messages listlengths 2 2 | extra_info dict |
|---|---|
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "$P(x)$ is a polynomial of degree $3n$ such that\n\\begin{eqnarray*} P(0) = P(3) = \\cdots &=& P(3n) = 2, \\\\ P(1) = P(4) = \\cdots &=& P(3n-2) = 1, \\\\ P(2) = P(5) = \\cdots &=& P(3n-1)... | {
"answer": "n = 4"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Diameter $AB$ of a circle has length a $2$-digit integer (base ten). Reversing the digits gives the length of the perpendicular chord $CD$. The distance from their intersection point $H$ ... | {
"answer": "65"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Consider all 1000-element subsets of the set $\\{1, 2, 3, ... , 2015\\}$. From each such subset choose the least element. The arithmetic mean of all of these least elements is $\\frac{p... | {
"answer": "431"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "In [right triangle](https://artofproblemsolving.com/wiki/index.php/Right_triangle) $ABC$ with [right angle](https://artofproblemsolving.com/wiki/index.php/Right_angle) $C$, $CA = 30$ and ... | {
"answer": "737"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Julie is preparing a speech for her class. Her speech must last between one-half hour and three-quarters of an hour. The ideal rate of speech is 150 words per minute. If Julie speaks a... | {
"answer": "5650"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Let $m$ be the least positive integer divisible by $17$ whose digits sum to $17$. Find $m$.",
"role": "user"
}
] | {
"answer": "476"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "A student council must select a two-person welcoming committee and a three-person planning committee from among its members. There are exactly $10$ ways to select a two-person team for th... | {
"answer": "10"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Five towns are connected by a system of roads. There is exactly one road connecting each pair of towns. Find the number of ways there are to make all the roads one-way in such a way that ... | {
"answer": "544"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "The complex number $z$ satisfies $z + |z| = 2 + 8i$. What is $|z|^{2}$? Note: if $z = a + bi$, then $|z| = \\sqrt{a^{2} + b^{2}}$.\n$\\textbf{(A)}\\ 68\\qquad \\textbf{(B)}\\ 100\\qquad \... | {
"answer": "289"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Let $P$ be an interior point of triangle $ABC$ and extend lines from the vertices through $P$ to the opposite sides. Let $a$, $b$, $c$, and $d$ denote the lengths of the segments indicat... | {
"answer": "441"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "A game of solitaire is played with $R$ red cards, $W$ white cards, and $B$ blue cards. A player plays all the cards one at a time. With each play he accumulates a penalty. If he plays a b... | {
"answer": "\\min(BW, 2WR, 3RB)"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and you will paint each of the six sections a solid color. ... | {
"answer": "732"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Find the sum of all positive integers $a=2^n3^m$ where $n$ and $m$ are non-negative integers, for which $a^6$ is not a divisor of $6^a$.",
"role": "user"
}
] | {
"answer": "42"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "The population of the United States in $1980$ was $226,504,825$. The area of the country is $3,615,122$ square miles. There are $(5280)^{2}$ \nsquare feet in one square mile. Which number... | {
"answer": "500,000"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "A cylindrical tank with radius $4$ feet and height $9$ feet is lying on its side. The tank is filled with water to a depth of $2$ feet. What is the volume of water, in cubic feet?\n$\\mat... | {
"answer": "48\\pi - 36\\sqrt{3}"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "The manager of a company planned to distribute a $$50$ bonus to each employee from the company fund, but the fund contained $$5$ less than what was needed. Instead the manager gave each ... | {
"answer": "995\\text{ dollars}"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Circle $C_1$ has its center $O$ lying on circle $C_2$. The two circles meet at $X$ and $Y$. Point $Z$ in the exterior of $C_1$ lies on circle $C_2$ and $XZ=13$, $OZ=11$, and $YZ=7$. Wh... | {
"answer": "\\sqrt{30}"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Let $m$ be the number of solutions in positive integers to the equation $4x+3y+2z=2009$, and let $n$ be the number of solutions in positive integers to the equation $4x+3y+2z=2000$. Find ... | {
"answer": "000"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "The number $n$ can be written in base $14$ as $\\underline{a}\\text{ }\\underline{b}\\text{ }\\underline{c}$, can be written in base $15$ as $\\underline{a}\\text{ }\\underline{c}\\text{ ... | {
"answer": "925"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "The [polygon](https://artofproblemsolving.com/wiki/index.php/Polygon) enclosed by the solid lines in the figure consists of 4 [congruent](https://artofproblemsolving.com/wiki/index.php/Co... | {
"answer": "6"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Find the number of $7$-tuples of positive integers $(a,b,c,d,e,f,g)$ that satisfy the following systems of equations:\n\\begin{align*} abc&=70,\\\\ cde&=71,\\\\ efg&=72. \\end{align*}",
... | {
"answer": "96"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Find the number of ordered triples $(a,b,c)$ where $a$, $b$, and $c$ are positive [integers](https://artofproblemsolving.com/wiki/index.php/Integer), $a$ is a [factor](https://artofproble... | {
"answer": "200"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Let $f(n)$ be the number of ways to write $n$ as a sum of powers of $2$, where we keep track of the order of the summation. For example, $f(4)=6$ because $4$ can be written as $4$, $2+2$... | {
"answer": "2047"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "A right prism with height $h$ has bases that are regular hexagons with sides of length $12$. A vertex $A$ of the prism and its three adjacent vertices are the vertices of a triangular pyr... | {
"answer": "108"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Let $x$, $y$ and $z$ all exceed $1$ and let $w$ be a positive number such that $\\log_x w = 24$, $\\log_y w = 40$ and $\\log_{xyz} w = 12$. Find $\\log_z w$.",
"role": "user"
}
] | {
"answer": "60"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Makarla attended two meetings during her $9$-hour work day. The first meeting took $45$ minutes and the second meeting took twice as long. What percent of her work day was spent attending... | {
"answer": "(C)"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Suppose that the measurement of time during the day is converted to the metric system so that each day has $10$ metric hours, and each metric hour has $100$ metric minutes. Digital clocks... | {
"answer": "275"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Suppose $n$ is a [positive integer](https://artofproblemsolving.com/wiki/index.php/Positive_integer) and $d$ is a single [digit](https://artofproblemsolving.com/wiki/index.php/Digit) in [... | {
"answer": "750"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Each of the $100$ students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. There are $42$ students ... | {
"answer": "64"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Two unit squares are selected at random without replacement from an $n \\times n$ grid of unit squares. Find the least positive integer $n$ such that the probability that the two selected... | {
"answer": "90"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "The vertices of $\\triangle ABC$ are $A = (0,0)\\,$, $B = (0,420)\\,$, and $C = (560,0)\\,$. The six faces of a die are labeled with two $A\\,$'s, two $B\\,$'s, and two $C\\,$'s. Point ... | {
"answer": "344"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "[Set](https://artofproblemsolving.com/wiki/index.php/Set) $A$ consists of $m$ consecutive integers whose sum is $2m$, and set $B$ consists of $2m$ consecutive integers whose sum is $m.$ T... | {
"answer": "201"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "The sum of two numbers is $S$. Suppose $3$ is added to each number and then\neach of the resulting numbers is doubled. What is the sum of the final two\nnumbers?\n$\\textbf{(A)}\\ 2S + 3\... | {
"answer": "2S + 12"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "The value of $x$ that satisfies $\\log_{2^x} 3^{20} = \\log_{2^{x+3}} 3^{2020}$ can be written as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.",
... | {
"answer": "103"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "For nonnegative integers $a$ and $b$ with $a + b \\leq 6$, let $T(a, b) = \\binom{6}{a} \\binom{6}{b} \\binom{6}{a + b}$. Let $S$ denote the sum of all $T(a, b)$, where $a$ and $b$ are ... | {
"answer": "564"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Jon and Steve ride their bicycles along a path that parallels two side-by-side train tracks running the east/west direction. Jon rides east at $20$ miles per hour, and Steve rides west at... | {
"answer": "49"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Assume that $x$ is a [positive](https://artofproblemsolving.com/wiki/index.php/Positive) [real number](https://artofproblemsolving.com/wiki/index.php/Real_number). Which is equivalent to ... | {
"answer": "(D)"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Before starting to paint, Bill had $130$ ounces of blue paint, $164$ ounces of red paint, and $188$ ounces of white paint. Bill painted four equally sized stripes on a wall, making a blue... | {
"answer": "114"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "For $t = 1, 2, 3, 4$, define $S_t = \\sum_{i = 1}^{350}a_i^t$, where $a_i \\in \\{1,2,3,4\\}$. If $S_1 = 513$ and $S_4 = 4745$, find the minimum possible value for $S_2$.",
"role": "u... | {
"answer": "905"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Determine the number of [ordered pairs](https://artofproblemsolving.com/wiki/index.php/Ordered_pair) $(a,b)$ of [integers](https://artofproblemsolving.com/wiki/index.php/Integer) such tha... | {
"answer": "54"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "A hexagon that is inscribed in a circle has side lengths $22$, $22$, $20$, $22$, $22$, and $20$ in that order. The radius of the circle can be written as $p+\\sqrt{q}$, where $p$ and $q$ ... | {
"answer": "272"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "The sum of two numbers is $S$. Suppose $3$ is added to each number and then\neach of the resulting numbers is doubled. What is the sum of the final two\nnumbers?\n$\\textbf{(A)}\\ 2S + 3\... | {
"answer": "2S + 12"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "A group of clerks is assigned the task of sorting $1775$ files. Each clerk sorts at a constant rate of $30$ files per hour. At the end of the first hour, some of the clerks are reassigned... | {
"answer": "945"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Let the [set](https://artofproblemsolving.com/wiki/index.php/Set) $\\mathcal{S} = \\{8, 5, 1, 13, 34, 3, 21, 2\\}.$ Susan makes a list as follows: for each two-element subset of $\\mathca... | {
"answer": "484"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "What is the smallest integer $n$, greater than one, for which the root-mean-square of the first $n$ positive integers is an integer?\n$\\mathbf{Note.}$ The root-mean-square of $n$ numbers... | {
"answer": "337"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "A triangular corner with side lengths $DB=EB=1$ is cut from equilateral triangle ABC of side length $3$. \nThe perimeter of the remaining quadrilateral is\n\n$\\text{(A)} \\ 6 \\qquad \\... | {
"answer": "8"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Two thousand points are given on a [circle](https://artofproblemsolving.com/wiki/index.php/Circle). Label one of the points $1$. From this point, count $2$ points in the clockwise directi... | {
"answer": "118"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "A group of children held a grape-eating contest. When the contest was over, the winner had eaten $n$ grapes, and the child in $k$-th place had eaten $n+2-2k$ grapes. The total number of ... | {
"answer": "89"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Find, with proof, all positive integers $n$ for which $2^n + 12^n + 2011^n$ is a perfect square.",
"role": "user"
}
] | {
"answer": "n = 1"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "The system of equations\n\\begin{eqnarray*}\\log_{10}(2000xy) - (\\log_{10}x)(\\log_{10}y) & = & 4 \\\\ \\log_{10}(2yz) - (\\log_{10}y)(\\log_{10}z) & = & 1 \\\\ \\log_{10}(zx) - (\\log_{... | {
"answer": "25"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "What is the smallest [positive](https://artofproblemsolving.com/wiki/index.php/Positive) [integer](https://artofproblemsolving.com/wiki/index.php/Integer) that can be expressed as the sum... | {
"answer": "495"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Rudolph bikes at a [constant](https://artofproblemsolving.com/wiki/index.php/Constant) rate and stops for a five-minute break at the end of every mile. Jennifer bikes at a constant rate w... | {
"answer": "620 \\text{ minutes}"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "In base $R_1$ the expanded fraction $F_1$ becomes $.373737\\cdots$, and the expanded fraction $F_2$ becomes $.737373\\cdots$. In base $R_2$ fraction $F_1$, when expanded, becomes $.252525... | {
"answer": "19"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "A pyramid has a triangular base with side lengths $20$, $20$, and $24$. The three edges of the pyramid from the three corners of the base to the fourth vertex of the pyramid all have leng... | {
"answer": "803"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Find the number of ordered pairs of positive integers $(m,n)$ such that ${m^2n = 20 ^{20}}$.",
"role": "user"
}
] | {
"answer": "231"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Angelina drove at an average rate of $80$ kmh and then stopped $20$ minutes for gas. After the stop, she drove at an average rate of $100$ kmh. Altogether she drove $250$ km in a total tr... | {
"answer": "(A)"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Let $(a,b,c)$ be the [real](https://artofproblemsolving.com/wiki/index.php/Real_number) solution of the system of equations $x^3 - xyz = 2$, $y^3 - xyz = 6$, $z^3 - xyz = 20$. The greates... | {
"answer": "158"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Triangle $ABC_0$ has a right angle at $C_0$. Its side lengths are pairwise relatively prime positive integers, and its perimeter is $p$. Let $C_1$ be the foot of the altitude to $\\overli... | {
"answer": "182"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Let $f(x)=(x^2+3x+2)^{\\cos(\\pi x)}$. Find the sum of all positive integers $n$ for which \n\\[\\left |\\sum_{k=1}^n\\log_{10}f(k)\\right|=1.\\]",
"role": "user"
}
] | {
"answer": "21"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Find the number of ordered pairs of positive integer solutions $(m, n)$ to the equation $20m + 12n = 2012$.",
"role": "user"
}
] | {
"answer": "34"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Find the number of positive integers less than $1000$ that can be expressed as the difference of two integral powers of $2.$",
"role": "user"
}
] | {
"answer": "50"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "For any positive integer $k$, let $f_1(k)$ denote the square of the sum of the digits of $k$. For $n \\ge 2$, let $f_n(k) = f_1(f_{n - 1}(k))$. Find $f_{1988}(11)$.",
"role": "user"... | {
"answer": "169"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "In a drawer Sandy has $5$ pairs of socks, each pair a different color. On Monday Sandy selects two individual socks at random from the $10$ socks in the drawer. On Tuesday Sandy selects... | {
"answer": "341"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Call a permutation $a_1, a_2, \\ldots, a_n$ of the integers $1, 2, \\ldots, n$ quasi-increasing if $a_k \\leq a_{k+1} + 2$ for each $1 \\leq k \\leq n-1$. For example, 53421 and 14253 are... | {
"answer": "486"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Randy drove the first third of his trip on a gravel road, the next $20$ miles on pavement, and the remaining one-fifth on a dirt road. In miles, how long was Randy's trip?\n$\\textbf{(A)}... | {
"answer": "\\frac{300}{7}"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Points $A$, $B$, and $C$ lie in that order along a straight path where the distance from $A$ to $C$ is $1800$ meters. Ina runs twice as fast as Eve, and Paul runs twice as fast as Ina. Th... | {
"answer": "800"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "There exist unique positive integers $x$ and $y$ that satisfy the equation $x^2 + 84x + 2008 = y^2$. Find $x + y$.",
"role": "user"
}
] | {
"answer": "80"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Sally has five red cards numbered $1$ through $5$ and four blue cards numbered $3$ through $6$. She stacks the cards so that the colors alternate and so that the number on each red card d... | {
"answer": "12"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "[Rectangle](https://artofproblemsolving.com/wiki/index.php/Rectangle) $ABCD$ is given with $AB=63$ and $BC=448.$ Points $E$ and $F$ lie on $AD$ and $BC$ respectively, such that $AE=CF=84.... | {
"answer": "259"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "In a new school $40$ percent of the students are freshmen, $30$ percent are sophomores, $20$ percent are juniors, and $10$ percent are seniors. All freshmen are required to take Latin, an... | {
"answer": "25"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "For each positive integer $n$, let \n\n$a_n = \\frac{(n+9)!}{(n-1)!}$\nLet $k$ denote the smallest positive integer for which the rightmost nonzero digit of $a_k$ is odd. The rightmost no... | {
"answer": "9"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Consider functions $f : [0, 1] \\rightarrow \\mathbb{R}$ which satisfy\n\n\n (i)$f(x)\\ge0$ for all $x$ in $[0, 1]$,\n\n (ii)$f(1) = 1$,\n\n (iii) $f(x) + f(y) \\le f(x + ... | {
"answer": "2"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Let $P(x) = x^2 - 3x - 7$, and let $Q(x)$ and $R(x)$ be two quadratic polynomials also with the coefficient of $x^2$ equal to $1$. David computes each of the three sums $P + Q$, $P + R$, ... | {
"answer": "71"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Find the sum of all positive integers $b < 1000$ such that the base-$b$ integer $36_{b}$ is a perfect square and the base-$b$ integer $27_{b}$ is a perfect cube.",
"role": "user"
}
... | {
"answer": "371"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\mathbb{Z}$ be the set of integers. Find all functions $f : \\mathbb{Z} \\rightarrow \\mathbb{Z}$ such that \\[xf(2f(y)-x)+y^2f(2x-f(y))=\\frac{f(x)^2}{x}+f(yf(y))\\] for all $x, y ... | {
"answer": "f(x) = x^2"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Al walks down to the bottom of an escalator that is moving up and he counts 150 steps. His friend, Bob, walks up to the top of the escalator and counts 75 steps. If Al's speed of walking ... | {
"answer": "120"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "The area of a rectangle remains unchanged when it is made $2 \\frac{1}{2}$ inches longer and $\\frac{2}{3}$ inch narrower, or when it is made $2 \\frac{1}{2}$ inches shorter and $\\frac{4... | {
"answer": "20"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "A permutation of the set of positive integers $[n] = \\{1, 2, \\ldots, n\\}$ is a sequence $(a_1, a_2, \\ldots, a_n)$ such that each element of $[n]$ appears precisely one time as a term ... | {
"answer": "4489"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Positive integers $a$ and $b$ satisfy the condition\n\\[\\log_2(\\log_{2^a}(\\log_{2^b}(2^{1000}))) = 0.\\]\nFind the sum of all possible values of $a+b$.",
"role": "user"
}
] | {
"answer": "881"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Let $C$ be the [graph](https://artofproblemsolving.com/wiki/index.php/Graph) of $xy = 1$, and denote by $C^*$ the [reflection](https://artofproblemsolving.com/wiki/index.php/Reflection) o... | {
"answer": "084"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "For any positive integer $a, \\sigma(a)$ denotes the sum of the positive integer divisors of $a$. Let $n$ be the least positive integer such that $\\sigma(a^n)-1$ is divisible by $2021$ f... | {
"answer": "125"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Let the set $S = \\{P_1, P_2, \\dots, P_{12}\\}$ consist of the twelve vertices of a regular $12$-gon. A subset $Q$ of $S$ is called \"communal\" if there is a circle such that all points... | {
"answer": "134"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "The diagram shows twenty congruent [circles](https://artofproblemsolving.com/wiki/index.php/Circle) arranged in three rows and enclosed in a rectangle. The circles are tangent to one anot... | {
"answer": "154"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Fifteen distinct points are designated on $\\triangle ABC$: the 3 vertices $A$, $B$, and $C$; $3$ other points on side $\\overline{AB}$; $4$ other points on side $\\overline{BC}$; and $5$... | {
"answer": "390"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "One commercially available ten-button lock may be opened by pressing -- in any order -- the correct five buttons. The sample shown below has $\\{1,2,3,6,9\\}$ as its [combination](https:/... | {
"answer": "770"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Let $N$ be the number of ordered triples $(A,B,C)$ of integers satisfying the conditions \n(a) $0\\le A<B<C\\le99$, \n(b) there exist integers $a$, $b$, and $c$, and prime $p$ where $0\\l... | {
"answer": "272"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Let $m$ be the number of five-element subsets that can be chosen from the set of the first $14$ natural numbers so that at least two of the five numbers are consecutive. Find the remainde... | {
"answer": "750"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Let $X_1, X_2, \\ldots, X_{100}$ be a sequence of mutually distinct nonempty subsets of a set $S$. Any two sets $X_i$ and $X_{i+1}$ are disjoint and their union is not the whole set $S$, ... | {
"answer": "8"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Let N = $69^{5} + 5\\cdot69^{4} + 10\\cdot69^{3} + 10\\cdot69^{2} + 5\\cdot69 + 1$. How many positive integers are factors of $N$?\n$\\textbf{(A)}\\ 3\\qquad \\textbf{(B)}\\ 5\\qquad \\te... | {
"answer": "216"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "In a Martian civilization, all logarithms whose bases are not specified are assumed to be base $b$, for some fixed $b\\ge2$. A Martian student writes down\n\\[3\\log(\\sqrt{x}\\log x)=56\... | {
"answer": "216"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\overline{AB}$ be a diameter in a circle of radius $5\\sqrt2.$ Let $\\overline{CD}$ be a chord in the circle that intersects $\\overline{AB}$ at a point $E$ such that $BE=2\\sqrt5$ ... | {
"answer": "100"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "For any finite set $S$, let $|S|$ denote the number of elements in $S$. Find the number of ordered pairs $(A,B)$ such that $A$ and $B$ are (not necessarily distinct) subsets of $\\{1,2,3,... | {
"answer": "454"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Find the number of positive integers $m$ for which there exist nonnegative integers $x_0$, $x_1$ , $\\dots$ , $x_{2011}$ such that\n\\[m^{x_0} = \\sum_{k = 1}^{2011} m^{x_k}.\\]",
"ro... | {
"answer": "16"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Initially Alex, Betty, and Charlie had a total of $444$ peanuts. Charlie had the most peanuts, and Alex had the least. The three numbers of peanuts that each person had formed a geometric... | {
"answer": "108"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "A bug walks all day and sleeps all night. On the first day, it starts at point $O$, faces east, and walks a distance of $5$ units due east. Each night the bug rotates $60^\\circ$ counterc... | {
"answer": "103"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Define a sequence recursively by $t_1 = 20$, $t_2 = 21$, and\\[t_n = \\frac{5t_{n-1}+1}{25t_{n-2}}\\]for all $n \\ge 3$. Then $t_{2020}$ can be expressed as $\\frac{p}{q}$, where $p$ and ... | {
"answer": "626"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Find the number of integers $c$ such that the equation \\[\\left||20|x|-x^2|-c\\right|=21\\]has $12$ distinct real solutions.",
"role": "user"
}
] | {
"answer": "57"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Abe can paint the room in 15 hours, Bea can paint 50 percent faster than Abe, and Coe can paint twice as fast as Abe. Abe begins to paint the room and works alone for the first hour and a... | {
"answer": "334"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "Anh read a book. On the first day she read $n$ pages in $t$ minutes, where $n$ and $t$ are positive integers. On the second day Anh read $n + 1$ pages in $t + 1$ minutes. Each day thereaf... | {
"answer": "53"
} |
[
{
"content": "You are a math solver. Give the final answer in the \\boxed{}.",
"role": "system"
},
{
"content": "What is the value of $[\\log_{10}(5\\log_{10}100)]^2$?\n$\\textbf{(A)}\\ \\log_{10}50 \\qquad \\textbf{(B)}\\ 25\\qquad \\textbf{(C)}\\ 10 \\qquad \\textbf{(D)}\\ 2\\qquad \\textbf{(E)}\\... | {
"answer": "1"
} |
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