system stringclasses 1
value | conversations listlengths 2 2 | task stringclasses 1
value |
|---|---|---|
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Once upon a time, a team of Knights and a team of Liars met in the park and decided to ride a circular carousel that can hold 40 people (the \"Chain\" carousel, where everyone sits one behind the other). When they took their seats, each person saw two others: one in front and on... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Find the biggest real number $ k$ such that for each right-angled triangle with sides $ a$ , $ b$ , $ c$ , we have\r\n\\[ a^{3}\\plus{}b^{3}\\plus{}c^{3}\\geq k\\left(a\\plus{}b\\plus{}c\\right)^{3}.\\]"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to ... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Given \\( \\theta_{1}, \\theta_{2}, \\theta_{3}, \\theta_{4} \\in \\mathbf{R}^{+} \\) and \\( \\theta_{1} + \\theta_{2} + \\theta_{3} + \\theta_{4} = \\pi \\), find the minimum value of \\( \\left(2 \\sin^{2} \\theta_{1} + \\frac{1}{\\sin^{2} \\theta_{1}}\\right)\\left(2 \\sin^{... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "The center of the circle inscribed in a trapezoid is at distances of 5 and 12 from the ends of one of the non-parallel sides. Find the length of this side."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this problem about a trapezoid with an inscribed circle... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "A word is an ordered, non-empty sequence of letters, such as $word$ or $wrod$ . How many distinct $3$ -letter words can be made from a subset of the letters $c, o, m, b, o$ , where each letter in the list is used no more than the number of times it appears?"
},
{
"f... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Given a parabola \\(C\\) with the center of ellipse \\(E\\) as its focus, the parabola \\(C\\) passes through the two foci of the ellipse \\(E\\), and intersects the ellipse \\(E\\) at exactly three points. Find the eccentricity of the ellipse \\(E\\)."
},
{
"from": "gpt... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "There exists a unique strictly increasing sequence of nonnegative integers $a_1 < a_2 < … < a_k$ such that \\[\\frac{2^{289}+1}{2^{17}+1} = 2^{a_1} + 2^{a_2} + … + 2^{a_k}.\\] What is $k?$"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out the valu... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Find all the roots of the equation\n\n\\[ 1 - \\frac{x}{1} + \\frac{x(x-1)}{2!} - \\frac{x(x-1)(x-2)}{3!} + \\frac{x(x-1)(x-2)(x-3)}{4!} - \\frac{x(x-1)(x-2)(x-3)(x-4)}{5!} + \\frac{x(x-1)(x-2)(x-3)(x-4)(x-5)}{6!} = 0 \\]\n\n(Where \\( n! = 1 \\cdot 2 \\cdot 3 \\cdots n \\))\n\n... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Let $\\overline{AB}$ be a chord of a circle $\\omega$ , and let $P$ be a point on the chord $\\overline{AB}$ . Circle $\\omega_1$ passes through $A$ and $P$ and is internally tangent to $\\omega$ . Circle $\\omega_2$ passes through $B$ and $P$ and is internally tangent to $\\ome... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Calculate \\( \\left[6 \\frac{3}{5}-\\left(8.5-\\frac{1}{3}\\right) \\div 3.5\\right] \\times\\left(2 \\frac{5}{18}+\\frac{11}{12}\\right) = \\)"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. I need to calculate this expression: \n\n\\[ \\left[6 \\frac{3}{5... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Given a parallelogram \\(ABCD\\) with \\(\\angle B = 60^\\circ\\). Point \\(O\\) is the center of the circumcircle of triangle \\(ABC\\). Line \\(BO\\) intersects the bisector of the exterior angle \\(\\angle D\\) at point \\(E\\). Find the ratio \\(\\frac{BO}{OE}\\)."
},
{
... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "The distance from the intersection point of the diameter of a circle with a chord of length 18 cm to the center of the circle is 7 cm. This point divides the chord in the ratio 2:1. Find the radius of the circle.\n\n$$\nAB = 18, EO = 7, AE = 2 BE, R = ?\n$$"
},
{
"from":... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Real numbers \\(a\\), \\(b\\), and \\(c\\) and positive number \\(\\lambda\\) make \\(f(x) = x^3 + ax^2 + b x + c\\) have three real roots \\(x_1\\), \\(x_2\\), \\(x_3\\), such that:\n(1) \\(x_2 - x_1 = \\lambda\\);\n(2) \\(x_3 > \\frac{1}{2}(x_1 + x_2)\\).\nFind the maximum val... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Find the number of permutations $x_1, x_2, x_3, x_4, x_5$ of numbers $1, 2, 3, 4, 5$ such that the sum of five products \\[x_1x_2x_3 + x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_1 + x_5x_1x_2\\] is divisible by $3$"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "There are three boxes \\(A\\), \\(B\\), and \\(C\\) containing 100, 50, and 80 balls respectively. Each box has a certain number of black balls. It is known that box \\(A\\) contains 15 black balls. If a box is chosen randomly and then a ball is randomly drawn from that box, the... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "In the triangle \\(ABC\\), it is known that \\(AB=BC\\) and \\(\\angle BAC=45^\\circ\\). The line \\(MN\\) intersects side \\(AC\\) at point \\(M\\), and side \\(BC\\) at point \\(N\\). Given that \\(AM=2 \\cdot MC\\) and \\(\\angle NMC=60^\\circ\\), find the ratio of the area o... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "The Dyslexian alphabet consists of consonants and vowels. It so happens that a finite sequence of letters is a word in Dyslexian precisely if it alternates between consonants and vowels (it may begin with either). There are 4800 five-letter words in Dyslexian. How many letters a... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "The sequence \\(a_{0}, a_{1}, a_{2}, \\cdots, a_{n}\\) satisfies \\(a_{0}=\\sqrt{3}\\), \\(a_{n+1}=\\left\\lfloor a_{n}\\right\\rfloor + \\frac{1}{\\left\\{a_{n}\\right\\}}\\), where \\(\\left\\lfloor a_{n}\\right\\rfloor\\) and \\(\\left\\{a_{n}\\right\\}\\) represent the integ... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "How many quadratic polynomials with real coefficients are there such that the set of roots equals the set of coefficients? (For clarification: If the polynomial is $ax^2+bx+c,a\\neq 0,$ and the roots are $r$ and $s,$ then the requirement is that $\\{a,b,c\\}=\\{r,s\\}$ .)"
},
... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Suppose $x$ is a random real number between $1$ and $4$ , and $y$ is a random real number between $1$ and $9$ . If the expected value of \\[ \\left\\lceil \\log_2 x \\right\\rceil - \\left\\lfloor \\log_3 y \\right\\rfloor \\] can be expressed as $\\frac mn$ where $... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "A triangle \\( ABC \\) is given. It is known that \\( AB=4 \\), \\( AC=2 \\), and \\( BC=3 \\). The angle bisector of \\( \\angle BAC \\) intersects side \\( BC \\) at point \\( K \\). A line passing through point \\( B \\) parallel to \\( AC \\) intersects the extension of the ... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Two points are drawn on each side of a square with an area of 81 square units, dividing the side into 3 congruent parts. Quarter-circle arcs connect the points on adjacent sides to create the figure shown. What is the length of the boundary of the bolded figure? Express your ... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "From points A and B, a motorcyclist and a cyclist respectively set off towards each other simultaneously and met at a distance of 4 km from B. At the moment the motorcyclist arrived in B, the cyclist was at a distance of 15 km from A. Find the distance AB."
},
{
"from": ... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "What is the largest integer that must divide $n^5-5n^3+4n$ for all integers $n$ ?\n\n*2016 CCA Math Bonanza Lightning #2.4*"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the largest integer that must divide the expression \\( n^5 - 5n^3 + 4n \\) ... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "How many six-digit numbers exist that have three even and three odd digits?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out how many six-digit numbers have exactly three even digits and three odd digits. Hmm, let's start by recalling that a six-... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Given the real numbers $a, x, y$ that satisfy the equation:\n$$\nx \\sqrt{a(x-a)}+y \\sqrt{a(y-a)}=\\sqrt{|\\lg (x-a)-\\lg (a-y)|},\n$$\n\nfind the value of the algebraic expression $\\frac{3 x^{2}+x y-y^{2}}{x^{2}-x y+y^{2}}$."
},
{
"from": "gpt",
"value": "<think>\... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Let $2000 < N < 2100$ be an integer. Suppose the last day of year $N$ is a Tuesday while the first day of year $N+2$ is a Friday. The fourth Sunday of year $N+3$ is the $m$ th day of January. What is $m$ ?\n\n*Based on a proposal by Neelabh Deka*"
},
{
"from": ... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Let \\( M_{n} = \\left\\{ 0 . \\overline{a_{1} a_{2} \\cdots a_{n}} \\mid a_{i} \\ \\text{is either 0 or 1 for} \\ i=1,2, \\cdots, n-1, \\ a_{n}=1 \\right\\} \\). \\( T_{n} \\) is the number of elements in \\( M_{n} \\) and \\( S_{n} \\) is the sum of all elements in \\( M_{n} \... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Given the sequence \\( a_{1}, a_{2}, \\cdots, a_{n}, \\cdots \\) that satisfies \\( a_{1}=a_{2}=1 \\) and \\( a_{3}=2 \\), and for any natural number \\( n \\), \\( a_{n} a_{n+1} a_{n+2} \\neq 1 \\), and \\( a_{n} a_{n+1} a_{n+2} a_{n+3}=a_{n}+a_{n+1}+a_{n+2}+a_{n+3} \\), find t... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Given that \\( x + y + z = xy + yz + zx \\), find the minimum value of \\( \\frac{x}{x^2 + 1} + \\frac{y}{y^2 + 1} + \\frac{z}{z^2 + 1} \\)."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the minimum value of the expression \\( \\frac{x}{x^2 + 1} + \... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Determine how many roots of the equation\n\n$$\n4 \\sin 2x + 3 \\cos 2x - 2 \\sin x - 4 \\cos x + 1 = 0\n$$\n\nare located in the interval \\(\\left[10^{2014!} \\pi, 10^{2014!+2015} \\pi\\right]\\). In the answer, write down the sum of all the digits of the found number."
},
... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Kathy has $5$ red cards and $5$ green cards. She shuffles the $10$ cards and lays out $5$ of the cards in a row in a random order. She will be happy if and only if all the red cards laid out are adjacent and all the green cards laid out are adjacent. For example, card orders RRG... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Let \\(a_n\\) be the sequence defined by \\(a_1 = 3\\) and \\(a_{n+1} = 3^{k}\\), where \\(k = a_n\\). Let \\(b_n\\) be the remainder when \\(a_n\\) is divided by 100. Which values \\(b_n\\) occur for infinitely many \\(n\\)?"
},
{
"from": "gpt",
"value": "<think>\nO... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Let $a,b,c,d$ be real numbers such that $b-d \\ge 5$ and all zeros $x_1, x_2, x_3,$ and $x_4$ of the polynomial $P(x)=x^4+ax^3+bx^2+cx+d$ are real. Find the smallest value the product $(x_1^2+1)(x_2^2+1)(x_3^2+1)(x_4^2+1)$ can take."
},
{
"from": "gpt",
"value": "<th... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "There is a basket of apples. After dividing them into three equal parts, 2 apples remain. Taking out two of these parts, and dividing them into three equal parts again, 2 apples remain. After taking out two of these parts again and dividing them into three equal parts, 2 apples ... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "There are 10 numbers written in a circle, and their sum is 100. It is known that the sum of any three consecutive numbers is not less than 29.\n\nDetermine the smallest number \\( A \\) such that in any such set of numbers, each number does not exceed \\( A \\)."
},
{
"f... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "With about six hours left on the van ride home from vacation, Wendy looks for something to do. She starts working on a project for the math team.\n\nThere are sixteen students, including Wendy, who are about to be sophomores on the math team. Elected as a math team officer, on... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Let $m > n$ be positive integers such that $3(3mn - 2)^2 - 2(3m -3n)^2 = 2019$ . Find $3m + n$ ."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to solve this equation: 3(3mn - 2)^2 - 2(3m - 3n)^2 = 2019, where m and n are positive integers with m > n. An... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "There are three identical red balls, three identical yellow balls, and three identical green balls. In how many different ways can they be split into three groups of three balls each?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this problem here: there ar... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Given a plane intersects all 12 edges of a cube at an angle $\\alpha$, find $\\sin \\alpha$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this problem here: A plane intersects all 12 edges of a cube at an angle α. I need to find sin α. Hmm, interesting. Le... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Given a square \\(ABCD\\) with side length 1, determine the maximum value of \\(PA \\cdot PB \\cdot PC \\cdot PD\\) where point \\(P\\) lies inside or on the boundary of the square."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this problem: Given a square ... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "In triangle \\( \\triangle ABC \\), the sides opposite to angles \\( A \\), \\( B \\), and \\( C \\) are \\( a \\), \\( b \\), and \\( c \\) respectively. Given that \\( a^2 - (b - c)^2 = (2 - \\sqrt{3})bc \\) and \\( \\sin A \\sin B = \\cos^2 \\frac{C}{2} \\), and the length of... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Two workers were assigned to produce a batch of identical parts; after the first worked for \\(a\\) hours and the second for \\(0.6a\\) hours, it turned out that they had completed \\(\\frac{5}{n}\\) of the entire job. After working together for another \\(0.6a\\) hours, they fo... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Find the infinite sum of \\(\\frac{1^{3}}{3^{1}}+\\frac{2^{3}}{3^{2}}+\\frac{3^{3}}{3^{3}}+\\frac{4^{3}}{3^{4}}+\\cdots\\).\n\n求 \\(\\frac{1^{3}}{3^{1}}+\\frac{2^{3}}{3^{2}}+\\frac{3^{3}}{3^{3}}+\\frac{4^{3}}{3^{4}}+\\cdots\\) 無限項之和。"
},
{
"from": "gpt",
"value": "<t... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Peter's most listened-to CD contains eleven tracks. His favorite is the eighth track. When he inserts the CD into the player and presses one button, the first track starts, and by pressing the button seven more times, he reaches his favorite song. If the device is in \"random\" ... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "A team consisting of Petya, Vasya, and a single-seat scooter is participating in a race. The distance is divided into 42 equal-length segments, with a checkpoint at the beginning of each segment. Petya completes a segment in 9 minutes, Vasya in 11 minutes, and either of them can... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "A convex pentagon $P=ABCDE$ is inscribed in a circle of radius $1$ . Find the maximum area of $P$ subject to the condition that the chords $AC$ and $BD$ are perpendicular."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the maximum area of a ... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Point \\( A \\) lies on the line \\( y = \\frac{5}{12} x - 7 \\), and point \\( B \\) lies on the parabola \\( y = x^2 \\). What is the minimum length of segment \\( AB \\)?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the minimum distance between ... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "A trapezoid \\(AEFG\\) (\\(EF \\parallel AG\\)) is positioned inside a square \\(ABCD\\) with a side length of 14, such that points \\(E\\), \\(F\\), and \\(G\\) lie on sides \\(AB\\), \\(BC\\), and \\(CD\\) respectively. The diagonals \\(AF\\) and \\(EG\\) are perpendicular, an... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "A target consists of five zones: the center circle (bullseye) and four colored rings. The width of each ring is equal to the radius of the bullseye. It is known that the score for hitting each zone is inversely proportional to the probability of hitting that zone, and hitting th... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "The hares are cutting the log again, but now both ends of the log are fixed. Ten middle logs fell, and the two end ones remained fixed. How many cuts did the hares make?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so the problem is about hares cutting a log, right?... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "The base of a rectangular parallelepiped is a square with a side length of \\(2 \\sqrt{3}\\). The diagonal of a lateral face forms an angle of \\(30^\\circ\\) with the plane of an adjacent lateral face. Find the volume of the parallelepiped."
},
{
"from": "gpt",
"val... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "For how many integer values of $m$ ,\n(i) $1\\le m \\le 5000$ (ii) $[\\sqrt{m}] =[\\sqrt{m+125}]$ Note: $[x]$ is the greatest integer function"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out how many integer values of m satisfy two conditio... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Lily and Sarah are playing a game. They each choose a real number at random between -1 and 1. They then add the squares of their numbers together. If the result is greater than or equal to 1, Lily wins, and if the result is less than 1, Sarah wins. What is the probability that S... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Let \\( f(x) = x^{9} + x^{8} + x^{7} + x^{6} + x^{5} + x^{4} + x^{3} + x^{2} + x + 1 \\). When \\( f(x^{10}) \\) is divided by \\( f(x) \\), the remainder is \\( b \\). Find the value of \\( b \\)."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this problem ... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Points $A$ , $B$ , $C$ , $D$ , and $E$ are on the same plane such that $A,E,C$ lie on a line in that order, $B,E,D$ lie on a line in that order, $AE = 1$ , $BE = 4$ , $CE = 3$ , $DE = 2$ , and $\\angle AEB = 60^\\circ$ . Let $AB$ and $CD$ intersect at $P$ . T... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Find the least real number $K$ such that for all real numbers $x$ and $y$ , we have $(1 + 20 x^2)(1 + 19 y^2) \\ge K xy$ ."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the least real number K such that for all real numbers x and y, the inequa... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "How many solutions does the system have: $ \\{\\begin{matrix}&(3x+2y) *(\\frac{3}{x}+\\frac{1}{y})=2 & x^2+y^2\\leq 2012 \\end{matrix} $ where $ x,y $ are non-zero integers"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this system of equations to solve. L... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Victor has $3$ piles of $3$ cards each. He draws all of the cards, but cannot draw a card until all the cards above it have been drawn. (For example, for his first card, Victor must draw the top card from one of the $3$ piles.) In how many orders can Victor draw the cards?... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Inside of the square $ABCD$ the point $P$ is given such that $|PA|:|PB|:|PC|=1:2:3$ . Find $\\angle APB$ ."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this geometry problem here: Inside square ABCD, there's a point P such that the distances from P t... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Find the least positive integer, which may not be represented as ${2^a-2^b\\over 2^c-2^d}$ , where $a,\\,b,\\,c,\\,d$ are positive integers."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the least positive integer that cannot be expressed in the ... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Two tangents $AT$ and $BT$ touch a circle at $A$ and $B$ , respectively, and meet perpendicularly at $T$ . $Q$ is on $AT$ , $S$ is on $BT$ , and $R$ is on the circle, so that $QRST$ is a rectangle with $QT = 8$ and $ST = 9$ . Determine the radius of the circ... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Let \\( A \\) and \\( B \\) be two sets, and \\((A, B)\\) be called a \"pair\". If \\( A \\neq B \\), then \\((A, B)\\) and \\((B, A)\\) are considered different \"pairs\". Find the number of different pairs \\((A, B)\\) that satisfy the condition \\( A \\cup B = \\{1,2,3,4\\} \... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "How many triangles are in the figure to the right? [asy]\ndefaultpen(linewidth(0.7));\npair hexcoords (real over, real upover)\n{\n\nreturn dir(0)*over+dir(60)*upover;\n}\nreal r = 0.3;\nint i,j;\nfor(i=0;i<=2;++i)\n\n{\n\nfor(j=0;j<=2-i;++j)\n\n{\n\ndraw(hexcoords(i,j)--hexcoor... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "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... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Let \\( A B C \\) be an isosceles triangle with \\( B \\) as the vertex of the equal angles. Let \\( F \\) be a point on the bisector of \\( \\angle A B C \\) such that \\( (A F) \\) is parallel to \\( (B C) \\). Let \\( E \\) be the midpoint of \\([B C]\\), and let \\( D \\) be... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Let the set \\( P = \\{1, 2, \\ldots, 2014\\} \\) and \\( A \\subseteq P \\). If the difference between any two numbers in the set \\( A \\) is not a multiple of 99, and the sum of any two numbers in the set \\( A \\) is also not a multiple of 99, then the set \\( A \\) can cont... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Two concentric circles $\\omega, \\Omega$ with radii $8,13$ are given. $AB$ is a diameter of $\\Omega$ and the tangent from $B$ to $\\omega$ touches $\\omega$ at $D$ . What is the length of $AD$ ."
},
{
"from": "gpt",
"value": "<think>\nOkay, let's tr... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "The sum of two nonzero natural numbers is 210, and their least common multiple is 1547. What is their product? $\\qquad$"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. I need to find two nonzero natural numbers whose sum is 210 and whose least common multip... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "What is the maximum value that the expression \\(\\frac{1}{a+\\frac{2010}{b+\\frac{1}{c}}}\\) can take, where \\(a, b, c\\) are distinct non-zero digits?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the maximum value of the expression \\(\\frac{1}{... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "What is the maximum number of rooks that can be placed on a $300 \\times 300$ chessboard such that each rook attacks at most one other rook? (A rook attacks all the squares it can reach according to chess rules without passing through other pieces.)"
},
{
"from": "gpt",
... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "A sequence is defined as follows $a_1=a_2=a_3=1,$ and, for all positive integers $n, a_{n+3}=a_{n+2}+a_{n+1}+a_n.$ Given that $a_{28}=6090307, a_{29}=11201821,$ and $a_{30}=20603361,$ find the remainder when $\\sum^{28}_{k=1} a_k$ is divided by 1000."
},
{
"from": "gpt",... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "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 $... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "A whole number was increased by 2, and its square decreased by 2016. What was the number initially (before the increase)?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to solve this problem: A whole number was increased by 2, and its square decreased by 201... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Find all real numbers \\( k \\) such that the inequality\n$$\na^{3}+b^{3}+c^{3}+d^{3}+1 \\geqslant k(a+b+c+d)\n$$\nholds for any \\( a, b, c, d \\in [-1, +\\infty) \\)."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find all real numbers k such that the i... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Let \\( f(x) = \\frac{1}{x^3 + 3x^2 + 2x} \\). Determine the smallest positive integer \\( n \\) such that \n\\[ f(1) + f(2) + f(3) + \\cdots + f(n) > \\frac{503}{2014}. \\]"
},
{
"from": "gpt",
"value": "<think>\nOkay, let me try to figure out this problem. So, we h... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "There are 294 distinct cards with numbers \\(7, 11, 7^{2}, 11^{2}, \\ldots, 7^{147}, 11^{147}\\) (each card has exactly one number, and each number appears exactly once). How many ways can two cards be selected so that the product of the numbers on the selected cards is a perfec... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Farm A has a total of 625 chickens and ducks, and Farm B has a total of 748 chickens and ducks. The number of chickens in Farm B is 24% more than in Farm A, and the number of ducks in Farm A is 15% less than in Farm B. How many chickens does Farm B have?"
},
{
"from": "g... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "The altitudes of an acute isosceles triangle, where \\(AB = BC\\), intersect at point \\(H\\). Find the area of triangle \\(ABC\\), given \\(AH = 5\\) and the altitude \\(AD\\) is 8."
},
{
"from": "gpt",
"value": "<think>\nOkay, let's try to solve this problem. So we... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Among the 9 natural numbers $1,2,3, \\cdots, 9$, if 3 numbers are chosen, let $x$ be the number of pairs of adjacent numbers among the chosen 3 numbers (for example, if the 3 chosen numbers are $1,2,3$, there are 2 pairs of adjacent numbers: 1,2 and 2,3, so the value of $x$ is 2... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Two cars travel along a circular track $n$ miles long, starting at the same point. One car travels $25$ miles along the track in some direction. The other car travels $3$ miles along the track in some direction. Then the two cars are once again at the same point along the ... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Find the biggest positive integer $n$ such that $n$ is $167$ times the amount of it's positive divisors."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the largest positive integer n such that n is 167 times the number of its positive divisors.... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Given \\( \\frac{1}{3} \\leqslant a \\leqslant 1 \\), if \\( f(x)=a x^{2}-2 x+1 \\) attains its maximum value \\( M(a) \\) and minimum value \\( N(a) \\) on the interval \\([1,3]\\), and let \\( g(a)=M(a)-N(a) \\), then the minimum value of \\( g(a) \\) is \\(\\quad\\) ."
},
... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "What is the largest possible area of a quadrilateral with sidelengths $1, 4, 7$ and $8$ ?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the largest possible area of a quadrilateral with side lengths 1, 4, 7, and 8. Hmm, quadrilaterals can be tri... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Let $D(n)$ denote the number of ways of writing the positive integer $n$ as a product \\[n = f_1\\cdot f_2\\cdots f_k,\\] where $k\\ge1$ , the $f_i$ are integers strictly greater than $1$ , and the order in which the factors are listed matters (that is, two representations that ... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Given a regular tetrahedron of volume 1, obtain a second regular tetrahedron by reflecting the given one through its center. What is the volume of their intersection?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this problem here: Given a regular tetrahedr... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Given \\( x_1, x_2, \\cdots, x_{1993} \\) that satisfy\n\\[\n\\begin{array}{l}\n\\left|x_1 - x_2\\right| + \\left|x_2 - x_3\\right| + \\cdots + \\left|x_{1992} - x_{1993}\\right| = 1993, \\\\\ny_k = \\frac{x_1 + x_2 + \\cdots + x_k}{k}, \\quad (k=1, 2, \\cdots, 1993)\n\\end{arra... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "If a positive integer multiple of 864 is picked randomly, with each multiple having the same probability of being picked, what is the probability that it is divisible by 1944?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the probability that a rand... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Find the maximum value of the expression \\( x^{2} + y^{2} \\) if \\( |x-y| \\leq 2 \\) and \\( |3x + y| \\leq 6 \\)."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the maximum value of \\( x^{2} + y^{2} \\) given the constraints \\( |x - y| \\leq 2 ... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "The equation \\( x^{2} + mx + 1 + 2i = 0 \\) has real roots. Find the minimum value of the modulus of the complex number \\( m \\)."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the minimum value of the modulus of the complex number \\( m \\) such t... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Let $L,E,T,M,$ and $O$ be digits that satisfy $LEET+LMT=TOOL.$ Given that $O$ has the value of $0,$ digits may be repeated, and $L\\neq0,$ what is the value of the $4$ -digit integer $ELMO?$"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to solv... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Points \\( P \\) and \\( Q \\) are located on side \\( BC \\) of triangle \\( ABC \\), with \\( BP: PQ: QC = 1: 2: 3 \\). Point \\( R \\) divides side \\( AC \\) of this triangle such that \\( AR: RC = 1: 2 \\). What is the ratio of the area of quadrilateral \\( PQST \\) to the ... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "A pedestrian is moving in a straight line towards a crosswalk at a constant speed of 3.6 km/h. Initially, the distance from the pedestrian to the crosswalk is 40 meters. The length of the crosswalk is 6 meters. What distance from the crosswalk will the pedestrian be after two mi... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Let $AC$ and $CE$ be two diagonals of a regular hexagon $ABCDEF$. Points $M$ and $N$ divide $AC$ and $CE$ internally such that $AM:AC = CN:CE = r$. If $B$, $M$, and $N$ are collinear, find $r$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the ratio... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "There is a unique angle $\\theta$ between $0^{\\circ}$ and $90^{\\circ}$ such that for nonnegative integers $n$ , the value of $\\tan{\\left(2^{n}\\theta\\right)}$ is positive when $n$ is a multiple of $3$ , and negative otherwise. The degree measure of $\\theta$ is $\\tfrac{p}{... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Let $S_n$ be the sum of the reciprocals of the non-zero digits of the integers from $1$ to $10^n$ inclusive. Find the smallest positive integer $n$ for which $S_n$ is an integer."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the smallest positive in... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "If $20\\%$ of a number is $12$ , what is $30\\%$ of the same number?\n$\\textbf{(A)}\\ 15\\qquad\\textbf{(B)}\\ 18\\qquad\\textbf{(C)}\\ 20\\qquad\\textbf{(D)}\\ 24\\qquad\\textbf{(E)}\\ 30$"
},
{
"from": "gpt",
"value": "<think>\nFirst, let's denote the unknown numb... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Let $z_1=18+83i,~z_2=18+39i,$ and $z_3=78+99i,$ where $i=\\sqrt{-1}.$ Let $z$ be the unique complex number with the properties that $\\frac{z_3-z_1}{z_2-z_1}~\\cdot~\\frac{z-z_2}{z-z_3}$ is a real number and the imaginary part of $z$ is the greatest possible. Find the real part ... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "In \\(\\triangle ABC\\), \\(AB = 8\\), \\(BC = 13\\), and \\(CA = 15\\). Let \\(H\\), \\(I\\), and \\(O\\) be the orthocenter, incenter, and circumcenter of \\(\\triangle ABC\\) respectively. Find \\(\\sin \\angle HIO\\)."
},
{
"from": "gpt",
"value": "<think>\nOkay,... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "$K, L, M$ and $N$ are points on sides $AB, BC, CD$ and $DA$ , respectively, of the unit square $ABCD$ such that $KM$ is parallel to $BC$ and $LN$ is parallel to $AB$ . The perimeter of triangle $KLB$ is equal to $1$ . What is the area of triangle $MND$ ?"
},... | Math_Reasoning |
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