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": "A tetrahedron $ABCD$ satisfies the following conditions: the edges $AB,AC$ and $AD$ are pairwise orthogonal, $AB=3$ and $CD=\\sqrt2$ . Find the minimum possible value of $$ BC^6+BD^6-AC^6-AD^6. $$"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "A rectangular piece of paper $ABCD$ has sides of lengths $AB = 1$ , $BC = 2$ . The rectangle is folded in half such that $AD$ coincides with $BC$ and $EF$ is the folding line. Then fold the paper along a line $BM$ such that the corner $A$ falls on line $EF$ . How ... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "The rodent control task force went into the woods one day and caught $200$ rabbits and $18$ squirrels. The next day they went into the woods and caught $3$ fewer rabbits and two more squirrels than the day before. Each day they went into the woods and caught $3$ fewer ra... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Given that \\( p \\) is a prime number and \\( r \\) is the remainder when \\( p \\) is divided by 210, if \\( r \\) is a composite number that can be expressed as the sum of two perfect squares, find \\( r \\)."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Determine the minimum of the following function defined in the interval $45^{\\circ}<x<90^{\\circ}$:\n\n$$\ny=\\tan x+\\frac{\\tan x}{\\sin \\left(2 x-90^{\\circ}\\right)}\n$$"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the minimum of the function... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Given set \\( A = \\{0, 1, 2, 3, 4, 5, 9\\} \\), and \\( a, b \\in A \\) where \\( a \\neq b \\). The number of functions of the form \\( y = -a x^2 + (4 - b)x \\) whose vertex lies in the first quadrant is ___."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Jeffrey jogs $3$ miles at $4$ miles per hour. Then, he walks $4$ miles at $3$ miles an hour. What was Jeffrey's average speed in miles per hour for the entire trip? Express your answer as a decimal to the nearest hundredth."
},
{
"from": "gpt",
"value": "<thi... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Construct a square \\(A B C D\\) with side length \\(6 \\text{ cm}\\). Construct a line \\(p\\) parallel to the diagonal \\(A C\\) passing through point \\(D\\). Construct a rectangle \\(A C E F\\) such that vertices \\(E\\) and \\(F\\) lie on the line \\(p\\).\n\nUsing the give... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "The vertex of a parabola is \\( O \\) and its focus is \\( F \\). When a point \\( P \\) moves along the parabola, find the maximum value of the ratio \\( \\left|\\frac{P O}{P F}\\right| \\)."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the maximum... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Let $S$ be the set of all 3-tuples $(a, b, c)$ of positive integers such that $a + b + c = 2013$ . Find $$ \\sum_{(a,b,c)\\in S} abc. $$"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the sum of abc for all possible 3-tuples (a, b, c) of posit... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "At the moment when Pierrot left the \"Commercial\" bar, heading to the \"Theatrical\" bar, Jeannot was leaving the \"Theatrical\" bar on his way to the \"Commercial\" bar. They were walking at constant (but different) speeds. When the vagabonds met, Pierrot proudly noted that he... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Real numbers $x$ $y$ , and $z$ are chosen independently and at random from the interval $[0,n]$ for some positive integer $n$ . The probability that no two of $x$ $y$ , and $z$ are within 1 unit of each other is greater than $\\frac {1}{2}$ . What is the smallest possible value ... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Cátia leaves school every day at the same time and rides her bicycle home. When she pedals at $20 \\mathrm{~km} / \\mathrm{h}$, she arrives home at $16 \\mathrm{~h} 30 \\mathrm{~m}$. If she pedals at $10 \\mathrm{~km} / \\mathrm{h}$, she arrives home at $17 \\mathrm{~h} 15 \\mat... | 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) \\) be the polynomial \\( (x - a_1)(x - a_2)(x - a_3)(x - a_4)(x - a_5) \\) where \\( a_1, a_2, a_3, a_4, \\) and \\( a_5 \\) are distinct integers. Given that \\( f(104) = 2012 \\), evaluate \\( a_1 + a_2 + a_3 + a_4 + a_5 \\)."
},
{
"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": "Piercarlo chooses \\( n \\) integers from 1 to 1000 inclusive. None of his integers is prime, and no two of them share a factor greater than 1. What is the greatest possible value of \\( n \\)?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so Piercarlo is choosing n ... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "You are playing a game in which you have $3$ envelopes, each containing a uniformly random amount of money between $0$ and $1000$ dollars. (That is, for any real $0 \\leq a < b \\leq 1000$ , the probability that the amount of money in a given envelope is between $a$ and... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "In a cube with edge length 1, two cross-sections in the form of regular hexagons are made. Find the length of the segment at which these cross-sections intersect."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this problem about a cube with edge length 1, an... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "At night, there was a heavy snowfall. In the morning, Xiao Long and his father measured the length of a circular path in the garden by walking. They started from the same point and walked in the same direction. Xiao Long's step length is 54 cm, and his father's step length is 72... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Let \\( A \\) be a subset of \\(\\{1, 2, 3, \\ldots, 2019\\}\\) having the property that the difference between any two of its elements is not a prime number. What is the largest possible number of elements in \\( A \\)?"
},
{
"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": "Given the complex numbers \\( z_{1} \\) and \\( z_{2} \\) such that \\( \\left| z_{2} \\right| = 4 \\) and \\( 4z_{1}^{2} - 2z_{1}z_{2} + z_{2}^{2} = 0 \\), find the maximum value of \\( \\left| \\left( z_{1} + 1 \\right)^{2} \\left( z_{1} - 2 \\right) \\right| \\)."
},
{
... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Let $A_1A_2A_3\\ldots A_{12}$ be a dodecagon ( $12$ -gon). Three frogs initially sit at $A_4,A_8,$ and $A_{12}$ . At the end of each minute, simultaneously, each of the three frogs jumps to one of the two vertices adjacent to its current position, chosen randomly and independent... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Juan wrote a natural number and Maria added a digit $ 1$ to the left and a digit $ 1$ to the right. Maria's number exceeds to the number of Juan in $14789$ . Find the number of Juan."
},
{
"from": "gpt",
"value": "<think>\nOkay, let's try to solve this problem. ... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Let $f(x) = 2^x + 3^x$ . For how many integers $1 \\leq n \\leq 2020$ is $f(n)$ relatively prime to all of $f(0), f(1), \\dots, f(n-1)$ ?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out how many integers n between 1 and 2020 inclusive have... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "On a rectangular table of size \\( x \\) cm \\(\\times 80\\) cm, identical sheets of paper of size 5 cm \\(\\times 8\\) cm are placed. The first sheet is placed in the bottom left corner, and each subsequent sheet is placed one centimeter higher and one centimeter to the right o... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Find the sum of all positive integers $n$ such that $\\sqrt{n^2+85n+2017}$ is an integer."
},
{
"from": "gpt",
"value": "<think>\nAlright, let me try to solve this problem. The question is asking for the sum of all positive integers \\( n \\) such that \\( \\sqrt{n^2... | 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 $ n$ so that the polynomial $ P(X)\\equal{}\\sqrt3\\cdot X^{n\\plus{}1}\\minus{}X^n\\minus{}1$ has at least one root of modulus $ 1$ ."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the smallest positive integer n... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "In the country of Taxland, everyone pays a percentage of their salary as tax that is equal to the number of thousands of Tuzrics their salary amounts to. What salary is the most advantageous to have?\n\n(Salary is measured in positive, not necessarily whole number, Tuzrics)"
}... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Oleg drew an empty $50 \\times 50$ table and wrote a non-zero number above each column and to the left of each row. It turned out that all 100 numbers written were different, with 50 of them being rational and the remaining 50 irrational. Then, in each cell of the table, he wrot... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Justine has two fair dice, one with sides labeled $1,2,\\ldots, m$ and one with sides labeled $1,2,\\ldots, n.$ She rolls both dice once. If $\\tfrac{3}{20}$ is the probability that at least one of the numbers showing is at most 3, find the sum of all distinct possible val... | 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} = 1 + 2 + \\cdots + n \\), where \\( n \\in \\mathbf{N}_{+} \\), and \\( S_{m} = a_{1} + a_{2} + \\cdots + a_{m} \\), \\( m = 1, 2, \\cdots, m \\). Find the number of values among \\( S_{1}, S_{2}, \\cdots, S_{2017} \\) that are divisible by 2 but not by 4."
},
... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Cities $A$ $B$ $C$ $D$ , and $E$ are connected by roads $\\widetilde{AB}$ $\\widetilde{AD}$ $\\widetilde{AE}$ $\\widetilde{BC}$ $\\widetilde{BD}$ $\\widetilde{CD}$ , and $\\widetilde{DE}$ . How many different routes are there from $A$ to $B$ that use each road exactly once? (Suc... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "How many integers $n$ are there such that $0 \\le n \\le 720$ and $n^2 \\equiv 1$ (mod $720$ )?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find how many integers n satisfy 0 ≤ n ≤ 720 and n² ≡ 1 mod 720. Hmm, let's start by understanding what ... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Integers a, b, c, d, and e satisfy the following three properties: \n(i) $2 \\le a < b <c <d <e <100$ \n(ii) $ \\gcd (a,e) = 1 $ (iii) a, b, c, d, e form a geometric sequence. \nWhat is the value of c?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. So, th... | 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 point of intersection of a circle's diameter 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.\n\n$$\nAB = 18, EO = 7, AE = 2BE, R = ?\n$$"
},
{
"from": "gpt",
"valu... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Ed has five identical green marbles, and a large supply of identical red marbles. He arranges the green marbles and some of the red ones in a row and finds that the number of marbles whose right hand neighbor is the same color as themselves is equal to the number of marbles whos... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "In trapezoid \\(A B C D\\), the base \\(A D\\) is four times larger than \\(B C\\). A line passing through the midpoint of diagonal \\(B D\\) and parallel to \\(A B\\) intersects side \\(C D\\) at point \\(K\\). Find the ratio \\(D K : K C\\)."
},
{
"from": "gpt",
"v... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Given a $5 \\times 5$ grid where the number in the $i$-th row and $j$-th column is denoted by \\( a_{ij} \\) (where \\( a_{ij} \\in \\{0, 1\\} \\)), with the condition that \\( a_{ij} = a_{ji} \\) for \\( 1 \\leq i, j \\leq 5 \\). Calculate the total number of ways to fill the g... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Given real numbers \\( x \\) and \\( y \\) satisfy\n\\[\n\\left\\{\n\\begin{array}{l}\nx - y \\leq 0, \\\\\nx + y - 5 \\geq 0, \\\\\ny - 3 \\leq 0\n\\end{array}\n\\right.\n\\]\n\nIf the inequality \\( a(x^2 + y^2) \\leq (x + y)^2 \\) always holds, then the maximum value of the r... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Three vertices of parallelogram $PQRS$ are $P(-3,-2), Q(1,-5), R(9,1)$ with $P$ and $R$ diagonally opposite. \nThe sum of the coordinates of vertex $S$ is:\n$\\textbf{(A)}\\ 13 \\qquad \\textbf{(B)}\\ 12 \\qquad \\textbf{(C)}\\ 11 \\qquad \\textbf{(D)}\\ 10 \\qquad \\textbf{(E)}... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Quadrilateral $ABCD$ satisfies $\\angle ABC = \\angle ACD = 90^{\\circ}, AC=20,$ and $CD=30.$ Diagonals $\\overline{AC}$ and $\\overline{BD}$ intersect at point $E,$ and $AE=5.$ What is the area of quadrilateral $ABCD?$"
},
{
"from": "gpt",
"value": "<think>\nAlright... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Let\n\\[f(x)=\\cos(x^3-4x^2+5x-2).\\]\nIf we let $f^{(k)}$ denote the $k$ th derivative of $f$ , compute $f^{(10)}(1)$ . For the sake of this problem, note that $10!=3628800$ ."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the 10th derivative ... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "A subset \\( H \\) of the set of numbers \\(\\{1, 2, \\ldots, 100\\}\\) has the property that if an element is in \\( H \\), then ten times that element is not in \\( H \\). What is the maximum number of elements that \\( H \\) can have?"
},
{
"from": "gpt",
"value":... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "The number \\( A \\) in decimal notation has the form \\( A = \\overline{7a631b} \\), where \\( a \\) and \\( b \\) are non-zero digits. The number \\( B \\) is obtained by summing all the six-digit, distinct numbers derived from \\( A \\) by cyclic permutations of its digits (t... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "In how many ways can a committee of three people be formed if the members are to be chosen from four married couples?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so the problem is asking: In how many ways can a committee of three people be formed if the members are... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "A thousand integer divisions are made: $2018$ is divided by each of the integers from $ 1$ to $1000$ . Thus, a thousand integer quotients are obtained with their respective remainders. Which of these thousand remainders is the bigger?"
},
{
"from": "gpt",
"value... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Rolling a die twice, let the points shown the first and second times be \\(a\\) and \\(b\\) respectively. Find the probability that the quadratic equation \\(x^{2} + a x + b = 0\\) has two distinct real roots both less than -1. (Answer with a number)."
},
{
"from": "gpt"... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Chester is traveling from Hualien to Lugang, Changhua, to participate in the Hua Luogeng Golden Cup Mathematics Competition. Before setting off, his father checked the car's odometer, which read a palindromic number of 69,696 kilometers (a palindromic number remains the same whe... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Given a triangle \\(ABC\\) with an area of 1. Points \\(P\\), \\(Q\\), and \\(R\\) are taken on the medians \\(AK\\), \\(BL\\), and \\(CN\\) respectively such that \\(AP = PK\\), \\(BQ : QL = 1 : 2\\), and \\(CR : RN = 5 : 4\\). Find the area of triangle \\(PQR\\)."
},
{
... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "If \\(\\cos ^{4} \\theta + \\sin ^{4} \\theta + (\\cos \\theta \\cdot \\sin \\theta)^{4} + \\frac{1}{\\cos ^{4} \\theta + \\sin ^{4} \\theta} = \\frac{41}{16}\\), find the value of \\(\\sin ^{2} \\theta\\)."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "For a particular value of the angle $\\theta$ we can take the product of the two complex numbers $(8+i)\\sin\\theta+(7+4i)\\cos\\theta$ and $(1+8i)\\sin\\theta+(4+7i)\\cos\\theta$ to get a complex number in the form $a+bi$ where $a$ and $b$ are real numbers. Find th... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Given \\( x, y, z \\in \\mathbf{R} \\) such that \\( x^2 + y^2 + xy = 1 \\), \\( y^2 + z^2 + yz = 2 \\), \\( x^2 + z^2 + xz = 3 \\), find \\( x + y + z \\)."
},
{
"from": "gpt",
"value": "<think>\nOkay, let me try to solve this problem. So, we have three equations wi... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Given the complex numbers \\( z_{1}, z_{2}, z_{3} \\) satisfying:\n\\[\n\\begin{array}{l}\n\\left|z_{1}\\right| \\leq 1, \\left|z_{2}\\right| \\leq 2, \\\\\n\\left|2z_{3} - z_{1} - z_{2}\\right| \\leq \\left|z_{1} - z_{2}\\right|.\n\\end{array}\n\\]\nWhat is the maximum value of... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "A regular hexagon with side length $1$ is given. Using a ruler construct points in such a way that among the given and constructed points there are two such points that the distance between them is $\\sqrt7$ .\n\nNotes: ''Using a ruler construct points $\\ldots$ '' means: Ne... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "A sequence of numbers starts with $1$ $2$ , and $3$ . The fourth number of the sequence is the sum of the previous three numbers in the sequence: $1+2+3=6$ . In the same way, every number after the fourth is the sum of the previous three numbers. What is the eighth number in the... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "A residential building has a construction cost of 250 yuan per square meter. Considering a useful life of 50 years and an annual interest rate of 5%, what monthly rent per square meter is required to recoup the entire investment?"
},
{
"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": "A person flips a coin, where the probability of heads up and tails up is $\\frac{1}{2}$ each. Construct a sequence $\\left\\{a_{n}\\right\\}$ such that\n$$\na_{n}=\\left\\{\n\\begin{array}{ll}\n1, & \\text{if the } n \\text{th flip is heads;} \\\\\n-1, & \\text{if the } n \\text... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Two circles with radii $\\sqrt{5}$ and $\\sqrt{2}$ intersect at point $A$. The distance between the centers of the circles is 3. A line through point $A$ intersects the circles at points $B$ and $C$ such that $A B = A C$ (point $B$ does not coincide with $C$). Find $A B$."
},
... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Let $S$ be the sum of the interior angles of a polygon $P$ for which each interior angle is $7\\frac{1}{2}$ times the \nexterior angle at the same vertex. Then\n$\\textbf{(A)}\\ S=2660^{\\circ} \\text{ } \\text{and} \\text{ } P \\text{ } \\text{may be regular}\\qquad \\\\ \\text... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "In a right triangle \\(ABC\\), the legs \\(AB\\) and \\(AC\\) measure 4 and 3 respectively. Point \\(D\\) bisects the hypotenuse \\(BC\\). Find the distance between the centers of the incircles of triangles \\(ADC\\) and \\(ABD\\)."
},
{
"from": "gpt",
"value": "<thi... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Let $ABC$ be an equilateral triangle. $A $ point $P$ is chosen at random within this triangle. What is the probability that the sum of the distances from point $P$ to the sides of triangle $ABC$ are measures of the sides of a triangle?"
},
{
"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 mouse is sitting in a toy car on a negligibly small turntable. The car cannot turn on its own, but the mouse can control when the car is launched and when the car stops (the car has brakes). When the mouse chooses to launch, the car will immediately leave the turntable on a st... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "A packet of seeds was passed around the table. The first person took 1 seed, the second took 2 seeds, the third took 3 seeds, and so on: each subsequent person took one more seed than the previous one. It is known that on the second round, a total of 100 more seeds were taken th... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "The UEFA Champions League playoffs is a 16-team soccer tournament in which Spanish teams always win against non-Spanish teams. In each of 4 rounds, each remaining team is randomly paired against one other team; the winner advances to the next round, and the loser is permanently ... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Given the three interior angles \\( A, B, C \\) of \\(\\triangle ABC\\) satisfy \\( A = 3B = 9C \\), find the value of \\( \\cos A \\cos B + \\cos B \\cos C + \\cos C \\cos A \\)."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this problem here: In triangle ... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "The function \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) is continuous. For every real number \\( x \\), the equation \\( f(x) \\cdot f(f(x)) = 1 \\) holds. It is known that \\( f(1000) = 999 \\). Find \\( f(500) \\)."
},
{
"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": "4.\tFind the biggest positive integer $n$ , lesser thar $2012$ , that has the following property:\nIf $p$ is a prime divisor of $n$ , then $p^2 - 1$ is a divisor of $n$ ."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the largest positive int... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Calculate: $1.23 \\times 67 + 8.2 \\times 12.3 - 90 \\times 0.123$"
},
{
"from": "gpt",
"value": "<think>\nOkay, let me try to solve this problem: 1.23 times 67 plus 8.2 times 12.3 minus 90 times 0.123. Hmm, that seems like a mix of multiplication and addition/subtra... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Four students from Harvard, one of them named Jack, and five students from MIT, one of them named Jill, are going to see a Boston Celtics game. However, they found out that only 5 tickets remain, so 4 of them must go back. Suppose that at least one student from each school must ... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Find the smallest value of $n$ for which the series \\[1\\cdot 3^1 + 2\\cdot 3^2 + 3\\cdot 3^3 + \\cdots + n\\cdot 3^n\\] exceeds $3^{2007}$ ."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the smallest value of n such that the series 1·3¹ + 2·3² ... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "A rectangular table of size \\( x \\) cm \\( \\times 80 \\) cm is covered with identical sheets of paper of size 5 cm \\( \\times 8 \\) cm. The first sheet is placed in the bottom-left corner, and each subsequent sheet is placed 1 cm higher and 1 cm to the right of the previous ... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "$x$ is a base- $10$ number such that when the digits of $x$ are interpreted as a base- $20$ number, the resulting number is twice the value as when they are interpreted as a base- $13$ number. Find the sum of all possible values of $x$ ."
},
{
"from": "gpt",
"... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "How many positive integers less that $200$ are relatively prime to either $15$ or $24$ ?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out how many positive integers less than 200 are relatively prime to either 15 or 24. Hmm, let's start by r... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Compute the prime factorization of $1007021035035021007001$ . (You should write your answer in the form $p_1^{e_1}p_2^{e_2}\\ldots p_k^{e_k}$ where $p_1,\\ldots,p_k$ are distinct prime numbers and $e_1,\\ldots,e_k$ are positive integers.)"
},
{
"from": "gpt",
... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "This century will mark the 200th anniversary of the birth of the famous Russian mathematician Pafnuty Lvovich Chebyshev, a native of Kaluga province. The sum of the digits in the hundreds and thousands places of the year he was born is 3 times the sum of the digits in the units ... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "A square in the coordinate plane has vertices whose $y$ -coordinates are $0$ $1$ $4$ , and $5$ . What is the area of the square?"
},
{
"from": "gpt",
"value": "<think>\nFirst, I note that the square has vertices with y-coordinates 0, 1, 4, and 5. Since these y-coordi... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Each number in the list $1,2,3,\\ldots,10$ is either colored red or blue. Numbers are colored independently, and both colors are equally probable. The expected value of the number of positive integers expressible as a sum of a red integer and a blue integer can be written as ... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "On a \\(6 \\times 6\\) chessboard, we randomly place counters on three different squares. What is the probability that no two counters are in the same row or column?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the probability that when I place thr... | 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 minimum number of sides of a regular polygon that approximates the area of its circumscribed circle with an error of less than 1 per thousand (0.1%)?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the minimum number of sides a regular pol... | 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 smallest positive odd number that has the same number of divisors as 360?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the smallest positive odd number that has the same number of divisors as 360. Hmm, let me start by figuring out how m... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "The largest divisor of a natural number \\( N \\), smaller than \\( N \\), was added to \\( N \\), producing a power of ten. Find all such \\( N \\)."
},
{
"from": "gpt",
"value": "<think>\nOkay, let me try to figure out this problem. So, we have a natural number N, ... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Define $\\lfloor x \\rfloor$ as the largest integer less than or equal to $x$ . Define $\\{x \\} = x - \\lfloor x \\rfloor$ . For example, $\\{ 3 \\} = 3-3 = 0$ , $\\{ \\pi \\} = \\pi - 3$ , and $\\{ - \\pi \\} = 4-\\pi$ . If $\\{n\\} + \\{ 3n\\} = 1.4$ , then find the s... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "In the isosceles triangle \\(ABC\\) with the sides \\(AB = BC\\), the angle \\(\\angle ABC\\) is \\(80^\\circ\\). Inside the triangle, a point \\(O\\) is taken such that \\(\\angle OAC = 10^\\circ\\) and \\(\\angle OCA = 30^\\circ\\). Find the angle \\(\\angle AOB\\)."
},
{
... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Let $R$ be a unit square region and $n \\geq 4$ an integer. A point $X$ in the interior of $R$ is called n-ray partitional if there are $n$ rays emanating from $X$ that divide $R$ into $n$ triangles of equal area. How many points are $100$ -ray partitional but not $60$ -ray part... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Determine the largest natural number $r$ with the property that among any five subsets with $500$ elements of the set $\\{1,2,\\ldots,1000\\}$ there exist two of them which share at least $r$ elements."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need t... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "$AB$ and $AC$ are tangents to a circle, $\\angle BAC = 60^{\\circ}$, the length of the broken line $BAC$ is 1. Find the distance between the points of tangency $B$ and $C$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this geometry problem here: AB and AC ... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Ottó decided to assign a number to each pair \\((x, y)\\) and denote it as \\((x \\circ y)\\). He wants the following relationships to hold:\na) \\(x \\circ y = y \\circ x\\)\nb) \\((x \\circ y) \\circ z = (x \\circ z) \\circ (y \\circ z)\\)\nc) \\((x \\circ y) + z = (x + z) \\c... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Nikita usually leaves home at 8:00 AM, gets into Uncle Vanya's car, and his uncle drives him to school at a certain time. But on Friday, Nikita left home at 7:10 AM and ran in the opposite direction. Uncle Vanya waited for him and at 8:10 AM drove after him, caught up with Nikit... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Find how many positive integers $k\\in\\{1,2,\\ldots,2022\\}$ have the following property: if $2022$ real numbers are written on a circle so that the sum of any $k$ consecutive numbers is equal to $2022$ then all of the $2022$ numbers are equal."
},
{
"from": "... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "David found four sticks of different lengths that can be used to form three non-congruent convex cyclic quadrilaterals, $A,\\text{ }B,\\text{ }C$ , which can each be inscribed in a circle with radius $1$ . Let $\\varphi_A$ denote the measure of the acute angle made by the diagon... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "A building contractor needs to pay his $108$ workers $\\$ 200 $ each. He is carrying $ 122 $ one hundred dollar bills and $ 188 $ fifty dollar bills. Only $ 45 $ workers get paid with two $ \\ $100$ bills. Find the number of workers who get paid with four $\\$ 50$ bills."
... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Given the quadratic function \\( f(x) = a x^{2} + b x + c \\) where \\( a, b, c \\in \\mathbf{R}_{+} \\), if the function has real roots, determine the maximum value of \\( \\min \\left\\{\\frac{b+c}{a}, \\frac{c+a}{b}, \\frac{a+b}{c}\\right\\} \\)."
},
{
"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 are 8 cards; one side of each card is blank, and the other side has the letters И, Я, Л, З, Г, О, О, О written on them. The cards are placed on the table with the blank side up, shuffled, and then turned over one by one in sequence. What is the probability that the letters... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Construct a square such that two adjacent vertices lie on a circle with a unit radius, and the side connecting the other two vertices is tangent to the circle. Calculate the sides of the square!"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. I need to const... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Let \\( D \\) be a point inside \\( \\triangle ABC \\) such that \\( \\angle BAD = \\angle BCD \\) and \\( \\angle BDC = 90^\\circ \\). If \\( AB = 5 \\), \\( BC = 6 \\), and \\( M \\) is the midpoint of \\( AC \\), find the length of \\( DM \\)."
},
{
"from": "gpt",
... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Evgeny is laying tiles on the floor of his living room, which measures 12 by 16 meters. He plans to place square tiles of size 1 m × 1 m along the border of the room and to lay the rest of the floor with square tiles of size 2 m × 2 m. How many tiles will he need in total?"
},... | 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. If the angles \\( A \\), \\( B \\), and \\( C \\) form a geometric progression, and \\( b^{2} - a^{2} = ac \\), then the rad... | 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 \\), \\(E\\) and \\(F\\) are the midpoints of \\(AC\\) and \\(AB\\) respectively, and \\( AB = \\frac{2}{3} AC \\). If \\( \\frac{BE}{CF} < t \\) always holds, then the minimum value of \\( t \\) is ______."
},
{
"from": "gpt",
"value":... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "$\\textbf{Problem 5.}$ Miguel has two clocks, one clock advances $1$ minute per day and the other one goes $15/10$ minutes per day. \nIf you put them at the same correct time, What is the least number of days that must pass for both to give the correct time simultaneously?"... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "A deck consists of 32 cards divided into 4 suits, each containing 8 cards. In how many ways can we choose 6 cards such that all four suits are represented among them?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out how many ways there are to cho... | Math_Reasoning |
You are a careful math assistant that reasons step-by-step and reports a final boxed answer. | [
{
"from": "human",
"value": "Calculate the lengths of the arcs of the curves given by the equations in the rectangular coordinate system.\n\n\\[ y = \\ln \\frac{5}{2 x}, \\quad \\sqrt{3} \\leq x \\leq \\sqrt{8} \\]"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the length of the... | Math_Reasoning |
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