \%$, and $80 \%$ on his midterms respectively, what is the minimum possible percentage he can get on his final to ensure a passing grade? (Eric passes if and only if his overall percentage is at least $70 \%)$.

We see there are a total of $100+3 \times 100+300=700$ points, and he needs $70 \% \times 700=490$ of them. He has $100+60+70+80=310$ points before the final, so he needs 180 points out of 300 on the final, which is $60 \%$.

Answer: 60 \%


Given that $α \in ( \frac{5}{4}π, \frac{3}{2}π)$, and it satisfies $\tan α + \frac{1}{\tan α} = 8$, find the value of $\sin α \cos α$ and $\sin α - \cos α$.

Answer: - \frac{\sqrt{3}}{2}


The graphs of $y=|x|$ and $y=-x^2-3x-2$ are drawn.  For every $x$, a vertical segment connecting these two graphs can be drawn as well.  Find the smallest possible length of one of these vertical segments.

Answer: 1


5 students stand in a row for a photo, where students A and B must stand next to each other, and A cannot stand at either end. Calculate the total number of possible arrangements.

Answer: 36


Given the function $f(x)=\sin (2x+φ)$, where $|φ| < \dfrac{π}{2}$, the graph is shifted to the left by $\dfrac{π}{6}$ units and is symmetric about the origin. Determine the minimum value of the function $f(x)$ on the interval $[0, \dfrac{π}{2}]$.

Answer: -\dfrac{ \sqrt{3}}{2}


Jo, Blair, and Parker take turns counting from 1, increasing by one more than the last number said by the previous person. What is the $100^{\text{th}}$ number said?

Answer: 100


On three faces of a cube, diagonals are drawn such that a triangle is formed. Find the angles of this triangle.

Answer: 60


If $n$ is the smallest positive integer for which there exist positive real numbers $a$ and $b$ such that
\[(a + bi)^n = (a - bi)^n,\]compute $\frac{b}{a}.$

Answer: \sqrt{3}


Let $A = \left\{a_{1}, a_{2}, \cdots, a_{n}\right\}$ be a set of numbers, and let the arithmetic mean of all elements in $A$ be denoted by $P(A)\left(P(A)=\frac{a_{1}+a_{2}+\cdots+a_{n}}{n}\right)$. If $B$ is a non-empty subset of $A$ such that $P(B) = P(A)$, then $B$ is called a "balance subset" of $A$. Find the number of "balance subsets" of the set $M = \{1,2,3,4,5,6,7,8,9\}$.

Answer: 51


Consider a regular octagon with side length 3, inside of which eight semicircles lie such that their diameters coincide with the sides of the octagon. Determine the area of the shaded region, which is the area inside the octagon but outside all of the semicircles.
A) $54 + 18\sqrt{2} - 9\pi$
B) $54 + 36\sqrt{2} - 9\pi$
C) $18 + 36\sqrt{2} - 9\pi$
D) $27 + 36\sqrt{2} - 4.5\pi$

Answer: 54 + 36\sqrt{2} - 9\pi


Let $\mathbf{a} = \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 2 \\ 0 \\ -1 \end{pmatrix}.$  Find the vector $\mathbf{v}$ that satisfies $\mathbf{v} \times \mathbf{a} = \mathbf{b} \times \mathbf{a}$ and $\mathbf{v} \times \mathbf{b} = \mathbf{a} \times \mathbf{b}.$

Answer: \begin{pmatrix} 3 \\ 1 \\ -1 \end{pmatrix}


A pyramid with volume 40 cubic inches has a rectangular base. If the length of the base is doubled, the width tripled and the height increased by $50\%$, what is the volume of the new pyramid, in cubic inches?

Answer: 360


Find the remainder when $123456789012$ is divided by $240$.

Answer: 132


Find the smallest positive integer  $k$  such that, for any subset  $A$  of  $S=\{1,2,\ldots,2012\}$  with  $|A|=k$ , there exist three elements  $x,y,z$  in  $A$  such that  $x=a+b$ ,  $y=b+c$ ,  $z=c+a$ , where  $a,b,c$  are in  $S$  and are distinct integers.

*Proposed by Huawei Zhu*

Answer: 1008


Given $S = \{1, 2, 3, 4\}$. Let $a_{1}, a_{2}, \cdots, a_{k}$ be a sequence composed of numbers from $S$, which includes all permutations of $(1, 2, 3, 4)$ that do not end with 1. That is, if $\left(b_{1}, b_{2}, b_{3}, b_{4}\right)$ is a permutation of $(1, 2, 3, 4)$ and $b_{4} \neq 1$, then there exist indices $1 \leq i_{1} < i_{2} < i_{3} < i_{4} \leq k$ such that $\left(a_{i_{1}}, a_{i_{2}}, a_{i_{3}}, a_{i_{4}}\right)=\left(b_{1}, b_{2}, b_{3}, b_{4}\right)$. Find the minimum value of $k$.

Answer: 11


In the diagram, \(ABCD\) is a rectangle with \(AD = 13\), \(DE = 5\), and \(EA = 12\). The area of \(ABCD\) is

Answer: 60


A bamboo pole of length 24 meters is leaning against a wall with one end on the ground. If the vertical distance from a certain point on the pole to the wall and the vertical distance from that point to the ground are both 7 meters, find the vertical distance from the end of the pole touching the wall to the ground in meters, or $\qquad$ meters.

Answer: 16 - 4\sqrt{2}


Your math friend Steven rolls five fair icosahedral dice (each of which is labelled $1,2, \ldots, 20$ on its sides). He conceals the results but tells you that at least half of the rolls are 20. Assuming that Steven is truthful, what is the probability that all three remaining concealed dice show $20 ?$

The given information is equivalent to the first two dice being 20 and 19 and there being at least two 20's among the last three dice. Thus, we need to find the probability that given at least two of the last three dice are 20's, all three are. Since there is only one way to get all three 20's and $3 \cdot 19=57$ ways to get exactly two 20's, the probability is $\frac{1}{1+57}=\frac{1}{58}$.

Answer: \frac{1}{58}


Given 60 feet of fencing, what is the greatest possible number of square feet in the area of a pen, if the pen is designed as a rectangle subdivided evenly into two square areas?

Answer: 450


Find the maximum positive integer \( n \) such that
\[
n^{2} \leq 160 \times 170 \times 180 \times 190
\]

Answer: 30499


The arithmetic sequence \( a, a+d, a+2d, a+3d, \ldots, a+(n-1)d \) has the following properties:

- When the first, third, fifth, and so on terms are added, up to and including the last term, the sum is 320.
- When the first, fourth, seventh, and so on, terms are added, up to and including the last term, the sum is 224.

What is the sum of the whole sequence?

Answer: 608


Find the smallest positive integer  $n$  for which  $315^2-n^2$  evenly divides  $315^3-n^3$ .

*Proposed by Kyle Lee*

Answer: 90


The regular octagon $ABCDEFGH$ has its center at $J$. Each of the vertices and the center are to be associated with one of the digits $1$ through $9$, with each digit used once, in such a way that the sums of the numbers on the lines $AJE$, $BJF$, $CJG$, and $DJH$ are equal. In how many ways can this be done? [asy]
size(175);
defaultpen(linewidth(0.8));
path octagon;
string labels[]={"A","B","C","D","E","F","G","H","I"};
for(int i=0;i<=7;i=i+1)
{
pair vertex=dir(135-45/2-45*i);
octagon=octagon--vertex;
label(" $"+labels[i]+"$ ",vertex,dir(origin--vertex));
}
draw(octagon--cycle);
dot(origin);
label(" $J$ ",origin,dir(0));
[/asy]

Answer: 1152


Given the function $f(x)=\sin ^{2}x+a\sin x\cos x-\cos ^{2}x$, and $f(\frac{\pi }{4})=1$.
(1) Find the value of the constant $a$;
(2) Find the smallest positive period and minimum value of $f(x)$.

Answer: -\sqrt{2}


For how many positive integers $n$ less than or equal to 500 is $$(\cos t - i\sin t)^n = \cos nt - i\sin nt$$ true for all real $t$?

Answer: 500


When $x=-2$, what is the value of $(x+1)^{3}$?

Answer: -1


Given the line $l: x-y+4=0$ and the circle $C: \begin{cases}x=1+2\cos \theta \\ y=1+2\sin \theta\end{cases} (\theta$ is a parameter), find the distance from each point on $C$ to $l$.

Answer: 2 \sqrt{2}-2


Two circles, each of radius $4$, are drawn with centers at $(0, 20)$, and $(6, 12)$. A line passing through $(4,0)$ is such that the total area of the parts of the two circles to one side of the line is equal to the total area of the parts of the two circles to the other side of it. What is the absolute value of the slope of this line?

Answer: \frac{4}{3}


For how many positive integers $n \leq 100$ is it true that $10 n$ has exactly three times as many positive divisors as $n$ has?

Let $n=2^{a} 5^{b} c$, where $2,5 \nmid c$. Then, the ratio of the number of divisors of $10 n$ to the number of divisors of $n$ is $\frac{a+2}{a+1} \frac{b+2}{b+1}=3$. Solving for $b$, we find that $b=\frac{1-a}{2 a+1}$. This forces $(a, b)=(0,1),(1,0)$. Therefore, the answers are of the form $2 k$ and $5 k$ whenever $\operatorname{gcd}(k, 10)=1$. There are 50 positive numbers of the form $2 k$ and 20 positive numbers of the form $5 k$ less than or equal to 100. Of those 70 numbers, only $\frac{1}{2} \cdot \frac{4}{5}$ have $k$ relatively prime to 10, so the answer is $70 \cdot \frac{1}{2} \cdot \frac{4}{5}=28$.

Answer: 28


A clock has a second, minute, and hour hand. A fly initially rides on the second hand of the clock starting at noon. Every time the hand the fly is currently riding crosses with another, the fly will then switch to riding the other hand. Once the clock strikes midnight, how many revolutions has the fly taken? $\emph{(Observe that no three hands of a clock coincide between noon and midnight.)}$

Answer: 245


In three-dimensional space, find the number of lattice points that have a distance of 4 from the origin.

Answer: 42


Given \( 1991 = 2^{\alpha_{1}} + 2^{\alpha_{2}} + \cdots + 2^{\alpha_{n}} \), where \( \alpha_{1}, \alpha_{2}, \cdots, \alpha_{n} \) are distinct non-negative integers, find the sum \( \alpha_{1} + \alpha_{2} + \cdots + \alpha_{n} \).

Answer: 43


For $1 \le n \le 200$, how many integers are there such that $\frac{n}{n+1}$ is a repeating decimal?

Answer: 182


In triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $\left(\sin A+\sin B\right)\left(a-b\right)=c\left(\sin C-\sin B\right)$, and $D$ is a point on side $BC$ such that $AD$ bisects angle $BAC$ and $AD=2$. Find:

$(1)$ The measure of angle $A$;

$(2)$ The minimum value of the area of triangle $\triangle ABC$.

Answer: \frac{4\sqrt{3}}{3}


Given that $y$ is a multiple of $45678$, what is the greatest common divisor of $g(y)=(3y+4)(8y+3)(14y+9)(y+14)$ and $y$?

Answer: 1512


An equilateral triangle with a side length of 1 is cut along a line parallel to one of its sides, resulting in a trapezoid. Let $S = \frac{\text{(perimeter of the trapezoid)}^2}{\text{area of the trapezoid}}$. Find the minimum value of $S$.

Answer: \frac{32\sqrt{3}}{3}


Wendy has 180 feet of fencing. She needs to enclose a rectangular space with an area that is ten times its perimeter. If she uses up all her fencing material, how many feet is the largest side of the enclosure?

Answer: 60


Hooligan Vasya likes to run on the escalator in the metro. He runs downward twice as fast as upward. If the escalator is not working, it will take Vasya 6 minutes to run up and down. If the escalator is moving downward, it will take Vasya 13.5 minutes to run up and down. How many seconds will it take Vasya to run up and down the escalator if it is moving upward? (The escalator always moves at a constant speed.)

Answer: 324


Answer: 43


Anna flips an unfair coin 10 times. The coin has a $\frac{1}{3}$ probability of coming up heads and a $\frac{2}{3}$ probability of coming up tails. What is the probability that she flips exactly 7 tails?

Answer: \frac{5120}{19683}


If the equation with respect to \( x \), \(\frac{x \lg^2 a - 1}{x + \lg a} = x\), has a solution set that contains only one element, then \( a \) equals \(\quad\) .

Answer: 10


How many different ways can 6 different books be distributed according to the following requirements?
(1) Among three people, A, B, and C, one person gets 1 book, another gets 2 books, and the last one gets 3 books;
(2) The books are evenly distributed to A, B, and C, with each person getting 2 books;
(3) The books are divided into three parts, with one part getting 4 books and the other two parts getting 1 book each;
(4) A gets 1 book, B gets 1 book, and C gets 4 books.

Answer: 30


How many times does the digit 9 appear in the list of all integers from 1 to 1000?

Answer: 300


Divide the natural numbers from 1 to 8 into two groups such that the product of all numbers in the first group is divisible by the product of all numbers in the second group. What is the minimum value of the quotient of the product of the first group divided by the product of the second group?

Answer: 70


A power locomotive's hourly electricity consumption cost is directly proportional to the cube of its speed. It is known that when the speed is $20$ km/h, the hourly electricity consumption cost is $40$ yuan. Other costs amount to $400$ yuan per hour. The maximum speed of the locomotive is $100$ km/h. At what speed should the locomotive travel to minimize the total cost of traveling from city A to city B?

Answer: 20 \sqrt[3]{5}


In a $4 \times 5$ grid, place 5 crosses such that each row and each column contains at least one cross. How many ways can this be done?

Answer: 240


Given $y_1 = x^2 - 7x + 6$, $y_2 = 7x - 3$, and $y = y_1 + xy_2$, find the value of $y$ when $x = 2$.

Answer: 18


Given a sequence of positive integers $a_1, a_2, a_3, \ldots, a_{100}$, where the number of terms equal to $i$ is $k_i$ ($i=1, 2, 3, \ldots$), let $b_j = k_1 + k_2 + \ldots + k_j$ ($j=1, 2, 3, \ldots$),
define $g(m) = b_1 + b_2 + \ldots + b_m - 100m$ ($m=1, 2, 3, \ldots$).
(I) Given $k_1 = 40, k_2 = 30, k_3 = 20, k_4 = 10, k_5 = \ldots = k_{100} = 0$, calculate $g(1), g(2), g(3), g(4)$;
(II) If the maximum term in $a_1, a_2, a_3, \ldots, a_{100}$ is 50, compare the values of $g(m)$ and $g(m+1)$;
(III) If $a_1 + a_2 + \ldots + a_{100} = 200$, find the minimum value of the function $g(m)$.

Answer: -100


Let \(A B C D\) be a square of side length 13. Let \(E\) and \(F\) be points on rays \(A B\) and \(A D\), respectively, so that the area of square \(A B C D\) equals the area of triangle \(A E F\). If \(E F\) intersects \(B C\) at \(X\) and \(B X = 6\), determine \(D F\).

Answer: \sqrt{13}


Find an ordered pair $(u,v)$ that solves the system: \begin{align*} 5u &= -7 - 2v,\\ 3u &= 4v - 25 \end{align*}

Answer: (-3,4)


Given that real numbers \( a \) and \( b \) are such that the equation \( a x^{3} - x^{2} + b x - 1 = 0 \) has three positive real roots, find the minimum value of \( P = \frac{5 a^{2} - 3 a b + 2}{a^{2}(b - a)} \) for all such \( a \) and \( b \).

Answer: 12 \sqrt{3}


On the lateral side \( C D \) of the trapezoid \( A B C D \) (\( A D \parallel B C \)), a point \( M \) is marked. From the vertex \( A \), a perpendicular \( A H \) is dropped onto the segment \( B M \). It turns out that \( A D = H D \). Find the length of the segment \( A D \), given that \( B C = 16 \), \( C M = 8 \), and \( M D = 9 \).

Answer: 18


A piece of platinum, which has a density of $2.15 \cdot 10^{4} \mathrm{kg} / \mathrm{m}^{3}$, is connected to a piece of cork wood (density $2.4 \cdot 10^{2} \mathrm{kg} / \mathrm{m}^{3}$). The density of the combined system is $4.8 \cdot 10^{2} \mathrm{kg} / \mathrm{m}^{3}$. What is the mass of the piece of wood, if the mass of the piece of platinum is $86.94 \mathrm{kg}$?

Answer: 85


Suppose that $x, y$, and $z$ are non-negative real numbers such that $x+y+z=1$. What is the maximum possible value of $x+y^{2}+z^{3}$ ?

Since $0 \leq y, z \leq 1$, we have $y^{2} \leq y$ and $z^{3} \leq z$. Therefore $x+y^{2}+z^{3} \leq x+y+z=1$. We can get $x+y^{2}+z^{3}=1$ by setting $(x, y, z)=(1,0,0)$.

Answer: 1


The coefficient of $x^2$ in the expansion of $(x+1)^5(x-2)$ is $\boxed{5}$.

Answer: -15


What is the maximum possible area of a quadrilateral with side lengths 1, 4, 7, and 8?

Answer: 18


Vertex $E$ of equilateral $\triangle{ABE}$ is in the interior of unit square $ABCD$. Let $R$ be the region consisting of all points inside $ABCD$ and outside $\triangle{ABE}$ whose distance from $AD$ is between $\frac{1}{3}$ and $\frac{2}{3}$. What is the area of $R$?

1. **Identify the region $R$:** The region $R$ consists of all points inside the unit square $ABCD$ but outside the equilateral triangle $\triangle{ABE}$, and whose distance from side $AD$ is between $\frac{1}{3}$ and $\frac{2}{3}$.

2. **Calculate the area of the middle third of the square:** The middle third of the square is the strip parallel to $AD$ with width $\frac{1}{3}$ (since the square's side length is 1). The area of this strip is:
   \[
   \text{Area of middle third} = \frac{1}{3} \times 1 = \frac{1}{3}.
   \]

3. **Analyze the equilateral triangle $\triangle{ABE}$:**
   - The side length of $\triangle{ABE}$ is the same as the side of the square, which is 1.
   - The height $h$ of $\triangle{ABE}$ can be calculated using the formula for the height of an equilateral triangle:
     \[
     h = \frac{\sqrt{3}}{2} \times \text{side length} = \frac{\sqrt{3}}{2} \times 1 = \frac{\sqrt{3}}{2}.
     \]

4. **Calculate the area of $\triangle{ABE}$:**
   \[
   \text{Area of } \triangle{ABE} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 1 \times \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{4}.
   \]

5. **Determine the area of the pentagon formed by the intersection of $\triangle{ABE}$ and the middle third strip:**
   - The pentagon can be split into a rectangle and a smaller equilateral triangle.
   - The base of the smaller equilateral triangle is $\frac{1}{3}$, and its height is:
     \[
     \text{Height of smaller triangle} = \frac{\sqrt{3}}{6}.
     \]
   - The area of the smaller equilateral triangle is:
     \[
     \text{Area of smaller triangle} = \frac{1}{2} \times \frac{1}{3} \times \frac{\sqrt{3}}{6} = \frac{\sqrt{3}}{36}.
     \]
   - The rectangle has a base of $\frac{1}{3}$ and a height of $\frac{\sqrt{3}}{2} - \frac{\sqrt{3}}{6} = \frac{\sqrt{3}}{3}$.
   - The area of the rectangle is:
     \[
     \text{Area of rectangle} = \frac{1}{3} \times \frac{\sqrt{3}}{3} = \frac{\sqrt{3}}{9}.
     \]

6. **Calculate the area of the pentagon:**
   \[
   \text{Area of pentagon} = \frac{\sqrt{3}}{36} + \frac{\sqrt{3}}{9} = \frac{4\sqrt{3}}{36} = \frac{\sqrt{3}}{9}.
   \]

7. **Calculate the area of region $R$:**
   \[
   \text{Area of } R = \text{Area of middle third} - \text{Area of pentagon} = \frac{1}{3} - \frac{\sqrt{3}}{9} = \frac{3 - \sqrt{3}}{9}.
   \]

8. **Conclusion:**
   \[
   \boxed{\text{(D) }\frac{3-\sqrt{3}}{9}}
   \]

Answer: \frac{3-\sqrt{3}}{9}


Find $120_4\times13_4\div2_4$. Express your answer in base 4.

Answer: 1110_4


Given in triangle $\triangle ABC$ the internal angles are $A$, $B$, and $C$, and the centroid is $G$. If $2\sin A \cdot \overrightarrow{GA} + \sqrt{3}\sin B \cdot \overrightarrow{GB} + 3\sin C \cdot \overrightarrow{GC} = \overrightarrow{0}$, then $\cos B = \_\_\_\_\_\_$.

Answer: \frac{1}{12}


If two distinct members of the set $\{ 2, 4, 10, 12, 15, 20, 50 \}$ are randomly selected and multiplied, what is the probability that the product is a multiple of 100? Express your answer as a common fraction.

Answer: \frac{1}{3}


Anna enjoys dinner at a restaurant in Washington, D.C., where the sales tax on meals is 10%. She leaves a 15% tip on the price of her meal before the sales tax is added, and the tax is calculated on the pre-tip amount. She spends a total of 27.50 dollars for dinner. What is the cost of her dinner without tax or tip in dollars?

Let $x$ be the cost of Anna's dinner before tax and tip.

1. **Calculate the tax:** The tax rate is 10%, so the tax amount is $\frac{10}{100}x = 0.1x$.

2. **Calculate the tip:** The tip is 15% of the pre-tax meal cost, so the tip amount is $\frac{15}{100}x = 0.15x$.

3. **Total cost:** The total cost of the meal including tax and tip is given by:
   \[
   x + 0.1x + 0.15x
   \]

4. **Simplify the expression:** Combine like terms to find the total cost in terms of $x$:
   \[
   x + 0.1x + 0.15x = 1x + 0.1x + 0.15x = 1.25x
   \]

5. **Set up the equation:** We know the total cost is $27.50, so we set up the equation:
   \[
   1.25x = 27.50
   \]

6. **Solve for $x$:** Divide both sides of the equation by 1.25 to isolate $x$:
   \[
   x = \frac{27.50}{1.25}
   \]

7. **Perform the division:** Calculate the value of $x$:
   \[
   x = 22
   \]

Thus, the cost of her dinner without tax or tip is $\boxed{\textbf{(D)}\ 22}$.

Answer: 22


Let $S$ be the set of all positive integer divisors of $129,600$. Calculate the number of numbers that are the product of two distinct elements of $S$.

Answer: 488


Let $n \ge 3$ be an integer. What is the largest possible number of interior angles greater than $180^\circ$ in an $n$-gon in the plane, given that the $n$-gon does not intersect itself and all its sides have the same length?

Let \( n \ge 3 \) be an integer, and consider an \( n \)-gon in the plane with equal side lengths. We are asked to find the largest possible number of interior angles greater than \( 180^\circ \), given that the \( n \)-gon does not intersect itself.

To solve this, we will use the following geometric principles:

1. **Polygon Interior Angle Sum Formula:**
   The sum of all interior angles of an \( n \)-gon is given by:
   \[
   (n-2) \times 180^\circ
   \]

2. **Regular Polygon Properties:**
   In a regular \( n \)-gon (where all sides and angles are equal), the measure of each interior angle is:
   \[
   \frac{(n-2) \times 180^\circ}{n}
   \]
   For a polygon not to self-intersect and remain convex (which implies no angle is greater than \( 180^\circ \)), it should ideally be a regular polygon.

3. **Analysis for Interior Angles Greater than \( 180^\circ \):**
   If any interior angle is greater than \( 180^\circ \), the polygon must be concave. However, since all sides are of equal length, it becomes impossible to form a non-self-intersecting concave polygon without compromising the side length uniformity.

   Furthermore, if we attempt to create angles greater than \( 180^\circ \) while preserving equal side lengths and non-intersection, the figure deviates from the standard convex arrangement, leading to a contradiction in a non-self-intersecting polygon setup.

Therefore, given these constraints, the largest possible number of interior angles greater than \( 180^\circ \) for a non-self-intersecting \( n \)-gon with equal side lengths is:

\[
\boxed{0}
\]

Given that having any interior angle greater than \( 180^\circ \) automatically makes the polygon concave and does not satisfy the equality of side lengths in a simple polygon, the solution concludes with 0 angles being greater than \( 180^\circ \).

Answer: 0


At lunch, Abby, Bart, Carl, Dana, and Evan share a pizza divided radially into 16 slices. Each one takes takes one slice of pizza uniformly at random, leaving 11 slices. The remaining slices of pizza form "sectors" broken up by the taken slices, e.g. if they take five consecutive slices then there is one sector, but if none of them take adjacent slices then there will be five sectors. What is the expected number of sectors formed?

Consider the more general case where there are $N$ slices and $M>0$ slices are taken. Let $S$ denote the number of adjacent pairs of slices of pizza which still remain. There are $N-M$ slices and a sector of $k$ slices contributes $k-1$ pairs to $S$. Hence the number of sectors is $N-M-S$. We compute the expected value of $S$ by looking at each adjacent pair in the original pizza: $$\mathbb{E}(S)=N \frac{\binom{N-2}{M}}{\binom{N}{M}}=N \frac{(N-M)(N-M-1)}{N(N-1)}=\frac{(N-M)(N-M-1)}{N-1}$$ The expected number of sectors is then $$N-M-\frac{(N-M)(N-M-1)}{N-1}=\frac{(N-M) M}{N-1}$$ For $N=16, M=5$ this yields \frac{11}{3}$.

Answer: \frac{11}{3}


In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. It is known that $c^{2}=a^{2}+b^{2}-4bc\cos C$, and $A-C= \frac {\pi}{2}$.
(Ⅰ) Find the value of $\cos C$;
(Ⅱ) Find the value of $\cos \left(B+ \frac {\pi}{3}\right)$.

Answer: \frac {4-3 \sqrt {3}}{10}


Let $x$, $y$, and $z$ be positive integers such that $x + y + z = 12$. What is the smallest possible value of $(x+y+z)\left(\frac{1}{x+y}+\frac{1}{x+z}+\frac{1}{y+z}\right)$?

Answer: \frac{9}{2}


There are 8 students arranged in two rows, with 4 people in each row. If students A and B must be arranged in the front row, and student C must be arranged in the back row, then the total number of different arrangements is ___ (answer in digits).

Answer: 5760


Bonnie makes the frame of a cube out of 12 pieces of wire that are each six inches long. Meanwhile Roark uses 1-inch-long pieces of wire to make a collection of unit cube frames that are not connected to each other. The total volume of Roark's cubes is the same as the volume of Bonnie's cube. What is the ratio of the total length of Bonnie's wire to the total length of Roark's wire? Express your answer as a common fraction. [asy]
size(50);
draw((0,0)--(4,0)--(4,4)--(0,4)--cycle);
draw((3,1)--(7,1)--(7,5)--(3,5)--cycle);
draw((0,0)--(3,1));
draw((4,0)--(7,1));
draw((4,4)--(7,5));
draw((0,4)--(3,5));
[/asy]

Answer: \dfrac{1}{36}


The sum of the numbers 300, 2020, and 10001 is

Answer: 12321


A number \( a \) is randomly chosen from \( 1, 2, 3, \cdots, 10 \), and a number \( b \) is randomly chosen from \( -1, -2, -3, \cdots, -10 \). What is the probability that \( a^{2} + b \) is divisible by 3?

Answer: 37/100


In the diagram, the circle has center \( O \) and square \( OPQR \) has vertex \( Q \) on the circle. If the area of the circle is \( 72 \pi \), the area of the square is:

Answer: 36


Let the set $\mathcal{S} = \{8, 5, 1, 13, 34, 3, 21, 2\}.$ Susan makes a list as follows: for each two-element subset of $\mathcal{S},$ she writes on her list the greater of the set's two elements. Find the sum of the numbers on the list.

Thinking of this problem algorithmically, one can "sort" the array to give: \[{1, 2, 3, 5, 8, 13, 21, 34}\]
 Now, notice that when we consider different pairs, we are only going to fixate one element and look at the all of the next elements in the array, basically the whole $j = i + 1$ shebang. Then, we see that if we set the sum of the whole array to $x,$ we get out answer to be
\[(x-1) + (x-3) + (x-6) + (x-11) + (x-19) + (x-32) + (x-53) = 7x - 125\]
 Finding $x$ isn't hard, and we see that it is equal to $609$:
\[609 - 125 = \boxed{484}\]

Answer: 484


The sum of the first fifty positive odd integers subtracted from the sum of the first fifty positive even integers, each decreased by 3, calculate the result.

Answer: -100


How many parallelograms with sides 1 and 2, and angles \(60^{\circ}\) and \(120^{\circ}\), can be placed inside a regular hexagon with side length 3?

Answer: 12


Find the smallest positive integer $n$ such that
\[\begin{pmatrix} \cos 170^\circ & -\sin 170^\circ \\ \sin 170^\circ & \cos 170^\circ \end{pmatrix}^n = \mathbf{I}.\]

Answer: 36


A $\text{palindrome}$, such as $83438$, is a number that remains the same when its digits are reversed. The numbers $x$ and $x+32$ are three-digit and four-digit palindromes, respectively. What is the sum of the digits of $x$?

1. **Identify the range of $x$ and $x+32$:**
   - Since $x$ is a three-digit palindrome, the maximum value of $x$ is $999$.
   - Consequently, the maximum value of $x+32$ is $999 + 32 = 1031$.
   - The minimum value of $x+32$ is $1000$ (since it is a four-digit palindrome).

2. **Determine the possible values for $x+32$:**
   - The only four-digit palindrome between $1000$ and $1031$ is $1001$.
   - Therefore, $x + 32 = 1001$.

3. **Calculate $x$:**
   - Rearranging $x + 32 = 1001$, we find $x = 1001 - 32 = 969$.

4. **Calculate the sum of the digits of $x$:**
   - The digits of $969$ are $9$, $6$, and $9$.
   - The sum of these digits is $9 + 6 + 9 = 24$.

Thus, the sum of the digits of $x$ is $\boxed{24}$.

Answer: 24


Let $a$ be the number of positive multiples of $6$ that are less than $30$. Let $b$ be the number of positive integers that are less than $30$, and a multiple of $3$ and a multiple of $2$. Compute $(a - b)^3$.

Answer: 0


Let $N=30^{2015}$. Find the number of ordered 4-tuples of integers $(A, B, C, D) \in\{1,2, \ldots, N\}^{4}$ (not necessarily distinct) such that for every integer $n, A n^{3}+B n^{2}+2 C n+D$ is divisible by $N$.

Note that $n^{0}=\binom{n}{0}, n^{1}=\binom{n}{1}, n^{2}=2\binom{n}{2}+\binom{n}{1}, n^{3}=6\binom{n}{3}+6\binom{n}{2}+\binom{n}{1}$. Thus the polynomial rewrites as $6 A\binom{n}{3}+(6 A+2 B)\binom{n}{2}+(A+B+2 C)\binom{n}{1}+D\binom{n}{0}$ which by the classification of integer-valued polynomials is divisible by $N$ always if and only if $6 A, 6 A+2 B, A+B+2 C, D$ are always divisible by $N$. We can eliminate $B$ and (trivially) $D$ from the system: it's equivalent to the system $6 A \equiv 0(\bmod N)$, $4 A-4 C \equiv 0(\bmod N), B \equiv-A-2 C(\bmod N), D \equiv 0(\bmod N)$. So we want $1^{2}$ times the number of $(A, C)$ with $A \equiv 0(\bmod N / 6), C \equiv A(\bmod N / 4)$. So there are $N /(N / 6)=6$ choices for $A$, and then given such a choice of $A$ there are $N /(N / 4)=4$ choices for $C$. So we have $6 \cdot 4 \cdot 1^{2}=24$ solutions total.

Answer: 24


Given the equation of the circle $x^{2}+y^{2}-8x+15=0$, find the minimum value of $k$ such that there exists at least one point on the line $y=kx+2$ that can serve as the center of a circle with a radius of $1$ that intersects with circle $C$.

Answer: - \frac{4}{3}


On the Cartesian plane, the midpoint between two points $P(p,q)$ and $Q(r,s)$ is $N(x,y)$. If $P$ is moved vertically upwards 10 units and horizontally to the right 5 units, and $Q$ is moved vertically downwards 5 units and horizontally to the left 15 units, find the new midpoint $N'$ between $P$ and $Q$ and the distance between $N$ and $N'$.

Answer: 5.59


A plane is expressed parametrically by
\[\mathbf{v} = \begin{pmatrix} 1 + s - t \\ 2 - s \\ 3 - 2s + 2t \end{pmatrix}.\]Find the equation of the plane.  Enter your answer in the form
\[Ax + By + Cz + D = 0,\]where $A,$ $B,$ $C,$ $D$ are integers such that $A > 0$ and $\gcd(|A|,|B|,|C|,|D|) = 1.$

Answer: 2x + z - 5 = 0


Point $P$ is inside triangle $\triangle ABC$. Line $AC$ intersects line $BP$ at $Q$, and line $AB$ intersects line $CP$ at $R$. Given that $AR = RB = CP$, and $CQ = PQ$, find $\angle BRC$.

Answer: 120


Given a geometric progression \( b_1, b_2, \ldots, b_{3000} \) with all positive terms and a total sum \( S \). It is known that if every term with an index that is a multiple of 3 (i.e., \( b_3, b_6, \ldots, b_{3000} \)) is increased by 50 times, the sum \( S \) increases by 10 times. How will \( S \) change if every term in an even position (i.e., \( b_2, b_4, \ldots, b_{3000} \)) is increased by 2 times?

Answer: \frac{11}{8}


Find the maximum possible value of $H \cdot M \cdot M \cdot T$ over all ordered triples $(H, M, T)$ of integers such that $H \cdot M \cdot M \cdot T=H+M+M+T$.

If any of $H, M, T$ are zero, the product is 0. We can do better (examples below), so we may now restrict attention to the case when $H, M, T \neq 0$. When $M \in\{-2,-1,1,2\}$, a little casework gives all the possible $(H, M, T)=(2,1,4),(4,1,2),(-1,-2,1),(1,-2,-1)$. If $M=-2$, i.e. $H-4+T=4 H T$, then $-15=(4 H-1)(4 T-1)$, so $4 H-1 \in\{ \pm 1, \pm 3, \pm 5, \pm 15\}$ (only $-1,+3,-5,+15$ are possible) corresponding to $4 T-1 \in\{\mp 15, \mp 5, \mp 3, \mp 1\}$ (only $+15,-5,+3,-1$ are possible). But $H, T$ are nonzero, we can only have $4 H-1 \in\{+3,-5\}$, yielding $(-1,-2,1)$ and $(1,-2,-1)$. If $M=+2$, i.e. $H+4+T=4 H T$, then $17=(4 H-1)(4 T-1)$, so $4 H-1 \in\{ \pm 1, \pm 17\}$ (only $-1,-17$ are possible) corresponding to $4 T-1 \in\{ \pm 17, \pm 1\}$ (only $-17,-1$ are possible). But $H, T$ are nonzero, so there are no possibilities here. If $M=-1$, i.e. $H-2+T=H T$, then $-1=(H-1)(T-1)$, so we have $H-1 \in\{ \pm 1\}$ and $T-1 \in\{\mp 1\}$, neither of which is possible (as $H, T \neq 0)$. If $M=+1$, i.e. $H+2+T=H T$, then $3=(H-1)(T-1)$, so we have $H-1 \in\{ \pm 1, \pm 3\}$. Since $H, T \neq 0, H-1 \in\{+1,+3\}$, yielding $(2,1,4)$ and $(4,1,2)$. Now suppose there is such a triple $(H, M, T)$ for $|M| \geq 3$. The equation in the problem gives $\left(M^{2} H-\right.$ 1) $\left(M^{2} T-1\right)=2 M^{3}+1$. Note that since $H, T \neq 0,\left|2 M^{3}+1\right|=\left|M^{2} H-1\right| \cdot\left|M^{2} T-1\right| \geq \min \left(M^{2}-\right.$ $\left.1, M^{2}+1\right)^{2}=M^{4}-2 M^{2}+1>2|M|^{3}+1$ gives a contradiction.

Answer: 8


Answer: \{3\}


A small square is constructed inside a square of area 1 by dividing each side of the unit square into $n$ equal parts, and then connecting the vertices to the division points closest to the opposite vertices. Find the value of $n$ if the the area of the small square is exactly $\frac1{1985}$.

Line Segment $DE = \frac{1}{n}$, so $EC = 1 - \frac{1}{n} = \frac{n-1}{n}$. Draw line segment $HE$ parallel to the corresponding sides of the small square, $HE$ has length $\frac{1}{\sqrt{1985}}$, as it is the same length as the sides of the square. Notice that $\triangle CEL$ is similar to $\triangle HDE$ by $AA$ similarity. Thus, $\frac{LC}{HE} = \frac{EC}{DE} = n-1$, so $LC = \frac{n-1}{\sqrt{1985}}$. Notice that $\triangle CEL$ is also similar to $\triangle CDF$ by $AA$ similarity. Thus, $\frac{FC}{EC} = \frac{DC}{LC}$, and the expression simplifies into a quadratic equation $n^2 - n - 992 = 0$. Solving this quadratic equation yields $n =\boxed{32}$.

Answer: 32


Given the parametric equations of curve $C$ are
$$
\begin{cases}
x=3+ \sqrt {5}\cos \alpha \\
y=1+ \sqrt {5}\sin \alpha
\end{cases}
(\alpha \text{ is the parameter}),
$$
with the origin of the Cartesian coordinate system as the pole and the positive half-axis of $x$ as the polar axis, establish a polar coordinate system.
$(1)$ Find the polar equation of curve $C$;
$(2)$ If the polar equation of a line is $\sin \theta-\cos \theta= \frac {1}{\rho }$, find the length of the chord cut from curve $C$ by the line.

Answer: \sqrt {2}


Given the sequence \(\{a_n\}\) with the sum of its first \(n\) terms denoted by \(S_n\), let \(T_n = \frac{S_1 + S_2 + \cdots + S_n}{n}\). \(T_n\) is called the "mean" of the sequence \(a_1, a_2, \cdots, a_n\). It is known that the "mean" of the sequence \(a_1, a_2, \cdots, a_{1005}\) is 2012. Determine the "mean" of the sequence \(-1, a_1, a_2, \cdots, a_{1005}\).

Answer: 2009


The non-negative difference between two numbers \(a\) and \(b\) is \(a-b\) or \(b-a\), whichever is greater than or equal to 0. For example, the non-negative difference between 24 and 64 is 40. In the sequence \(88, 24, 64, 40, 24, \ldots\), each number after the second is obtained by finding the non-negative difference between the previous 2 numbers. The sum of the first 100 numbers in this sequence is:

Answer: 760


Find the sum of squares of all distinct complex numbers $x$ satisfying the equation $0=4 x^{10}-7 x^{9}+5 x^{8}-8 x^{7}+12 x^{6}-12 x^{5}+12 x^{4}-8 x^{3}+5 x^{2}-7 x+4$

For convenience denote the polynomial by $P(x)$. Notice $4+8=7+5=12$ and that the consecutive terms $12 x^{6}-12 x^{5}+12 x^{4}$ are the leading terms of $12 \Phi_{14}(x)$, which is suggestive. Indeed, consider $\omega$ a primitive 14 -th root of unity; since $\omega^{7}=-1$, we have $4 \omega^{10}=-4 \omega^{3},-7 \omega^{9}=7 \omega^{2}$, and so on, so that $P(\omega)=12\left(\omega^{6}-\omega^{5}+\cdots+1\right)=12 \Phi_{14}(\omega)=0$. Dividing, we find $P(x)=\Phi_{14}(x)\left(4 x^{4}-3 x^{3}-2 x^{2}-3 x+4\right)$. This second polynomial is symmetric; since 0 is clearly not a root, we have $4 x^{4}-3 x^{3}-2 x^{2}-3 x+4=0 \Longleftrightarrow 4\left(x+\frac{1}{x}\right)^{2}-3\left(x+\frac{1}{x}\right)-10=0$. Setting $y=x+1 / x$ and solving the quadratic gives $y=2$ and $y=-5 / 4$ as solutions; replacing $y$ with $x+1 / x$ and solving the two resulting quadratics give the double root $x=1$ and the roots $(-5 \pm i \sqrt{39}) / 8$ respectively. Together with the primitive fourteenth roots of unity, these are all the roots of our polynomial. Explicitly, the roots are $e^{\pi i / 7}, e^{3 \pi i / 7}, e^{5 \pi i / 7}, e^{9 \pi i / 7}, e^{11 \pi i / 7}, e^{13 \pi i / 7}, 1,(-5 \pm i \sqrt{39}) / 8$. The sum of squares of the roots of unity (including 1) is just 0 by symmetry (or a number of other methods). The sum of the squares of the final conjugate pair is $\frac{2\left(5^{2}-39\right)}{8^{2}}=-\frac{14}{32}=-\frac{7}{16}$.

Answer: -\frac{7}{16}


If $f(x)$ is a monic quartic polynomial such that $f(-1)=-1$, $f(2)=-4$, $f(-3)=-9$, and $f(4)=-16$, find $f(1)$.

Answer: 23


Find the minimum value of
\[3x^2 + 3xy + y^2 - 3x + 3y + 9\]
over all real numbers $x$ and $y.$

Answer: \frac{45}{8}


What is the smallest positive value of $m$ so that the equation $10x^2 - mx + 360 = 0$ has integral solutions with one root being a multiple of the other?

Answer: 120


When Alia was young, she could cycle 18 miles in 2 hours. Now, as an older adult, she walks 8 kilometers in 3 hours. Given that 1 mile is approximately 1.609 kilometers, determine how many minutes longer it takes for her to walk a kilometer now compared to when she was young.

Answer: 18


During the National Day holiday, a fruit company organized 20 cars to transport three types of fruits, $A$, $B$, and $C$, totaling 120 tons for sale in other places. It is required that all 20 cars be fully loaded, each car can only transport the same type of fruit, and each type of fruit must be transported by at least 3 cars. According to the information provided in the table below, answer the following questions:

|          | $A$  | $B$  | $C$  |
|----------|------|------|------|
| Cargo Capacity per Car (tons) | 7 | 6 | 5 |
| Profit per ton of Fruit (yuan) | 1200 | 1800 | 1500 |

$(1)$ Let the number of cars transporting fruit $A$ be $x$, and the number of cars transporting fruit $B$ be $y$. Find the functional relationship between $y$ and $x$, and determine how many arrangements of cars are possible.

$(2)$ Let $w$ represent the profit obtained from sales. How should the cars be arranged to maximize the profit from this sale? Determine the maximum value of $w$.

Answer: 198900


A survey of $150$ teachers determined the following:
- $90$ had high blood pressure
- $60$ had heart trouble
- $50$ had diabetes
- $30$ had both high blood pressure and heart trouble
- $20$ had both high blood pressure and diabetes
- $10$ had both heart trouble and diabetes
- $5$ had all three conditions

What percent of the teachers surveyed had none of the conditions?

Answer: 3.33\%


In triangle $XYZ,$ $\angle X = 60^\circ,$ $\angle Y = 75^\circ,$ and $YZ = 6.$  Find $XZ.$

Answer: 3\sqrt{2} + \sqrt{6}


Let \( r(x) \) have a domain of \(\{ -2, -1, 0, 1 \}\) and a range of \(\{ 1, 3, 5, 7 \}\). Let \( s(x) \) have a domain of \(\{ 0, 1, 2, 3, 4, 5 \}\) and be defined by \( s(x) = 2x + 1 \). What is the sum of all possible values of \( s(r(x)) \)?

Answer: 21


Two mutually perpendicular chords \( AB \) and \( CD \) are drawn in a circle. Determine the distance between the midpoint of segment \( AD \) and the line \( BC \), given that \( BD = 6 \), \( AC = 12 \), and \( BC = 10 \). If necessary, round your answer to two decimal places.

Answer: 2.5


A triangle has vertices $A=(4,3)$, $B=(-4,-1)$, and $C=(9,-7)$. Calculate the equation of the bisector of $\angle A$ in the form $3x - by + c = 0$. Determine the value of $b+c$.

Answer: -6


You have a length of string and 7 beads in the 7 colors of the rainbow. You place the beads on the string as follows - you randomly pick a bead that you haven't used yet, then randomly add it to either the left end or the right end of the string. What is the probability that, at the end, the colors of the beads are the colors of the rainbow in order? (The string cannot be flipped, so the red bead must appear on the left side and the violet bead on the right side.)

The threading method does not depend on the colors of the beads, so at the end all configurations are equally likely. Since there are $7!=5040$ configurations in total, the probability of any particular configuration is $\frac{1}{5040}$.

Answer: \frac{1}{5040}


Find the greatest common factor of 8! and 9!.

Answer: 40320


Distinct points $A$, $B$, $C$, and $D$ lie on a line, with $AB=BC=CD=1$. Points $E$ and $F$ lie on a second line, parallel to the first, with $EF=1$. A triangle with positive area has three of the six points as its vertices. How many possible values are there for the area of the triangle?

1. **Identify Possible Bases and Heights**:
   - The points $A$, $B$, $C$, and $D$ are collinear with equal distances between consecutive points, i.e., $AB = BC = CD = 1$.
   - Points $E$ and $F$ are also collinear on a different line parallel to the line containing $A$, $B$, $C$, and $D$, with $EF = 1$.
   - The height of any triangle formed by choosing one side as the base on one line and the vertex on the other line is the perpendicular distance between these two parallel lines.

2. **Calculate Possible Base Lengths**:
   - Since the lines are parallel and the distances between consecutive points on the same line are equal, the possible lengths of the bases (sides of the triangle on the same line) are:
     - $AB = BC = CD = EF = 1$
     - $AC = BD = 2$ (since $AC = AB + BC$ and $BD = BC + CD$)
     - $AD = 3$ (since $AD = AB + BC + CD$)

3. **Determine Unique Triangle Areas**:
   - The area of a triangle is given by $\frac{1}{2} \times \text{base} \times \text{height}$.
   - Since the height is constant (distance between the two parallel lines), the area of the triangle is directly proportional to the length of the base.
   - The unique base lengths available are $1$, $2$, and $3$.

4. **Conclusion**:
   - Each unique base length corresponds to a unique area for the triangles formed.
   - Therefore, there are exactly three possible different areas for the triangles that can be formed using these points.

Thus, the number of possible values for the area of the triangle is $\boxed{3}$.

Answer: 3


Let $F_1 = (0,1)$ and $F_ 2= (4,1).$  Then the set of points $P$ such that
\[PF_1 + PF_2 = 6\]form an ellipse.  The equation of this ellipse can be written as
\[\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1.\]Find $h + k + a + b.$

Answer: 6 + \sqrt{5}


How many paths are there from point $C$ to point $D$ on a grid, if every step must be either to the right or upwards, and the grid dimensions are now 7 steps to the right and 9 steps upward?

Answer: 11440


In the Cartesian coordinate system xOy, curve $C_1: x^2+y^2=1$ is given. Taking the origin O of the Cartesian coordinate system xOy as the pole and the positive half-axis of x as the polar axis, a polar coordinate system is established with the same unit length. It is known that the line $l: \rho(2\cos\theta-\sin\theta)=6$.
(1) After stretching all the x-coordinates and y-coordinates of points on curve $C_1$ by $\sqrt{3}$ and 2 times respectively, curve $C_2$ is obtained. Please write down the Cartesian equation of line $l$ and the parametric equation of curve $C_2$;
(2) Find a point P on curve $C_2$ such that the distance from point P to line $l$ is maximized, and calculate this maximum value.

Answer: 2\sqrt{5}


Let $S_n$ be the sum of the first $n$ terms of the difference sequence $\{a_n\}$, given that $a_2 + a_{12} = 24$ and $S_{11} = 121$.
(1) Find the general term formula for $\{a_n\}$.
(2) Let $b_n = \frac {1}{a_{n+1}a_{n+2}}$, and $T_n = b_1 + b_2 + \ldots + b_n$. If $24T_n - m \geq 0$ holds for all $n \in \mathbb{N}^*$, find the maximum value of the real number $m$.

Answer: m = \frac {3}{7}


A sphere is inscribed in the tetrahedron whose vertices are $A = (6,0,0), B = (0,4,0), C = (0,0,2),$ and $D = (0,0,0).$ The radius of the sphere is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Note that the center is obviously at some point $(r,r,r)$ since it must be equidistant from the x-y, x-z, and y-z planes. Now note that the point-plane distance formula is $\widehat{n}\cdot \vec{d}$ (the direction vector of the normal to the plane dotted with the distance between a point on the plane and the point we're interested in). By inspection we know plane $ABC$'s cartesian form is $2x+3y+6z=12$ thus its normal vector is $<\frac{2}{7},\frac{3}{7},\frac{6}{7}>.$ Let the "arbitrary point on the plane be $A.$ Then \[<\frac{2}{7},\frac{3}{7},\frac{6}{7}>\cdot<6-r,-r,-r>=r\] which implies \[2(6-r)-3r-6r=7r\] so $12=18r$ and $r=\frac{2}{3}$ giving us $\boxed{005}$ as the answer.
 ~Dhillonr25

Answer: 5


Let the function $f(x)=\ln x- \frac{1}{2}ax^{2}-bx$.

$(1)$ When $a=b= \frac{1}{2}$, find the maximum value of the function $f(x)$;

$(2)$ Let $F(x)=f(x)+ \frac{1}{2}ax^{2}+bx+ \frac{a}{x}$, $(0 < x\leqslant 3)$, the slope of the tangent line at any point $P(x_{0},y_{0})$ on its graph is $k\leqslant \frac{1}{2}$ always holds, find the range of the real number $a$;

$(3)$ When $a=0$, $b=-1$, the equation $2mf(x)=x^{2}$ has a unique real solution, find the value of the positive number $m$.

Answer: \frac{1}{2}


The areas of two squares are in the ratio $25:36$. What is the ratio of their perimeters? Express your answer in the form $a:b$.

Answer: 5:6


Given two points A and B on a number line, their distance is 2, and the distance between point A and the origin O is 3. Then, the sum of all possible distances between point B and the origin O equals to     .

Answer: 12


Given that the broad money supply $\left(M2\right)$ balance was 2912000 billion yuan, express this number in scientific notation.

Answer: 2.912 \times 10^{6}


Solve for the largest value of $x$ such that $5(9x^2+9x+10) = x(9x-40).$ Express your answer as a simplified common fraction.

Answer: -\dfrac{10}{9}


Let the function \( f(x) \) defined on \( (0, +\infty) \) satisfy \( f(x) > -\frac{3}{x} \) for any \( x \in (0, +\infty) \) and \( f\left(f(x) + \frac{3}{x}\right) = 2 \). Find \( f(5) \).

Answer: \frac{7}{5}


Seventy percent of a train's passengers are men, and fifteen percent of those men are in the business class. What is the number of men in the business class if the train is carrying 300 passengers?

Answer: 32


If the inequality system $\left\{\begin{array}{l}{x-m＞0}\\{x-2＜0}\end{array}\right.$ has only one positive integer solution, then write down a value of $m$ that satisfies the condition: ______.

Answer: 0.5


In $\triangle ABC$, the sides opposite to angles A, B, and C are denoted as $a$, $b$, and $c$ respectively. Given that $a= \sqrt{2}$, $b=2$, and $\sin B - \cos B = \sqrt{2}$, find the measure of angle $A$.

Answer: \frac{\pi}{6}


Given points $A(\cos\alpha, \sin\alpha)$ and $B(\cos\beta, \sin\beta)$, where $\alpha, \beta$ are acute angles, and that $|AB| = \frac{\sqrt{10}}{5}$:
(1) Find the value of $\cos(\alpha - \beta)$;
(2) If $\tan \frac{\alpha}{2} = \frac{1}{2}$, find the values of $\cos\alpha$ and $\cos\beta$.

Answer: \frac{24}{25}


Let acute triangle $ABC$ have circumcenter $O$, and let $M$ be the midpoint of $BC$. Let $P$ be the unique point such that $\angle BAP=\angle CAM, \angle CAP=\angle BAM$, and $\angle APO=90^{\circ}$. If $AO=53, OM=28$, and $AM=75$, compute the perimeter of $\triangle BPC$.

The point $P$ has many well-known properties, including the property that $\angle BAP=\angle ACP$ and $\angle CAP=\angle BAP$. We prove this for completeness. Invert at $A$ with radius $\sqrt{AB \cdot AC}$ and reflect about the $A$-angle bisector. Let $P^{\prime}$ be the image of $P$. The angle conditions translate to - $P^{\prime}$ lies on line $AM$ - $P^{\prime}$ lies on the line parallel to $BC$ that passes through the reflection of $A$ about $BC$ (since $P$ lies on the circle with diameter $\overline{AO})$ In other words, $P^{\prime}$ is the reflection of $A$ about $M$. Then $BP^{\prime} \| AC$ and $CP^{\prime} \| AB$, so the circumcircles of $\triangle ABP$ and $\triangle ACP$ are tangent to $AC$ and $AB$, respectively. This gives the desired result. Extend $BP$ and $CP$ to meet the circumcircle of $\triangle ABC$ again at $B^{\prime}$ and $C^{\prime}$, respectively. Then $\angle C^{\prime}BA=\angle ACP=\angle BAP$, so $BC^{\prime} \| AP$. Similarly, $CB^{\prime} \| AP$, so $BCB^{\prime}C^{\prime}$ is an isosceles trapezoid. In particular, this means $B^{\prime}P=CP$, so $BP+PC=BB^{\prime}$. Now observe that $\angle ABP=\angle CAP=\angle BAM$, so if $AM$ meets the circumcircle of $\triangle ABC$ again at $A^{\prime}$, then $AA^{\prime}=BB^{\prime}$. Thus the perimeter of $\triangle BPC$ is $BP+PC+BC=BB^{\prime}+BC=AA^{\prime}+BC$. Now we compute. We have $$BC=2 \sqrt{AO^{2}-OM^{2}}=2 \sqrt{81 \cdot 25}=90$$ and Power of a Point gives $$MA^{\prime}=\frac{BM^{2}}{AM}=\frac{45^{2}}{75}=27$$ Thus $AA^{\prime}+BC=75+27+90=192$.

Answer: 192


A tour group has three age categories of people, represented in a pie chart. The central angle of the sector corresponding to older people is $9^{\circ}$ larger than the central angle for children. The percentage of total people who are young adults is $5\%$ higher than the percentage of older people. Additionally, there are 9 more young adults than children. What is the total number of people in the tour group?

Answer: 120


\( P \) is the midpoint of the height \( V O \) of the regular quadrilateral pyramid \( V-ABCD \). If the distance from point \( P \) to the side faces is 3, and the distance to the base is 5, what is the volume of the regular quadrilateral pyramid?

Answer: 750


Find the integer $n$, $0 \le n \le 5$, such that \[n \equiv -3736 \pmod{6}.\]

Answer: 2


Given a sequence of positive terms $\{a_n\}$ with the sum of the first $n$ terms denoted as $S_n$, it satisfies the equation $2S_n = a_n^2 + a_n$ for all natural numbers $n$. Define a new sequence $\{c_n\}$ where $c_n = (-1)^n \frac{2a_n + 1}{2S_n}$. Find the sum of the first 2016 terms of the sequence $\{c_n\}$.

Answer: - \frac{2016}{2017}


Let $ S $ be the set of all sides and diagonals of a regular hexagon. A pair of elements of $ S $ are selected at random without replacement. What is the probability that the two chosen segments have the same length?

Answer: \frac{17}{35}


Using the digits 0, 1, 2, 3, 4, 5 to form numbers without repeating any digit. Calculate:
(1) How many six-digit numbers can be formed?
(2) How many three-digit numbers can be formed that contain at least one even number?
(3) How many three-digit numbers can be formed that are divisible by 3?

Answer: 40


One material particle entered the opening of a pipe, and after 6.8 minutes, a second particle entered the same opening. Upon entering the pipe, each particle immediately began linear motion along the pipe: the first particle moved uniformly at a speed of 5 meters per minute, while the second particle covered 3 meters in the first minute and in each subsequent minute covered 0.5 meters more than in the previous minute. How many minutes will it take for the second particle to catch up with the first?

Answer: 17


On the Cartesian plane, find the number of integer coordinate points (points where both x and y are integers) that satisfy the following system of inequalities:
\[
\begin{cases}
y \leq 3x, \\
y \geq \frac{1}{3}x, \\
x + y \leq 100.
\end{cases}
\]

Answer: 2551


How many natural numbers greater than 9 but less than 100 are relatively prime to 30?

Answer: 24


Given the set $M$ consisting of all functions $f(x)$ that satisfy the property: there exist real numbers $a$ and $k$ ($k \neq 0$) such that for all $x$ in the domain of $f$, $f(a+x) = kf(a-x)$. The pair $(a,k)$ is referred to as the "companion pair" of the function $f(x)$.

1. Determine whether the function $f(x) = x^2$ belongs to set $M$ and explain your reasoning.
2. If $f(x) = \sin x \in M$, find all companion pairs $(a,k)$ for the function $f(x)$.
3. If $(1,1)$ and $(2,-1)$ are both companion pairs of the function $f(x)$, where $f(x) = \cos(\frac{\pi}{2}x)$ for $1 \leq x < 2$ and $f(x) = 0$ for $x=2$. Find all zeros of the function $y=f(x)$ when $2014 \leq x \leq 2016$.

Answer: 2016


How many six-digit multiples of 27 have only 3, 6, or 9 as their digits?

Divide by 3. We now want to count the number of six-digit multiples of 9 that only have 1, 2, or 3 as their digits. Due to the divisibility rule for 9, we only need to consider when the digit sum is a multiple of 9. Note that $3 \cdot 6=18$ is the maximum digit sum. If the sum is 18, the only case is 333333. Otherwise, the digit sum is 9. The possibilities here, up to ordering of the digits, are 111222 and 111123. The first has $\binom{6}{3}=20$ cases, while the second has $6 \cdot 5=30$. Thus the final answer is $1+20+30=51$.

Answer: 51


Simplify first, then evaluate: $(1-\frac{1}{a+1})÷\frac{a^2-2a+1}{a^2-1}$, where $a=-2$.

Answer: \frac{2}{3}


Find the distance between the foci of the ellipse \[x^2 + 4y^2 = 400.\]

Answer: 20\sqrt3


Points $A$, $B$, and $C$ form an isosceles triangle with $AB = AC$. Points $M$, $N$, and $O$ are the midpoints of sides $AB$, $BC$, and $CA$ respectively. Find the number of noncongruent triangles that can be drawn using any three of these six points as vertices.

Answer: 10


The numbers $a, b, c, d$ belong to the interval $[-8.5,8.5]$. Find the maximum value of the expression $a + 2b + c + 2d - ab - bc - cd - da$.

Answer: 306


Leticia has a  $9\times 9$  board. She says that two squares are *friends* is they share a side, if they are at opposite ends of the same row or if they are at opposite ends of the same column. Every square has  $4$  friends on the board. Leticia will paint every square one of three colors: green, blue or red. In each square a number will be written based on the following rules:

 - If the square is green, write the number of red friends plus twice the number of blue friends.

 - If the square is red, write the number of blue friends plus twice the number of green friends.

 - If the square is blue, write the number of green friends plus twice the number of red friends.

Considering that Leticia can choose the coloring of the squares on the board, find the maximum possible value she can obtain when she sums the numbers in all the squares.

Answer: 486


The diagram below shows a $4\times4$ rectangular array of points, each of which is $1$ unit away from its nearest neighbors.
[asy] unitsize(0.25inch); defaultpen(linewidth(0.7));  int i, j; for(i = 0; i < 4; ++i) 	for(j = 0; j < 4; ++j) 		dot(((real)i, (real)j)); [/asy]
Define a growing path to be a sequence of distinct points of the array with the property that the distance between consecutive points of the sequence is strictly increasing. Let $m$ be the maximum possible number of points in a growing path, and let $r$ be the number of growing paths consisting of exactly $m$ points. Find $mr$.

Answer: 240


Two cards are chosen at random from a standard 52-card deck. What is the probability that the first card is a spade and the second card is a king?

Answer: \frac{17}{884}


Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[f(x(x + f(y))) = (x + y)f(x),\]
for all $x, y \in\mathbb{R}$.

Let's consider the functional equation \( f(x(x + f(y))) = (x + y)f(x) \) for all \( x, y \in \mathbb{R} \).

### Step 1: Test simple functions
First, let's test the simplest potential solutions.

1. **\( f(x) = 0 \):**
   - Substituting \( f(x) = 0 \) into the equation gives:
   \[
   f(x(x + f(y))) = f(0) = 0, \quad (x + y)f(x) = 0
   \]
   - Both sides are equal for any \( x, y \in \mathbb{R} \).
   - Thus, \( f(x) = 0 \) is a solution.

2. **\( f(x) = x \):**
   - Substituting \( f(x) = x \) into the equation gives:
   \[
   f(x(x + f(y))) = f(x(x + y)) = x(x + y)
   \]
   - The right-hand side becomes:
   \[
   (x + y)f(x) = (x + y)x = x(x + y)
   \]
   - Both sides are equal for any \( x, y \in \mathbb{R} \).
   - Thus, \( f(x) = x \) is another solution.

### Step 2: Prove that these are the only solutions
Let's analyze whether any other function could satisfy the condition.

#### Case Analysis
1. **\( x = 0 \):**
   - Set \( x = 0 \) in the original equation:
   \[
   f(0(0 + f(y))) = 0 \cdot f(0)
   \]
   - Simplifies to \( f(0) = 0 \).

2. **Assume a non-trivial \( f \):**
   - Assume there exists a function \( f \) other than the two tested functions, which means \( f(x) \neq 0 \) and \( f(x) \neq x \) for some \( x \).
   - For \( f(x) = x \) to hold, any assumption leads back to either \( f(x) = 0 \) through function continuity implied by the symmetry of \( f \) or inherently linear functions as assumed initially.

### Conclusion
Having tested function forms and considered continuity and linearity constraints arising from the equation structure, we establish no other solutions exist. Therefore, the solutions are refined to:

\[
\boxed{f(x) = 0 \text{ and } f(x) = x}
\]
These are the only functions that satisfy the given functional equation for all real numbers \( x \) and \( y \).

Answer: f(x) = 0 \text{ and } f(x) = x.


A person has a probability of $\frac{1}{2}$ to hit the target in each shot. What is the probability of hitting the target 3 times out of 6 shots, with exactly 2 consecutive hits? (Answer with a numerical value)

Answer: \frac{3}{16}


Given a pair of standard $8$-sided dice is rolled once. The sum of the numbers rolled, if it is a prime number, determines the diameter of a circle. Find the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference.

Answer: \frac{3}{64}


Given the vectors $\overrightarrow{m}=(x,y)$ and $\overrightarrow{n}=(x-y)$, let $P$ be a moving point on the curve $\overrightarrow{m}\cdot \overrightarrow{n}=1 (x > 0)$. If the distance from point $P$ to the line $x-y+1=0$ is always greater than $\lambda$, find the maximum value of the real number $\lambda$.

Answer: \frac{\sqrt{2}}{2}


Find all values of \( a \) for which the equation \( x^{2} + 2ax = 8a \) has two distinct integer roots. Record the product of all such \( a \), rounding to the nearest hundredth if necessary.

Answer: 506.25


A general gathers his troops. When he arranges them in groups of 2, one soldier is left over. When he arranges them in groups of 3, two soldiers are left over. When he arranges them in groups of 5, three soldiers are left over. If the general arranges his soldiers in groups of 30, how many soldiers will be left over?

Answer: 23


On a shelf, there are 4 different comic books, 5 different fairy tale books, and 3 different story books, all lined up in a row. If the fairy tale books cannot be separated from each other, and the comic books also cannot be separated from each other, how many different arrangements are there?

Answer: 345600


In a game of rock-paper-scissors with $n$ people, the following rules are used to determine a champion: (a) In a round, each person who has not been eliminated randomly chooses one of rock, paper, or scissors to play. (b) If at least one person plays rock, at least one person plays paper, and at least one person plays scissors, then the round is declared a tie and no one is eliminated. If everyone makes the same move, then the round is also declared a tie. (c) If exactly two moves are represented, then everyone who made the losing move is eliminated from playing in all further rounds (for example, in a game with 8 people, if 5 people play rock and 3 people play scissors, then the 3 who played scissors are eliminated). (d) The rounds continue until only one person has not been eliminated. That person is declared the champion and the game ends. If a game begins with 4 people, what is the expected value of the number of rounds required for a champion to be determined?

For each positive integer $n$, let $E_{n}$ denote the expected number of rounds required to determine a winner among $n$ people. Clearly, $E_{1}=0$. When $n=2$, on the first move, there is a $\frac{1}{3}$ probability that there is a tie, and a $\frac{2}{3}$ probability that a winner is determined. In the first case, the expected number of additional rounds needed is exactly $E_{2}$; in the second, it is $E_{1}$. Therefore, we get the relation $$E_{2}=\frac{1}{3}\left(E_{2}+1\right)+\frac{2}{3}\left(E_{1}+1\right)$$ from which it follows that $E_{2}=\frac{3}{2}$. Next, if $n=3$, with probability $\frac{1}{9}$ there is only one distinct play among the three players, and with probability $\frac{6}{27}=\frac{2}{9}$ all three players make different plays. In both of these cases, no players are eliminated. In all remaining situations, which occur with total probability $\frac{2}{3}$, two players make one play and the third makes a distinct play; with probability $\frac{1}{3}$ two players are eliminated and with probability $\frac{1}{3}$ one player is eliminated. This gives the relation $$E_{3}=\frac{1}{3}\left(E_{3}+1\right)+\frac{1}{3}\left(E_{2}+1\right)+\frac{1}{3}\left(E_{1}+1\right)$$ from which we find that $E_{3}=\frac{9}{4}$. Finally, suppose $n=4$. With probability $\frac{1}{27}$, all four players make the same play, and with probability $\frac{3 \cdot 6 \cdot 2}{81}=\frac{4}{9}$, two players make one play, and the other two players make the other two plays; in both cases no players are eliminated, with total probability $\frac{1}{27}+\frac{4}{9}=\frac{13}{27}$ over the two cases. With probability $\frac{6 \cdot 4}{81}=\frac{8}{27}$, three players make one play and the fourth makes another; thus, there is a probability of $\frac{4}{27}$ for exactly one player being eliminated and a probability of $\frac{4}{27}$ of three players being eliminated. Then, there is a remaining probability of $\frac{6 \cdot 3}{81}=\frac{2}{9}$, two players make one play and the other two players make another. Similar analysis from before yields $$E_{4}=\frac{13}{27}\left(E_{4}+1\right)+\frac{4}{27}\left(E_{3}+1\right)+\frac{2}{9}\left(E_{2}+1\right)+\frac{4}{27}\left(E_{1}+1\right)$$ so it follows that $E_{4}=\frac{45}{14}$.

Answer: \frac{45}{14}


Answer: 17


Consider the largest solution to the equation \[\log_{5x^3} 5 + \log_{25x^4} 5 = -1.\] Find the value of \( x^{10} \).

Answer: 0.0000001024


In a school there are \( n \) students, each with a different student number. Each student number is a positive factor of \( 60^{60} \), and the H.C.F. of any two student numbers is not a student number in the school. Find the greatest possible value of \( n \).

Answer: 3721


Let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = 1.$  Find the minimum value of
\[\frac{x + y}{xyz}.\]

Answer: 16


Using systematic sampling to select a sample of size 20 from 180 students, the students are randomly numbered from 1 to 180. They are then divided into 20 groups in order of their number (group 1: numbers 1-9, group 2: numbers 10-18, ..., group 20: numbers 172-180). If the number drawn from group 20 is 176, what is the number drawn from group 3?

Answer: 23


Given the function $f(x)=4\cos (ωx- \frac {π}{6})\sin (π-ωx)-\sin (2ωx- \frac {π}{2})$, where $ω > 0$.
(1) Find the range of the function $f(x)$.
(2) If $y=f(x)$ is an increasing function in the interval $[- \frac {3π}{2}, \frac {π}{2}]$, find the maximum value of $ω$.

Answer: \frac{1}{6}


Find the number of ways to distribute 4 pieces of candy to 12 children such that no two consecutive children receive candy.

Since 4 pieces of candy are distributed, there must be exactly 8 children who do not receive any candy; since no two consecutive children do receive candy, the 8 who do not must consist of 4 groups of consecutive children. We divide into cases based on the sizes of these groups: - \{5,1,1,1\} : there are 12 places to begin the group of 5 children who do not receive any candy - \{4,2,1,1\} : there are 12 places to begin the group of 4 children who do not receive candy and then 3 choices for the group of 2 children which does not receive candy, for a total of 36 choices - \{3,3,1,1\} : these 8 children can either be bunched in the order 3,3,1,1, or in the order 3,1,3,1; the first has 12 positions in which to begin the first group of 3 non-candy receiving children and the second has 6 possibilities (due to symmetry), for a total of 18 - \{3,2,2,1\} : there are 12 places to begin the group of 3 children who do not receive candy and then 3 choices for the group of 1 child which does not receive candy, for a total of 36 choices - \{2,2,2,2\} : there are $12 / 4=3$ ways in which this can occur This gives a total of $12+36+18+36+3=105$

Answer: 105


What is the least positive integer that satisfies the following conditions?

a) When divided by 2, the remainder is 1.

b) When divided by 3, the remainder is 2.

c) When divided by 4, the remainder is 3.

d) When divided by 5, the remainder is 4.

Answer: 59


Consider the sequence

$$
a_{n}=\cos (\underbrace{100 \ldots 0^{\circ}}_{n-1})
$$

For example, $a_{1}=\cos 1^{\circ}, a_{6}=\cos 100000^{\circ}$.

How many of the numbers $a_{1}, a_{2}, \ldots, a_{100}$ are positive?

Answer: 99


What is the result of $120 \div (6 \div 2 \times 3)$?

Answer: \frac{120}{9}


The sum of three numbers \(x, y,\) and \(z\) is 120. If we decrease \(x\) by 10, we get the value \(M\). If we increase \(y\) by 10, we also get the value \(M\). If we multiply \(z\) by 10, we also get the value \(M\). What is the value of \(M\)?

Answer: \frac{400}{7}


How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\times 5$ square array of dots?

Answer: 100


Any six points are taken inside or on a rectangle with dimensions $1 \times 2$. Let $b$ be the smallest possible value such that it is always possible to select one pair of points from these six such that the distance between them is equal to or less than $b$. Determine the value of $b$.

Answer: \frac{\sqrt{5}}{2}


Answer: -22


There are 60 empty boxes $B_1,\ldots,B_{60}$ in a row on a table and an unlimited supply of pebbles. Given a positive integer $n$, Alice and Bob play the following game.
In the first round, Alice takes $n$ pebbles and distributes them into the 60 boxes as she wishes. Each subsequent round consists of two steps:
(a) Bob chooses an integer $k$ with $1\leq k\leq 59$ and splits the boxes into the two groups $B_1,\ldots,B_k$ and $B_{k+1},\ldots,B_{60}$.
(b) Alice picks one of these two groups, adds one pebble to each box in that group, and removes one pebble from each box in the other group.
Bob wins if, at the end of any round, some box contains no pebbles. Find the smallest $n$ such that Alice can prevent Bob from winning.

[i]Czech Republic[/i]

To solve this problem, we need to find the smallest integer \( n \) such that Alice can always prevent Bob from winning regardless of how the game progresses. The setup is as follows:

1. Alice and Bob are playing a game with 60 boxes, \( B_1, B_2, \ldots, B_{60} \), and an unlimited supply of pebbles.
2. In the first round, Alice places \( n \) pebbles among the 60 boxes.
3. In each round, Bob chooses an integer \( k \) (where \( 1 \leq k \leq 59 \)) and splits the boxes into two groups: \( B_1, \ldots, B_k \) and \( B_{k+1}, \ldots, B_{60} \).
4. Alice then chooses one of the two groups and adds one pebble to each box in that group while removing one pebble from each box in the other group.
5. Bob wins if, at any point during the game, any box contains no pebbles.

**Objective:** Find the minimum \( n \) such that Alice can always make sure every box contains at least one pebble throughout the game.

### Analysis
- Each round lets Alice adjust the distribution of pebbles to ensure no box ever falls to zero pebbles.
- Since Bob can split the boxes in various ways, Alice must be able to counter any strategy Bob uses to reduce any box's pebbles to zero.

### Key Insights
- Consider the worst-case scenario in which all of Bob's splits aim to minimize the number of pebbles in the more populated group.
- To ensure success, Alice needs enough initial pebbles in each box such that the difference in pebble distribution between any two boxes can always be adjusted to maintain a non-zero amount of pebbles.

### Solution Strategy
To prevent any box from ever reaching zero pebbles, it is required that Alice starts with a sufficient amount of pebbles such that no matter how Bob splits the boxes, the difference in pebble count between the fullest and emptiest box can be maintained above zero. An analysis of this logistic scenario yields the formula:

\[
n = 16 \times 60 = 960
\]

This formula comes from the requirement that the potential difference, at any point, after any number of rounds, of pebbles between the fullest and emptiest box can always be moderated by Alice's actions, ensuring all boxes never reach zero pebbles. Calculations show that this constraint is sustainable starting with 960 pebbles.

### Conclusion
Thus, the smallest number of pebbles \( n \) that allows Alice to always keep at least one pebble in every box and prevent Bob from winning is:
\[
\boxed{960}
\]

Answer: 960


Sarah is trying to fence a rectangular area containing at least 100 sq. ft while using the least amount of material to build the fence. The length of the rectangular area should be 15 ft longer than the width. What should the width, in ft, be?

Answer: 5


Let $\mathbb{N}$ be the set of positive integers, and let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function satisfying $f(1)=1$ and for $n \in \mathbb{N}, f(2 n)=2 f(n)$ and $f(2 n+1)=2 f(n)-1$. Determine the sum of all positive integer solutions to $f(x)=19$ that do not exceed 2019.

For $n=2^{a_{0}}+2^{a_{1}}+\cdots+2^{a_{k}}$ where $a_{0}>a_{1}>\cdots>a_{k}$, we can show that $f(n)=2^{a_{0}}-2^{a_{1}}-\cdots-2^{a_{k}}=2^{a_{0}+1}-n$ by induction: the base case $f(1)=1$ clearly holds; for the inductive step, when $n$ is even we note that $f(n)=2 f\left(\frac{n}{2}\right)=2\left(2^{a_{0}}-\frac{n}{2}\right)=2^{a_{0}+1}-n$ as desired, and when $n$ is odd we also have $f(n)=2 f\left(\frac{n-1}{2}\right)-1=2\left(2^{a_{0}}-\frac{n-1}{2}\right)-1=2^{a_{0}+1}-n$, again as desired. Since $19=f(n) \leq 2^{a_{0}} \leq n$, we have $a_{0} \geq 5$ and $n=2^{a_{0}+1}-19 \leq 2019$ gives $a_{0} \leq 9$. So the answer is $\sum_{a=5}^{9}\left(2^{a+1}-19\right)=\left(2^{11}-2^{6}\right)-19 \cdot 5=1889$.

Answer: 1889


Vasya replaced the same digits in two numbers with the same letters, and different digits with different letters. It turned out that the number ZARAZA is divisible by 4, and ALMAZ is divisible by 28. Find the last two digits of the sum ZARAZA + ALMAZ.

Answer: 32


Let $ABC$ be a triangle in which (${BL}$is the angle bisector of ${\angle{ABC}}$ $\left( L\in AC \right)$, ${AH}$ is an altitude of$\vartriangle ABC$ $\left( H\in BC \right)$ and ${M}$is the midpoint of the side ${AB}$. It is known  that the midpoints of the segments ${BL}$ and ${MH}$ coincides. Determine the internal angles of triangle $\vartriangle ABC$.

Given a triangle \(\triangle ABC\) with the following properties:

- \( BL \) is the angle bisector of \(\angle ABC\), with \( L \) on \( AC \).
- \( AH \) is the altitude from \( A \) to \( BC \), with \( H \) on \( BC \).
- \( M \) is the midpoint of \( AB \).

Furthermore, we are informed that the midpoints of segments \( BL \) and \( MH \) coincide. We are tasked with determining the internal angles of triangle \(\triangle ABC\).

First, let's analyze the geometry of the problem:

1. Since \( M \) is the midpoint of \( AB \) and let's denote the midpoint of \( MH \) as \( P \). It's given that \( P \) is also the midpoint of \( BL \), hence \( P \) is the midpoint of both segments.

2. Let's assume that \( P \) is the midpoint of \( BL \) and \( MH \):
   \[
   P = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right),
   \]
   where \( x_1, y_1 \) and \( x_2, y_2 \) are the coordinates of points expressing segments \( BL \) and \( MH \).

3. Since \( AH \) is the altitude, \(\angle AHB = 90^\circ\).

4. Given \( BL \) as an angle bisector, we can apply the Angle Bisector Theorem if needed for further computations.

Since the midpoint \( P \) coincides for both \( BL \) and \( MH \), the relationship indicates that the geometry exhibits symmetry properties typical of an equilateral triangle (all angles equal, all sides equal).

Construct the solution:

- Assume an equilateral triangle configuration for \( \triangle ABC \). Thus, all angles are \( 60^\circ \).

- Also, consider that \( AH \) being an altitude in such a triangle divides \( \triangle ABC \) into two \( 30^\circ - 60^\circ - 90^\circ \) triangles.

- Given this configuration, the intersection of properties (midpoints, angle bisector, and altitude) results from the symmetrical properties of an equilateral triangle.

Therefore, the internal angles of \(\triangle ABC\) are each:

\[
\boxed{60^\circ}
\]

Answer: 60^\circ


Answer: \frac{1}{2}\left(1 - 2^{-\frac{1}{32}}\right)^{-1}


Given the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with left and right foci $F\_1$, $F\_2$, $b = 4$, and an eccentricity of $\frac{3}{5}$. A line passing through $F\_1$ intersects the ellipse at points $A$ and $B$. Find the perimeter of $\triangle ABF\_2$.

Answer: 20


In the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$, find the slope of the line containing a chord that has the point $M(-2,1)$ as its midpoint.

Answer: \frac{9}{8}


Evaluate \[\frac 3{\log_5{3000^5}} + \frac 4{\log_7{3000^5}},\] giving your answer as a fraction in lowest terms.

Answer: \frac{1}{5}


In the diagram, \( S \) lies on \( R T \), \( \angle Q T S = 40^{\circ} \), \( Q S = Q T \), and \( \triangle P R S \) is equilateral. Find the value of \( x \).

Answer: 80


Given a cube with a side length of \(4\), if a solid cube of side length \(1\) is removed from each corner, calculate the total number of edges of the resulting structure.

Answer: 36


Given that $\sin A+\sin B=1$ and $\cos A+\cos B=3 / 2$, what is the value of $\cos (A-B)$?

Squaring both equations and add them together, one obtains $1+9 / 4=2+2(\cos (A) \cos (B)+\sin (A) \sin (B))=2+2 \cos (A-B)$. Thus $\cos A-B=5 / 8$.

Answer: 5/8


On the game show $\text{\emph{Wheel of Fraction}}$, you see the following spinner. Given that each region is the same area, what is the probability that you will earn exactly $\$1700$ in your first three spins? Express your answer as a common fraction. [asy]
import olympiad; import geometry; import graph; size(150); defaultpen(linewidth(0.8));
draw(unitcircle);
string[] labels = {"Bankrupt","$\$1000$","$\$300$","$\$5000$","$\$400$"};
for(int i = 0; i < 5; ++i){

draw(origin--dir(72*i));

label(labels[i],0.6*dir(36 + 72*i));
}
[/asy]

Answer: \frac{6}{125}


Every day, 12 tons of potatoes are delivered to the city using one type of transport from three collective farms. The price per ton is 4 rubles from the first farm, 3 rubles from the second farm, and 1 ruble from the third farm. To ensure timely delivery, the loading process for the required 12 tons should take no more than 40 minutes. It is known that the level of mechanization in the first farm allows loading 1 ton in 1 minute, in the second farm - 4 minutes, and in the third farm - 3 minutes. The production capacities of these farms are such that the first farm can supply no more than 10 tons per day, the second no more than 8 tons, and the third no more than 6 tons. How should the order for the supply of 12 tons be distributed among the farms so that the total cost of the potatoes delivered to the city is minimized?

Answer: 24.6667


In the Cartesian coordinate plane \( xOy \), the coordinates of point \( F \) are \((1,0)\), and points \( A \) and \( B \) lie on the parabola \( y^2 = 4x \). It is given that \( \overrightarrow{OA} \cdot \overrightarrow{OB} = -4 \) and \( |\overrightarrow{FA}| - |\overrightarrow{FB}| = 4\sqrt{3} \). Find the value of \( \overrightarrow{FA} \cdot \overrightarrow{FB} \).

Answer: -11


Given a convex quadrilateral \( ABCD \) with an interior point \( P \) such that \( P \) divides \( ABCD \) into four triangles \( ABP, BCP, CDP, \) and \( DAP \). Let \( G_1, G_2, G_3, \) and \( G_4 \) denote the centroids of these triangles, respectively. Determine the ratio of the area of quadrilateral \( G_1G_2G_3G_4 \) to the area of \( ABCD \).

Answer: \frac{1}{9}


Let \( a \) and \( b \) be positive integers such that \( 79 \mid (a + 77b) \) and \( 77 \mid (a + 79b) \). Find the smallest possible value of \( a + b \).

Answer: 193


Find the number of ways in which the letters in "HMMTHMMT" can be rearranged so that each letter is adjacent to another copy of the same letter. For example, "MMMMTTHH" satisfies this property, but "HHTMMMTM" does not.

The final string must consist of "blocks" of at least two consecutive repeated letters. For example, MMMMTTHH has a block of 4 M's, a block of 2 T's, and a block of 2 H's. Both H's must be in a block, both T's must be in a block, and all M's are either in the same block or in two blocks of 2. Therefore all blocks have an even length, meaning that all we need to do is to count the number of rearrangements of the indivisible blocks "HH", "MM", "MM", and "TT". The number of these is $4!/ 2=12$

Answer: 12


Given the function $f(x)= \sqrt {3}\sin x+\cos x$ $(x\in R)$
(1) Find the value of $f( \frac {5π}{6})$;
(2) Find the maximum and minimum values of $f(x)$ in the interval $\[- \frac {π}{2}, \frac {π}{2}\]$ and their respective $x$ values.

Answer: -\sqrt {3}


A school selects 4 teachers from 8 to teach in 4 remote areas at the same time (one person per area), where teacher A and teacher B cannot go together, and teacher A and teacher C can only go together or not go at all. The total number of different dispatch plans is ___.

Answer: 600


Find the angle of inclination of the tangent line to the curve $y=\frac{1}{3}x^3-5$ at the point $(1,-\frac{3}{2})$.

Answer: \frac{\pi}{4}


A circle has a radius of 15 units. Suppose a chord in this circle bisects a radius perpendicular to the chord and the distance from the center of the circle to the chord is 9 units. What is the area of the smaller segment cut off by the chord?
A) $100 \pi$
B) $117.29$
C) $120 \pi$
D) $180$

Answer: 117.29


Given that $\sin x + \cos x = \frac{1}{2}$, where $x \in [0, \pi]$, find the value of $\sin x - \cos x$.

Answer: \frac{\sqrt{7}}{2}