data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "As shown in the figure below, six semicircles lie in the interior of a regular hexagon with side length 2 so that the diameters of the semicircles coincide with the sides of the hexagon. What is the area of the shaded region ---- inside the hexagon but outside all of the semici... | math | {
"style": "rule",
"ground_truth": "6\\sqrt{3} - 3\\pi"
} | {
"split": "train",
"index": 6800,
"solution": "1. **Identify the Geometry and Dimensions:**\n - The problem involves a regular hexagon with side length 2.\n - Inside the hexagon, there are six semicircles, each with a diameter equal to the side of the hexagon. Thus, each semicircle has a radius of 1.\n\n2. *... |
deepscaler | [
{
"role": "user",
"content": "In 60 chandeliers (each with 4 shades), the shades need to be replaced. Each electrician takes 5 minutes to replace one shade. A total of 48 electricians will be working. No more than one shade can be replaced in a chandelier at the same time. What is the minimum time required ... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 6801,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve for $Q$ if $\\sqrt{Q^3} = 16\\sqrt[8]{16}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 6802,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Oleg drew an empty 50×50 table and wrote a number above each column and to the left of each row. It turned out that all 100 written numbers are different, 50 of which are rational and the remaining 50 are irrational. Then, in each cell of the table, he recorded the product of t... | math | {
"style": "rule",
"ground_truth": "1250"
} | {
"split": "train",
"index": 6803,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The following image is 1024 pixels by 1024 pixels, and each pixel is either black or white. The border defines the boundaries of the image, but is not part of the image. Let $a$ be the proportion of pixels that are black. Estimate $A=\\lfloor 10000 a\\rfloor$. An estimate of $E... | math | {
"style": "rule",
"ground_truth": "3633"
} | {
"split": "train",
"index": 6804,
"solution": "This is an area estimation problem. A good place to start is to focus on the jacket. The hair adds about as much area as the hand takes away; the jacket seems to occupy about $\\frac{2}{3}$ of the width of the square and $\\frac{1}{2}$ of the height. A crude estimat... |
deepscaler | [
{
"role": "user",
"content": "In triangle $A B C$ with $A B=8$ and $A C=10$, the incenter $I$ is reflected across side $A B$ to point $X$ and across side $A C$ to point $Y$. Given that segment $X Y$ bisects $A I$, compute $B C^{2}$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "84"
} | {
"split": "train",
"index": 6805,
"solution": "Let $E, F$ be the tangency points of the incircle to sides $A C, A B$, respectively. Due to symmetry around line $A I, A X I Y$ is a rhombus. Therefore $$\\angle X A I=2 \\angle E A I=2\\left(90^{\\circ}-\\angle E I A\\right)=180^{\\circ}-2 \\angle X A I$$ which imp... |
deepscaler | [
{
"role": "user",
"content": "Define $g$ by $g(x)=5x-4$. If $g(x)=f^{-1}(x)-3$ and $f^{-1}(x)$ is the inverse of the function $f(x)=ax+b$, find $5a+5b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 6806,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Denis has cards with numbers from 1 to 50. How many ways are there to choose two cards such that the difference of the numbers on the cards is 11, and their product is divisible by 5?\n\nThe order of the selected cards does not matter: for example, selecting cards with numbers ... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 6807,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sequence of positive integers \\(a_{1}, a_{2}, \\ldots\\) is such that for each \\(m\\) and \\(n\\) the following holds: if \\(m\\) is a divisor of \\(n\\) and \\(m < n\\), then \\(a_{m}\\) is a divisor of \\(a_{n}\\) and \\(a_{m} < a_{n}\\). Find the least possible value of ... | math | {
"style": "rule",
"ground_truth": "128"
} | {
"split": "train",
"index": 6808,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Fold a piece of graph paper once so that the point (0, 2) coincides with the point (4, 0), and the point (9, 5) coincides with the point (m, n). The value of m+n is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 6809,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the red light lasts for $45$ seconds, determine the probability that a pedestrian will have to wait at least $20$ seconds before the light turns green. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{5}{9}"
} | {
"split": "train",
"index": 6810,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five fair ten-sided dice are rolled. Calculate the probability that at least four of the five dice show the same value. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{23}{5000}"
} | {
"split": "train",
"index": 6811,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the set of all nonzero real numbers. Let $f : S \\to S$ be a function such that\n\\[f(x) + f(y) = f(xyf(x + y))\\]for all $x,$ $y \\in S$ such that $x + y \\neq 0.$\n\nLet $n$ be the number of possible values of $f(4),$ and let $s$ be the sum of all possible values... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 6812,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The quadratic equation $ax^2+20x+c=0$ has exactly one solution. If $a+c=29$, and $a<c$ find the ordered pair $(a,c)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(4,25)"
} | {
"split": "train",
"index": 6813,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "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 | {
"style": "rule",
"ground_truth": "89"
} | {
"split": "train",
"index": 6814,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the discrete random variable \\\\(\\xi\\\\) follows a normal distribution \\\\(N \\sim (2,1)\\\\), and \\\\(P(\\xi < 3) = 0.968\\\\), then \\\\(P(1 < \\xi < 3) =\\\\) \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.936"
} | {
"split": "train",
"index": 6815,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given 6 parking spaces in a row and 3 cars that need to be parked such that no two cars are next to each other, calculate the number of different parking methods. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 6816,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the complex number \\( z \\) satisfy \\( |z|=1 \\). Given that the equation \\( zx^2 + 2\\bar{z}x + 2 = 0 \\) in terms of \\( x \\) has a real root, find the sum of all such complex numbers \\( z \\). Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{2}"
} | {
"split": "train",
"index": 6817,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nPoints \\( C_1 \\), \\( A_1 \\), and \\( B_1 \\) are taken on the sides \\( AB \\), \\( BC \\), and \\( AC \\) of triangle \\( ABC \\) respectively, such that\n\n\\[\n\\frac{AC_1}{C_1B} = \\frac{BA_1}{A_1C} = \\frac{CB_1}{B_1A} = 2.\n\\]\n\nFind the area of triangle \\( A_1B_... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 6818,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence $a_1, a_2, \\ldots$ is geometric with $a_1=a$ and common ratio $r,$ where $a$ and $r$ are positive integers. Given that $\\log_8 a_1+\\log_8 a_2+\\cdots+\\log_8 a_{12} = 2006,$ find the number of possible ordered pairs $(a,r).$\n Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "46"
} | {
"split": "train",
"index": 6819,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is an unlimited supply of congruent equilateral triangles made of colored paper. Each triangle is a solid color with the same color on both sides of the paper. A large equilateral triangle is constructed from four of these paper triangles. Two large triangles are consider... | math | {
"style": "rule",
"ground_truth": "336"
} | {
"split": "train",
"index": 6820,
"solution": "If two of our big equilateral triangles have the same color for their center triangle and the same multiset of colors for their outer three triangles, we can carry one onto the other by a combination of rotation and reflection. Thus, to make two triangles distinct, ... |
deepscaler | [
{
"role": "user",
"content": "A solid right prism $ABCDEF$ has a height of 20, as illustrated. The bases are equilateral triangles with a side length of 14. Points $X$, $Y$, and $Z$ are the midpoints of edges $AC$, $BC$, and $DC$, respectively. Determine the perimeter of triangle $XYZ$. Let's think step by ... | math | {
"style": "rule",
"ground_truth": "7 + 2\\sqrt{149}"
} | {
"split": "train",
"index": 6821,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Quadratic polynomials $P(x)$ and $Q(x)$ have leading coefficients $2$ and $-2,$ respectively. The graphs of both polynomials pass through the two points $(16,54)$ and $(20,53).$ Find $P(0) + Q(0).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "116"
} | {
"split": "train",
"index": 6822,
"solution": "Let $R(x)=P(x)+Q(x).$ Since the $x^2$-terms of $P(x)$ and $Q(x)$ cancel, we conclude that $R(x)$ is a linear polynomial.\n Note that \\begin{alignat*}{8} R(16) &= P(16)+Q(16) &&= 54+54 &&= 108, \\\\ R(20) &= P(20)+Q(20) &&= 53+53 &&= 106, \\end{alignat*} so the slop... |
deepscaler | [
{
"role": "user",
"content": "The axial section of a cone is an equilateral triangle with a side length of 1. Find the radius of the sphere that is tangent to the axis of the cone, its base, and its lateral surface. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3} - 1}{4}"
} | {
"split": "train",
"index": 6823,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The midpoint of a line segment is located at $(1, -2)$. If one of the endpoints is $(4, 5)$, what is the other endpoint? Express your answer as an ordered pair. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(-2,-9)"
} | {
"split": "train",
"index": 6824,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the average daily high temperature in Addington from September 15th, 2008, through September 21st, 2008, inclusive. The high temperatures for the days are as follows: 51, 64, 60, 59, 48, 55, 57 degrees Fahrenheit. Express your answer as a decimal to the nearest tenth.... | math | {
"style": "rule",
"ground_truth": "56.3"
} | {
"split": "train",
"index": 6825,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, we have $AB=1$ and $AC=2$. Side $\\overline{BC}$ and the median from $A$ to $\\overline{BC}$ have the same length. What is $BC$? Express your answer in simplest radical form. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 6826,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A man buys a house for $10,000 and rents it. He puts $12\\frac{1}{2}\\%$ of each month's rent aside for repairs and upkeep; pays $325 a year taxes and realizes $5\\frac{1}{2}\\%$ on his investment. The monthly rent (in dollars) is: Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "83.33"
} | {
"split": "train",
"index": 6827,
"solution": "1. **Understanding the problem**: A man buys a house for $10,000 and aims to realize a $5\\frac{1}{2}\\%$ return on his investment. This means he wants to earn $5.5\\%$ of $10,000$ annually:\n \\[\n 0.055 \\times 10,000 = 550 \\text{ dollars per year}\n \\]\n\... |
deepscaler | [
{
"role": "user",
"content": "How many ten-digit numbers exist in which there are at least two identical digits? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8996734080"
} | {
"split": "train",
"index": 6828,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $$f(x)= \\begin{cases} f(x+1), x < 4 \\\\ ( \\frac {1}{2})^{x}, x\\geq4\\end{cases}$$, find $f(\\log_{2}3)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {1}{24}"
} | {
"split": "train",
"index": 6829,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Add $956_{12} + 273_{12}$. Express your answer in base $12$, using $A$ for $10$ and $B$ for $11$ if necessary. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1009_{12}"
} | {
"split": "train",
"index": 6830,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the equation about $x$, $(x-2)(x^2-4x+m)=0$ has three real roots. \n(1) Find the range of values for $m$. \n(2) If these three real roots can exactly be the lengths of the sides of a triangle, find the range of values for $m$. \n(3) If the triangle formed by these thre... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 6831,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow {a}$=(2,6) and $\\overrightarrow {b}$=(m,-1), find the value of m when $\\overrightarrow {a}$ is perpendicular to $\\overrightarrow {b}$ and when $\\overrightarrow {a}$ is parallel to $\\overrightarrow {b}$. Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{3}"
} | {
"split": "train",
"index": 6832,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $x^{3}-a x^{2}+b x-48$ is a polynomial with three positive roots $p, q$, and $r$ such that $p<q<r$. What is the minimum possible value of $1 / p+2 / q+3 / r$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3 / 2"
} | {
"split": "train",
"index": 6833,
"solution": "We know $p q r=48$ since the product of the roots of a cubic is the constant term. Now, $$ \\frac{1}{p}+\\frac{2}{q}+\\frac{3}{r} \\geq 3 \\sqrt[3]{\\frac{6}{p q r}}=\\frac{3}{2} $$ by AM-GM, with equality when $1 / p=2 / q=3 / r$. This occurs when $p=2, q=4$, $r=6$... |
deepscaler | [
{
"role": "user",
"content": "Determine all \\(\\alpha \\in \\mathbb{R}\\) such that for every continuous function \\(f:[0,1] \\rightarrow \\mathbb{R}\\), differentiable on \\((0,1)\\), with \\(f(0)=0\\) and \\(f(1)=1\\), there exists some \\(\\xi \\in(0,1)\\) such that \\(f(\\xi)+\\alpha=f^{\\prime}(\\xi)\... | math | {
"style": "rule",
"ground_truth": "\\alpha = \\frac{1}{e-1}"
} | {
"split": "train",
"index": 6834,
"solution": "First consider the function \\(h(x)=\\frac{e^{x}-1}{e-1}\\), which has the property that \\(h^{\\prime}(x)=\\frac{e^{x}}{e-1}\\). Note that \\(h \\in V\\) and that \\(h^{\\prime}(x)-h(x)=1 /(e-1)\\) is constant. As such, \\(\\alpha=1 /(e-1)\\) is the only possible v... |
deepscaler | [
{
"role": "user",
"content": "$A B C D$ is a rectangle with $A B=20$ and $B C=3$. A circle with radius 5, centered at the midpoint of $D C$, meets the rectangle at four points: $W, X, Y$, and $Z$. Find the area of quadrilateral $W X Y Z$. Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 6835,
"solution": "Suppose that $X$ and $Y$ are located on $A B$ with $X$ closer to $A$ than $B$. Let $O$ be the center of the circle, and let $P$ be the midpoint of $A B$. We have $O P \\perp A B$ so $O P X$ and $O P Y$ are right triangles with right angles at $P$. Because $O X=O Y=5... |
deepscaler | [
{
"role": "user",
"content": "A regular octagon has a side length of 8 cm. What is the number of square centimeters in the area of the shaded region formed by diagonals connecting alternate vertices (forming a square in the center)? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "192 + 128\\sqrt{2}"
} | {
"split": "train",
"index": 6836,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be the product of the first 10 primes, and let $$S=\\sum_{x y \\mid n} \\varphi(x) \\cdot y$$ where $\\varphi(x)$ denotes the number of positive integers less than or equal to $x$ that are relatively prime to $x$, and the sum is taken over ordered pairs $(x, y)$ of posi... | math | {
"style": "rule",
"ground_truth": "1024"
} | {
"split": "train",
"index": 6837,
"solution": "Solution 1: We see that, for any positive integer $n$, $$S=\\sum_{x y \\mid n} \\varphi(x) \\cdot y=\\sum_{x \\mid n} \\varphi(x)\\left(\\sum_{y \\left\\lvert\\, \\frac{n}{x}\\right.} y\\right)=\\sum_{x \\mid n} \\varphi(x) \\sigma\\left(\\frac{n}{x}\\right)$$ Since... |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse $C: \\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 (a > b > 0)$ with its left and right foci being $F_1$ and $F_2$ respectively, and passing through the point $P(0, \\sqrt{5})$, with an eccentricity of $\\frac{2}{3}$, and $A$ being a moving point on the line $x=4$.\n... | math | {
"style": "rule",
"ground_truth": "\\sqrt{21}"
} | {
"split": "train",
"index": 6838,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nCalculate the volume of a tetrahedron with vertices at points $A_{1}, A_{2}, A_{3}, A_{4}$, and its height dropped from vertex $A_{4}$ to the face $A_{1} A_{2} A_{3}$.\n\n$A_{1}(-2, 0, -4)$\n\n$A_{2}(-1, 7, 1)$\n\n$A_{3}(4, -8, -4)$\n\n$A_{4}(1, -4, 6)$ Let's think step by st... | math | {
"style": "rule",
"ground_truth": "5\\sqrt{2}"
} | {
"split": "train",
"index": 6839,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Nicholas is counting the sheep in a flock as they cross a road. The sheep begin to cross the road at 2:00 p.m. and cross at a constant rate of three sheep per minute. After counting 42 sheep, Nicholas falls asleep. He wakes up an hour and a half later, at which point exactly ha... | math | {
"style": "rule",
"ground_truth": "624"
} | {
"split": "train",
"index": 6840,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "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 $\\displaystyle \\sum_{k=1}^{28}a_k$ is divided by 1000. Let's ... | math | {
"style": "rule",
"ground_truth": "834"
} | {
"split": "train",
"index": 6841,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In Zuminglish-Advanced, all words still consist only of the letters $M, O,$ and $P$; however, there is a new rule that any occurrence of $M$ must be immediately followed by $P$ before any $O$ can occur again. Also, between any two $O's$, there must appear at least two consonant... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 6842,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the largest of all integers $n$ with the property that $n$ is divisible by all positive integers that are less than $\\sqrt[3]{n}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "420"
} | {
"split": "train",
"index": 6843,
"solution": "Observation from that $\\operatorname{lcm}(2,3,4,5,6,7)=420$ is divisible by every integer less than or equal to $7=[\\sqrt[3]{420}]$ and that $\\operatorname{lcm}(2,3,4,5,6,7,8)=840$ is not divisible by $9=[\\sqrt[3]{840}]$. One may guess 420 is the required intege... |
deepscaler | [
{
"role": "user",
"content": "Let $S_{n}$ and $T_{n}$ represent the sum of the first $n$ terms of the arithmetic sequences ${a_{n}}$ and ${b_{n}}$, respectively. Given that $\\frac{S_{n}}{T_{n}} = \\frac{2n+1}{4n-2}$ for all positive integers $n$, find the value of $\\frac{a_{10}}{b_{3}+b_{18}} + \\frac{a_{... | math | {
"style": "rule",
"ground_truth": "\\frac{41}{78}"
} | {
"split": "train",
"index": 6844,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A caterpillar is climbing a 20-meter pole. During the day, it climbs 5 meters, and during the night, it slides down 4 meters. How long will it take for the caterpillar to reach the top of the pole? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 6845,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "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)--hexcoo... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 6846,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When $555_{10}$ is expressed in this base, it has 4 digits, in the form ABAB, where A and B are two different digits. What base is it? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 6847,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, $\\angle PQR = 90^\\circ$. What is the value of $x$?\n\n[asy]\nsize(100);\ndraw((0,1)--(0,0)--(1,0));\ndraw((0,0)--(.9,.47));\ndraw((0,.1)--(.1,.1)--(.1,0));\nlabel(\"$P$\",(0,1),N); label(\"$Q$\",(0,0),SW); label(\"$R$\",(1,0),E); label(\"$S$\",(.9,.47),NE);\n... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 6848,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( \\triangle ABC \\) be an acute triangle, with \\( M \\) being the midpoint of \\( \\overline{BC} \\), such that \\( AM = BC \\). Let \\( D \\) and \\( E \\) be the intersection of the internal angle bisectors of \\( \\angle AMB \\) and \\( \\angle AMC \\) with \\( AB \\... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{9}"
} | {
"split": "train",
"index": 6849,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Julia is learning how to write the letter C. She has 6 differently-colored crayons, and wants to write Cc Cc Cc Cc Cc. In how many ways can she write the ten Cs, in such a way that each upper case C is a different color, each lower case C is a different color, and in each pair ... | math | {
"style": "rule",
"ground_truth": "222480"
} | {
"split": "train",
"index": 6850,
"solution": "Suppose Julia writes Cc a sixth time, coloring the upper-case C with the unique color different from that of the first five upper-case Cs, and doing the same with the lower-case C (note: we allow the sixth upper-case C and lower-case c to be the same color). Note th... |
deepscaler | [
{
"role": "user",
"content": "A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 7 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{10}"
} | {
"split": "train",
"index": 6851,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a geometric sequence $\\{a_n\\}$ where the sum of the first $n$ terms $S_n = a \\cdot 3^n - 2$, find the value of $a_2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 6852,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In pentagon $MATHS$, $\\angle M \\cong \\angle T \\cong \\angle H$ and $\\angle A$ is supplementary to $\\angle S$. How many degrees are in the measure of $\\angle H$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "120^\\circ"
} | {
"split": "train",
"index": 6853,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the $100\\text{th}$ number in the arithmetic sequence: $1,5,9,13,17,21,25,...$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "397"
} | {
"split": "train",
"index": 6854,
"solution": "1. **Identify the first term and common difference**: \n The given arithmetic sequence is $1, 5, 9, 13, 17, 21, 25, \\ldots$. The first term ($a_1$) is $1$. The common difference ($d$) can be calculated as the difference between any two consecutive terms:\n \\[\... |
deepscaler | [
{
"role": "user",
"content": "Given a circle circumscribed around triangle \\(FDC\\), a tangent \\(FK\\) is drawn such that \\(\\angle KFC = 58^\\circ\\). Points \\(K\\) and \\(D\\) lie on opposite sides of line \\(FC\\) as shown in the diagram. Find the acute angle between the angle bisectors of \\(\\angle... | math | {
"style": "rule",
"ground_truth": "61"
} | {
"split": "train",
"index": 6855,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the function $f(x)=\\sin (ωx+φ)(ω > 0,0 < φ < π)$ has a distance of $\\frac {π}{2}$ between adjacent symmetry axes, and the function $y=f(x+ \\frac {π}{2})$ is an even function.\n1. Find the analytical expression of $f(x)$.\n2. If $α$ is an acute angle, and $f(\\frac... | math | {
"style": "rule",
"ground_truth": "\\frac {24+7 \\sqrt {3}}{50}"
} | {
"split": "train",
"index": 6856,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A = (-1,1,2),$ $B = (1,2,3),$ and $C = (t,1,1),$ where $t$ is a real number. Find the smallest possible area of triangle $ABC.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 6857,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$, respectively, and $C= \\frac{3}{4}\\pi$, $\\sin A= \\frac{\\sqrt{5}}{5}$.\n(I) Find the value of $\\sin B$;\n(II) If $c-a=5-\\sqrt{10}$, find the area of $\\triangle ABC$. Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}"
} | {
"split": "train",
"index": 6858,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the function be $$f(x)=1-2\\sin^{2}x-\\cos(2x+ \\frac {\\pi}{3})$$ \n(1) Find the smallest positive period of the function $f(x)$. \n(2) For triangle ABC, the sides $a$, $b$, $c$ are opposite to the angles $A$, $B$, $C$, respectively. Given $b=5$, and $$f\\left( \\frac {B... | math | {
"style": "rule",
"ground_truth": "\\frac {25 \\sqrt {3}}{4}"
} | {
"split": "train",
"index": 6859,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence: 1, 3, 6, 10, 15, ..., find the 6th term. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 6860,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the internal angles of $\\triangle ABC$ satisfy $\\sin A + 2\\sin B = 3\\sin C$, then the minimum value of $\\cos C$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2 \\sqrt{10} - 2}{9}"
} | {
"split": "train",
"index": 6861,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest integer less than or equal to $\\frac{4^{50}+3^{50}}{4^{47}+3^{47}}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 6862,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is $(a^3+b^3)\\div(a^2-ab+b^2)$ when $a=5$ and $b=4$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 6863,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve for $n$, if $8^n\\cdot8^n\\cdot8^n=64^3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 6864,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bridge needs to be constructed over a river with a required elevation of 800 feet from one side to the other. Determine the additional bridge length required if the gradient is reduced from 2% to 1.5%. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "13333.33"
} | {
"split": "train",
"index": 6865,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x = (\\log_82)^{(\\log_28)}$, then $\\log_3x$ equals: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 6866,
"solution": "1. **Evaluate $x$:**\n Given $x = (\\log_8 2)^{(\\log_2 8)}$. We start by simplifying each logarithm:\n - $\\log_8 2$ can be rewritten using the change of base formula:\n \\[\n \\log_8 2 = \\frac{\\log_2 2}{\\log_2 8} = \\frac{1}{3}\n \\]\n since... |
deepscaler | [
{
"role": "user",
"content": "Let \\( M = \\{1, 2, \\ldots, 20\\} \\) and \\( A_1, A_2, \\ldots, A_n \\) be distinct non-empty subsets of \\( M \\). When \\( i \\neq j \\), the intersection of \\( A_i \\) and \\( A_j \\) has at most two elements. Find the maximum value of \\( n \\). Let's think step by step... | math | {
"style": "rule",
"ground_truth": "1350"
} | {
"split": "train",
"index": 6867,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On an algebra quiz, $10\\%$ of the students scored $70$ points, $35\\%$ scored $80$ points, $30\\%$ scored $90$ points, and the rest scored $100$ points. What is the difference between the mean and median score of the students' scores on this quiz? Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 6868,
"solution": "1. **Calculate the percentage of students scoring 100 points**: \n Given that $10\\%$ scored $70$ points, $35\\%$ scored $80$ points, and $30\\%$ scored $90$ points, the percentage of students scoring $100$ points is:\n \\[\n 100\\% - (10\\% + 35\\% + 30\\%) =... |
deepscaler | [
{
"role": "user",
"content": "The area of the region in the $xy$ -plane satisfying the inequality \\[\\min_{1 \\le n \\le 10} \\max\\left(\\frac{x^2+y^2}{4n^2}, \\, 2 - \\frac{x^2+y^2}{4n^2-4n+1}\\right) \\le 1\\] is $k\\pi$ , for some integer $k$ . Find $k$ .\n\n*Proposed by Michael Tang* Let's think s... | math | {
"style": "rule",
"ground_truth": "210"
} | {
"split": "train",
"index": 6869,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the polar coordinate equation of curve C is $\\rho = \\sqrt{3}$, and the parametric equations of line l are $x = 1 + \\frac{\\sqrt{2}}{2}t$, $y = \\frac{\\sqrt{2}}{2}t$ (where t is the parameter), and the parametric equations of curve M are $x = \\cos \\theta$, $y = ... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{5}}{2}"
} | {
"split": "train",
"index": 6870,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$2.46 \\times 8.163 \\times (5.17 + 4.829)$ is closest to Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 6871,
"solution": "To solve the problem, we will approximate the values and calculate the product.\n\n1. **Approximate the values:**\n - $2.46 \\approx 2.5$\n - $8.163 \\approx 8$\n - $5.17 + 4.829 = 10$ (This is exact since $5.17 + 4.829$ is designed to sum to $10$).\n\n2. **Ca... |
deepscaler | [
{
"role": "user",
"content": "Engineers use the formula $L=\\frac{25T^4}{H^2}$ to find the crushing load for square pillars. If $T = 4$ and $H = 8$, what is the value of $L$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 6872,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A geometric sequence of positive integers is formed for which the first term is 3 and the fourth term is 192. What is the third term of the sequence? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 6873,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the set of positive real numbers. Let $f : S \\to \\mathbb{R}$ be a function such that\n\\[f(x) f(y) = f(xy) + 2023 \\left( \\frac{2}{x} + \\frac{2}{y} + 2022 \\right)\\] for all $x, y > 0.$\n\nLet $n$ be the number of possible values of $f(2)$, and let $s$ be the su... | math | {
"style": "rule",
"ground_truth": "2023"
} | {
"split": "train",
"index": 6874,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence $\\{a\\_n\\}$, where $a\\_1=1$, $a\\_2=3$, and $2nS\\_n=(n+1)S\\_{n+1}+(n-1)S\\_{n-1}$ ($n\\geqslant 2,n\\in N$), find $S\\_{30}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{34}{5}"
} | {
"split": "train",
"index": 6875,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Although I am certain that my clock is 5 minutes fast, it is actually 10 minutes slow. On the other hand, my friend's clock is really 5 minutes fast, even though he thinks it is correct. We scheduled a meeting for 10 o'clock and plan to arrive on time. Who will arrive first? Af... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 6876,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When written out in full, the number \\((10^{2020}+2020)^{2}\\) has 4041 digits. What is the sum of the digits of this 4041-digit number?\nA) 9 \nB) 17 \nC) 25 \nD) 2048 \nE) 4041 Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 6877,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The right triangles \\(MDC\\) and \\(ADK\\) have a common right angle at \\(D\\). Point \\(K\\) lies on \\(CD\\) and divides it in the ratio \\(2:3\\) from point \\(C\\). Point \\(M\\) is the midpoint of side \\(AD\\). Find the sum of the degree measures of angles \\(AKD\\) and... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 6878,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a quiz, every question is solved by exactly four students, every pair of questions is solved by exactly one student, and none of the students solved all of the questions. Find the maximum possible number of questions on the quiz. Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 6879,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $a \\in \\mathbb{R}$, if the real part and the imaginary part of the complex number $\\frac{a + i}{1 + i}$ (where $i$ is the imaginary unit) are equal, then $\\_\\_\\_\\_\\_\\_$, $| \\overline{z}| = \\_\\_\\_\\_\\_\\_$. Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 6880,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the parallelepiped $ABCD-{A'}{B'}{C'}{D'}$, the base $ABCD$ is a square with side length $2$, the length of the side edge $AA'$ is $3$, and $\\angle {A'}AB=\\angle {A'}AD=60^{\\circ}$. Find the length of $AC'$. Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "\\sqrt{29}"
} | {
"split": "train",
"index": 6881,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two fair dice, each with at least $6$ faces are rolled. On each face of each die is printed a distinct integer from $1$ to the number of faces on that die, inclusive. The probability of rolling a sum of $7$ is $\\frac34$ of the probability of rolling a sum of $10,$ and the prob... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 6882,
"solution": "1. **Define Variables:**\n Let the number of faces on the two dice be $a$ and $b$, respectively, with $a \\geq b$. Assume each die has distinct integers from $1$ to the number of faces on that die.\n\n2. **Analyze the Probability of Rolling a Sum of $7$:**\n Sin... |
deepscaler | [
{
"role": "user",
"content": "Points \\( M \\) and \\( N \\) divide side \\( AC \\) of triangle \\( ABC \\) into three equal parts, each of which is 5, with \\( AB \\perp BM \\) and \\( BC \\perp BN \\). Find the area of triangle \\( ABC \\). Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "\\frac{75 \\sqrt{3}}{4}"
} | {
"split": "train",
"index": 6883,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle passes through the three vertices of an isosceles triangle that has two sides of length 3 and a base of length 2. What is the area of this circle? Express your answer in terms of $\\pi$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{81}{32}\\pi"
} | {
"split": "train",
"index": 6884,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 5 different books to be distributed among three people, with each person receiving at least 1 book and at most 2 books. Calculate the total number of different distribution methods. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 6885,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $b_n$ be the number obtained by writing the integers $1$ to $n$ from left to right, and then reversing the sequence. For example, $b_4 = 43211234$ and $b_{12} = 121110987654321123456789101112$. For $1 \\le k \\le 100$, how many $b_k$ are divisible by 9? Let's think step by ... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 6886,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square is divided into three congruent rectangles. The middle rectangle is removed and replaced on the side of the original square to form an octagon as shown.\nWhat is the ratio of the length of the perimeter of the square to the length of the perimeter of the octagon?\nA $3... | math | {
"style": "rule",
"ground_truth": "3:5"
} | {
"split": "train",
"index": 6887,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = x^2 + ax + b$ ($a, b \\in \\mathbb{R}$).\n\n(Ⅰ) Given $x \\in [0, 1]$,\n\n(i) If $a = b = 1$, find the range of the function $f(x)$;\n\n(ii) If the range of the function $f(x)$ is $[0, 1]$, find the values of $a$ and $b$;\n\n(Ⅱ) When $|x| \\geq 2$, it... | math | {
"style": "rule",
"ground_truth": "74"
} | {
"split": "train",
"index": 6888,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are six identical red balls and three identical green balls in a pail. Four of these balls are selected at random and then these four balls are arranged in a line in some order. Find the number of different-looking arrangements of the selected balls. Let's think step by s... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 6889,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For any natural values \\( m \\) and \\( n \\) (with \\( n > 1 \\)), a function is defined as \\( f\\left(\\frac{m}{n}\\right)=\\frac{\\sqrt[n]{3^{m}}}{\\sqrt[n]{3^{m}}+3} \\). Compute the sum\n\n\\[ f\\left(\\frac{1}{2020}\\right)+f\\left(\\frac{2}{2020}\\right)+f\\left(\\frac... | math | {
"style": "rule",
"ground_truth": "2020"
} | {
"split": "train",
"index": 6890,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For every $n$ the sum of $n$ terms of an arithmetic progression is $2n + 3n^2$. The $r$th term is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6r - 1"
} | {
"split": "train",
"index": 6891,
"solution": "1. **Identify the formula for the sum of the first $n$ terms**: Given that the sum of the first $n$ terms of an arithmetic progression is $S_n = 2n + 3n^2$.\n\n2. **Expression for the $r$th term**: The $r$th term of an arithmetic sequence can be found by subtracting... |
deepscaler | [
{
"role": "user",
"content": "Given that the first four terms of a geometric sequence $\\{a\\_n\\}$ have a sum of $S\\_4=5$, and $4a\\_1,\\;\\; \\frac {3}{2}a\\_2\\;,\\;a\\_2$ form an arithmetic sequence.\n(I) Find the general term formula for $\\{a\\_n\\}$;\n(II) Let $\\{b\\_n\\}$ be an arithmetic sequence... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 6892,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Following the directions of the arrows, how many different paths are there from $A$ to $C$?\n\n[asy]\npair A,B,C;\nA=(0,0);\nB=(5,0);\nC=(10,0);\ndot(A);\ndot(B);\ndot(C);\nlabel(\"$A$\",A,S);\nlabel(\"$B$\",B,S);\nlabel(\"$C$\",C,S);\ndraw((0,0)--(2.5,1)--(5,0),Arrow);\ndraw((... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 6893,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the numbers $\\{24,27,55,64,x\\}$ . Given that the mean of these five numbers is prime and the median is a multiple of $3$ , compute the sum of all possible positive integral values of $x$ . Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 6894,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the hyperbola $\\frac{x^{2}}{a^{2}} - \\frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$ with its right focus $F$, draw a line perpendicular to the $x$-axis passing through $F$, intersecting the two asymptotes at points $A$ and $B$, and intersecting the hyperbola at point $P$ in th... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 6895,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The mean, median, and unique mode of the positive integers 3, 4, 5, 6, 6, 7, and $x$ are all equal. What is the value of $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 6896,
"solution": "To solve this problem, we need to find the value of $x$ such that the mean, median, and mode of the set $\\{3, 4, 5, 6, 6, 7, x\\}$ are all equal.\n\n1. **Mode**: The mode is the number that appears most frequently in the set. Since $6$ appears twice and no other nu... |
deepscaler | [
{
"role": "user",
"content": "For a triangular piece of land with sides measuring 198 meters, 180 meters, and 210 meters, plant trees equidistantly along the three sides with 6 meters between each pair of consecutive trees, also requiring one tree planted at each of the three vertices. Determine the total n... | math | {
"style": "rule",
"ground_truth": "98"
} | {
"split": "train",
"index": 6897,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be a positive integer. All numbers $m$ which are coprime to $n$ all satisfy $m^6\\equiv 1\\pmod n$ . Find the maximum possible value of $n$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "504"
} | {
"split": "train",
"index": 6898,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number 3003 is the only number known to appear eight times in Pascal's triangle, at positions $\\binom{3003}{1},\\binom{3003}{3002},\\binom{a}{2},\\binom{a}{a-2},\\binom{15}{b},\\binom{15}{15-b},\\binom{14}{6},\\binom{14}{8}$. Compute $a+b(15-b)$. Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "128"
} | {
"split": "train",
"index": 6899,
"solution": "We first solve for $a$. Note that $3003=3 \\cdot 7 \\cdot 11 \\cdot 13$. We have $3003=\\binom{a}{2}=\\frac{a(a-1)}{2} \\approx \\frac{a^{2}}{2}$. This means we can estimate $a \\approx \\sqrt{3003 \\cdot 2}$, so $a$ is a little less than 80. Furthermore, $11 \\mid ... |
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