data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Determine all four-digit numbers $\\overline{abcd}$ which are perfect squares and for which the equality holds: $\\overline{ab}=3 \\cdot \\overline{cd} + 1$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2809"
} | {
"split": "train",
"index": 7800,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Geoff and Trevor each roll a fair six-sided die. What is the probability that the product of the numbers they roll is even? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac34"
} | {
"split": "train",
"index": 7801,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the graph of the power function $y=f(x)$ passes through the point $\\left( -2,-\\frac{1}{8} \\right)$, find the value(s) of $x$ that satisfy $f(x)=27$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 7802,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x) = 3x + 3$ and $g(x) = 4x + 3.$ What is $f(g(f(2)))$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 7803,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a triangle \\( ABC \\), \\( X \\) and \\( Y \\) are points on side \\( AB \\), with \\( X \\) closer to \\( A \\) than \\( Y \\), and \\( Z \\) is a point on side \\( AC \\) such that \\( XZ \\) is parallel to \\( YC \\) and \\( YZ \\) is parallel to \\( BC \\). Suppose \... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 7804,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $f(x)=\\sin \\omega x \\cos \\omega x - \\sqrt{3} \\cos^2 \\omega x$, where $(\\omega > 0)$, ${x}_{1}$ and ${x}_{2}$ are the two zeros of the function $y=f(x)+\\frac{2+\\sqrt{3}}{2}$, and $|x_{1}-x_{2}|_{\\min }=\\pi$. When $x\\in[0,\\frac{7\\pi}{12}]$, the sum of the min... | math | {
"style": "rule",
"ground_truth": "\\frac{2-3\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 7805,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If digits $A$ , $B$ , and $C$ (between $0$ and $9$ inclusive) satisfy\n \\begin{tabular}{c@{\\,}c@{\\,}c@{\\,}c}\n & $C$ & $C$ & $A$ \n + & $B$ & $2$ & $B$ \\hline\n & $A$ & $8$ & $8$ \n \\end{tabular}\n what is $A \\cdot B \\cd... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 7806,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jacqueline has 2 liters of soda. Liliane has 60% more soda than Jacqueline, and Alice has 40% more soda than Jacqueline. Calculate the percentage difference between the amount of soda Liliane has compared to Alice. Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "14.29\\%"
} | {
"split": "train",
"index": 7807,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "After reading the following solution, answer the question: Xiaofang found in the simplification of $\\sqrt{7+4\\sqrt{3}}$ that first, $\\sqrt{7+4\\sqrt{3}}$ can be simplified to $\\sqrt{7+2\\sqrt{12}}$. Since $4+3=7$ and $4\\times 3=12$, that is, ${(\\sqrt{4})^2}+{(\\sqrt{3})^2... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{10}}{2}-\\frac{\\sqrt{6}}{2}"
} | {
"split": "train",
"index": 7808,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In acute triangle $\\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively, and $4\\sin ^{2} \\frac {B+C}{2}-\\cos 2A= \\frac {7}{2}$.\n\n1. Find the measure of angle $A$.\n2. If the altitude on side $BC$ is $1$, find the minimum area ... | math | {
"style": "rule",
"ground_truth": "\\frac { \\sqrt {3}}{3}"
} | {
"split": "train",
"index": 7809,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\(ABC\\), \\(AC = 8\\) and \\(BC = 5\\). A line parallel to the bisector of the external angle at \\(C\\) passes through the midpoint of side \\(AB\\) and point \\(E\\) on side \\(AC\\). Find \\(AE\\). Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "1.5"
} | {
"split": "train",
"index": 7810,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse $C$: $\\frac{x^{2}}{a^{2}} + \\frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with an eccentricity of $\\frac{\\sqrt{2}}{2}$, and it passes through point $M(-2, 0)$.\n(I) Find the standard equation of ellipse $C$;\n(II) Let line $l$ with a slope of $1$ intersect ellipse ... | math | {
"style": "rule",
"ground_truth": "S = \\sqrt{10}"
} | {
"split": "train",
"index": 7811,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( x \\) and \\( y \\) be positive real numbers, and \\( x + y = 1 \\). Find the minimum value of \\( \\frac{x^2}{x+2} + \\frac{y^2}{y+1} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1/4"
} | {
"split": "train",
"index": 7812,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $XYZ$, $XY=153$, $XZ=147$, and $YZ=140$. The angle bisector of angle $X$ intersects $\\overline{YZ}$ at point $D$, and the angle bisector of angle $Y$ intersects $\\overline{XZ}$ at point $E$. Let $P$ and $Q$ be the feet of the perpendiculars from $Z$ to $\\overline... | math | {
"style": "rule",
"ground_truth": "67"
} | {
"split": "train",
"index": 7813,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the coefficient of $x^3$ in the expansion of $$(x+2\\sqrt3)^7?$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5040"
} | {
"split": "train",
"index": 7814,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The ratio of the interior angles of two regular polygons with sides of unit length is $3: 2$. How many such pairs are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 7815,
"solution": "1. **Identify the formula for the interior angle of a regular polygon**: The measure of an interior angle of a regular polygon with $n$ sides is given by the formula:\n \\[\n 180^\\circ - \\frac{360^\\circ}{n}\n \\]\n This formula arises from the fact that t... |
deepscaler | [
{
"role": "user",
"content": "The product of four different positive integers is 360. What is the maximum possible sum of these four integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "66"
} | {
"split": "train",
"index": 7816,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of $2x^2 + xy + 3y^2 - 11x - 20y + 40 = 0$ is an ellipse in the first quadrant of the $xy$-plane. Let $a$ and $b$ be the maximum and minimum values of $\\frac{y}{x}$ over all points $(x,y)$ on the ellipse. What is the value of $a+b$? Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{2}"
} | {
"split": "train",
"index": 7817,
"solution": "1. **Identify the problem**: We need to find the maximum and minimum values of $\\frac{y}{x}$ for points $(x, y)$ on the ellipse given by the equation $2x^2 + xy + 3y^2 - 11x - 20y + 40 = 0$. The ratio $\\frac{y}{x}$ represents the slope of a line passing through th... |
deepscaler | [
{
"role": "user",
"content": "Given that $\\sin 2α = - \\frac{24}{25}$, and $α ∈ (\\frac{3π}{4},π)$, find the value of $\\sin α$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"split": "train",
"index": 7818,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate $3 \\cdot 7^{-1} + 9 \\cdot 13^{-1} \\pmod{60}$.\n\nExpress your answer as an integer from $0$ to $59$, inclusive. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 7819,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many solutions in integers $x$ and $y$ does the inequality\n\n$$\n|x| + |y| < 10\n$$\n\nhave? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "181"
} | {
"split": "train",
"index": 7820,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The mean of one set of four numbers is 15, and the mean of a separate set of eight numbers is 20. What is the mean of the set of all twelve numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{55}{3}"
} | {
"split": "train",
"index": 7821,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A quadrilateral is divided by its diagonals into four triangles. The areas of three of them are 10, 20, and 30, each being less than the area of the fourth triangle. Find the area of the given quadrilateral. Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 7822,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the function \\( f(x) = x^3 + a x^2 + b x + c \\), where \\( a \\), \\( b \\), and \\( c \\) are non-zero integers. If \\( f(a) = a^3 \\) and \\( f(b) = b^3 \\), then the value of \\( c \\) is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 7823,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From the consecutive natural numbers 1, 2, 3, …, 2014, select $n$ numbers such that for any two numbers chosen, one is not five times the other. Find the maximum value of $n$ and explain the reason. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1665"
} | {
"split": "train",
"index": 7824,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the infinite sum $\\sum_{n=1}^{\\infty} \\frac{n}{n^{4}+4}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{8}"
} | {
"split": "train",
"index": 7825,
"solution": "We have $$\\begin{aligned} \\sum_{n=1}^{\\infty} \\frac{n}{n^{4}+4} & =\\sum_{n=1}^{\\infty} \\frac{n}{\\left(n^{2}+2 n+2\\right)\\left(n^{2}-2 n+2\\right)} \\\\ & =\\frac{1}{4} \\sum_{n=1}^{\\infty}\\left(\\frac{1}{n^{2}-2 n+2}-\\frac{1}{n^{2}+2 n+2}\\right) \\\\ &... |
deepscaler | [
{
"role": "user",
"content": "Three noncollinear points and a line $\\ell$ are given in the plane. Suppose no two of the points lie on a line parallel to $\\ell$ (or $\\ell$ itself). There are exactly $n$ lines perpendicular to $\\ell$ with the following property: the three circles with centers at the given... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 7826,
"solution": "The condition for the line is that each of the three points lies at an equal distance from the line as from some fixed point; in other words, the line is the directrix of a parabola containing the three points. Three noncollinear points in the coordinate plane deter... |
deepscaler | [
{
"role": "user",
"content": "Use Horner's Rule to find the value of $v_2$ when the polynomial $f(x) = x^5 + 4x^4 + x^2 + 20x + 16$ is evaluated at $x = -2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 7827,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the rectangular coordinate system $(xOy)$, there are two curves $C_1: x + y = 4$ and $C_2: \\begin{cases} x = 1 + \\cos \\theta \\\\ y = \\sin \\theta \\end{cases}$ (where $\\theta$ is a parameter). Establish a polar coordinate system with the coordinate origin $O$ as the po... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}(\\sqrt{2} + 1)"
} | {
"split": "train",
"index": 7828,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest possible three-digit number that is divisible by 4 and divisible by 5? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "980"
} | {
"split": "train",
"index": 7829,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ has side lengths $AB=4$, $BC=5$, and $CA=6$. Points $D$ and $E$ are on ray $AB$ with $AB<AD<AE$. The point $F \\neq C$ is a point of intersection of the circumcircles of $\\triangle ACD$ and $\\triangle EBC$ satisfying $DF=2$ and $EF=7$. Then $BE$ can be expresse... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 7830,
"solution": "Let $P=AE\\cap CF$. Let $CP=5x$ and $BP=5y$; from $\\triangle{CBP}\\sim\\triangle{EFP}$ we have $EP=7x$ and $FP=7y$. From $\\triangle{CAP}\\sim\\triangle{DFP}$ we have $\\frac{6}{4+5y}=\\frac{2}{7y}$ giving $y=\\frac{1}{4}$. So $BP=\\frac{5}{4}$ and $FP=\\frac{7}{4}... |
deepscaler | [
{
"role": "user",
"content": "Given the planar vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$, with $|\\overrightarrow{a}| = 1$, $|\\overrightarrow{b}| = \\sqrt{2}$, and $\\overrightarrow{a} \\cdot \\overrightarrow{b} = 1$, find the angle between vectors $\\overrightarrow{a}$ and $\\overrightarrow{... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"split": "train",
"index": 7831,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sequence of positive integers with $a_1=1$ and $a_9+a_{10}=646$ is formed so that the first three terms are in geometric progression, the second, third, and fourth terms are in arithmetic progression, and, in general, for all $n\\ge1,$ the terms $a_{2n-1}, a_{2n}, a_{2n+1}$ a... | math | {
"style": "rule",
"ground_truth": "973"
} | {
"split": "train",
"index": 7832,
"solution": "Let $x = a_2$. It is apparent that the sequence grows relatively fast, so we start trying positive integers to see what $x$ can be. Finding that $x = 5$ works, after bashing out the rest of the terms we find that $a_{16} = 957$ and $a_{17} = 1089$, hence our answer ... |
deepscaler | [
{
"role": "user",
"content": "Let \\(a,\\) \\(b,\\) and \\(c\\) be positive real numbers such that \\(a + b + c = 3.\\) Find the minimum value of\n\\[\\frac{a + b}{abc}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{16}{9}"
} | {
"split": "train",
"index": 7833,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a,$ $b,$ $c$ be nonzero real numbers such that $a + b + c = 0,$ and $ab + ac + bc \\neq 0.$ Find all possible values of\n\\[\\frac{a^7 + b^7 + c^7}{abc (ab + ac + bc)}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"split": "train",
"index": 7834,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $f(x)= \\frac{\\ln x+2^{x}}{x^{2}}$, find $f'(1)=$ ___. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\ln 2 - 3"
} | {
"split": "train",
"index": 7835,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $v$ and $w$ be distinct, randomly chosen roots of the equation $z^{1997}-1=0.$ Find the probability that\n\\[\\sqrt{2+\\sqrt{3}}\\le\\left|v+w\\right|.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{83}{499}"
} | {
"split": "train",
"index": 7836,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An object in the plane moves from the origin and takes a ten-step path, where at each step the object may move one unit to the right, one unit to the left, one unit up, or one unit down. How many different points could be the final point? Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "221"
} | {
"split": "train",
"index": 7837,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle of radius $10$ inches has its center at the vertex $C$ of an equilateral triangle $ABC$ and passes through the other two vertices. The side $AC$ extended through $C$ intersects the circle at $D$. The number of degrees of angle $ADB$ is: Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 7838,
"solution": "1. **Identify the properties of triangle $ABC$ and circle properties**: \n - Since $ABC$ is an equilateral triangle, each angle in the triangle is $60^\\circ$. Therefore, $\\angle ACB = 60^\\circ$.\n - The circle has its center at $C$ and passes through $A$ and ... |
deepscaler | [
{
"role": "user",
"content": "A square is cut into red and blue rectangles. The sum of areas of red triangles is equal to the sum of areas of the blue ones. For each blue rectangle, we write the ratio of the length of its vertical side to the length of its horizontal one and for each red rectangle, the rati... | math | {
"style": "rule",
"ground_truth": "5/2"
} | {
"split": "train",
"index": 7839,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We call a pair of natural numbers \\((a, p)\\) good if the number \\(a^3 + p^3\\) is divisible by \\(a^2 - p^2\\), with \\(a > p\\).\n\n(a) (1 point) Specify any possible value of \\(a\\) for which the pair \\((a, 13)\\) is good.\n\n(b) (3 points) Find the number of good pairs ... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 7840,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The city administration of a certain city launched a campaign that allows the exchange of four empty 1-liter bottles for one full 1-liter bottle of milk. How many liters of milk can a person obtain if they have 43 empty 1-liter bottles by making several of these exchanges? Let'... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 7841,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $f(\\alpha)=\\frac{2\\sin(2\\pi-\\alpha)\\cos(2\\pi+\\alpha)-\\cos(-\\alpha)}{1+\\sin^{2}\\alpha+\\sin(2\\pi+\\alpha)-\\cos^{2}(4\\pi-\\alpha)}$, find the value of $f\\left(-\\frac{23}{6}\\pi \\right)$. Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "-\\sqrt{3}"
} | {
"split": "train",
"index": 7842,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If a number is randomly selected from the set $\\left\\{ \\frac{1}{3}, \\frac{1}{4}, 3, 4 \\right\\}$ and denoted as $a$, and another number is randomly selected from the set $\\left\\{ -1, 1, -2, 2 \\right\\}$ and denoted as $b$, then the probability that the graph of the func... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{8}"
} | {
"split": "train",
"index": 7843,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points $A, B, C$, and $D$ are on a line in that order. The distance from $A$ to $D$ is 24. The distance from $B$ to $D$ is 3 times the distance from $A$ to $B$. Point $C$ is halfway between $B$ and $D$. What is the distance from $A$ to $C$? Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 7844,
"solution": "Since $B$ is between $A$ and $D$ and $B D=3 A B$, then $B$ splits $A D$ in the ratio $1: 3$. Since $A D=24$, then $A B=6$ and $B D=18$. Since $C$ is halfway between $B$ and $D$, then $B C=\frac{1}{2} B D=9$. Thus, $A C=A B+B C=6+9=15$."
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest prime $p$ for which there exist positive integers $a,b$ such that \n\t\\[\n\t\ta^{2} + p^{3} = b^{4}.\n\t\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 7845,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the number of distinct three-digit numbers formed using the digits 0, 1, 2, 3, 4, and 5 without repetition that are divisible by 9. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 7846,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\( a_{n} = \\mathrm{C}_{200}^{n} \\cdot (\\sqrt[3]{6})^{200-n} \\cdot \\left( \\frac{1}{\\sqrt{2}} \\right)^{n} \\) for \\( n = 1, 2, \\ldots, 95 \\), find the number of integer terms in the sequence \\(\\{a_{n}\\}\\). Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 7847,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many zeroes does $10!$ end with, when $10!$ is written in base 9? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 7848,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that circle $A$ has radius $150$, and circle $B$, with an integer radius $r$, is externally tangent to circle $A$ and rolls once around the circumference of circle $A$, determine the number of possible integer values of $r$. Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 7849,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $f(x) = 5x-4$, what is $f(f(f(2)))$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "126"
} | {
"split": "train",
"index": 7850,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ten points are spaced around at intervals of one unit around a modified $2 \\times 2$ square such that each vertex of the square and midpoints on each side of the square are included, along with two additional points, each located midway on a diagonal extension from opposite co... | math | {
"style": "rule",
"ground_truth": "\\frac{14}{45}"
} | {
"split": "train",
"index": 7851,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three lines were drawn through the point $X$ in space. These lines crossed some sphere at six points. It turned out that the distances from point $X$ to some five of them are equal to $2$ cm, $3$ cm, $4$ cm, $5$ cm, $6$ cm. What can be the distance from point $X$... | math | {
"style": "rule",
"ground_truth": "2.4"
} | {
"split": "train",
"index": 7852,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the sum of all single-digit replacements for $z$ such that the number ${36{,}z72}$ is divisible by both 6 and 4. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 7853,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At the mall's food court, Crystal wants to buy a meal consisting of one entree, one drink and one dessert. The table below lists Crystal's favorite foods in the food court. How many distinct possible meals can she buy from these options?\n\n\\begin{tabular}{ |c | c | c | }\n\\h... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 7854,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a row, there are 99 people - knights and liars (knights always tell the truth, and liars always lie). Each of them said one of two phrases: \"To my left, there are twice as many knights as liars\" or \"To my left, there are as many knights as liars.\" In reality, there were ... | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 7855,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $b_1, b_2, \\ldots$ be a sequence determined by the rule $b_n = \\frac{b_{n-1}}{3}$ if $b_{n-1}$ is divisible by 3, and $b_n = 2b_{n-1} + 2$ if $b_{n-1}$ is not divisible by 3. For how many positive integers $b_1 \\le 1500$ is it true that $b_1$ is less than each of $b_2$, ... | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"split": "train",
"index": 7856,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the middle of a vast prairie, a firetruck is stationed at the intersection of two perpendicular straight highways. The truck travels at $50$ miles per hour along the highways and at $14$ miles per hour across the prairie. Consider the set of points that can be reached by the... | math | {
"style": "rule",
"ground_truth": "731"
} | {
"split": "train",
"index": 7857,
"solution": "Let the intersection of the highways be at the origin $O$, and let the highways be the x and y axes. We consider the case where the truck moves in the positive x direction.\n After going $x$ miles, $t=\\frac{d}{r}=\\frac{x}{50}$ hours has passed. If the truck leaves... |
deepscaler | [
{
"role": "user",
"content": "The pony wants to cross a bridge where there are two monsters, A and B. Monster A is awake for 2 hours and rests for 1 hour. Monster B is awake for 3 hours and rests for 2 hours. The pony can only cross the bridge when both monsters are resting; otherwise, it will be eaten by t... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 7858,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A basketball player scored 18, 22, 15, and 20 points respectively in her first four games of a season. Her points-per-game average was higher after eight games than it was after these four games. If her average after nine games was greater than 19, determine the least number of... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 7859,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Luke wants to fence a square piece of land of at least 400 square feet. What should the side of the square be if he wants to use the least amount of fencing? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 7860,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the largest integer \\( k \\) such that \\( 2010^k \\) divides \\( 2010! \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 7861,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each of $b_1, b_2, \\dots, b_{150}$ is equal to $2$ or $-2$. Find the minimum positive value of\n\\[\\sum_{1 \\le i < j \\le 150} b_i b_j.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "38"
} | {
"split": "train",
"index": 7862,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Urn A contains 4 white balls and 2 red balls. Urn B contains 3 red balls and 3 black balls. An urn is randomly selected, and then a ball inside of that urn is removed. We then repeat the process of selecting an urn and drawing out a ball, without returning the first ball. What ... | math | {
"style": "rule",
"ground_truth": "7/15"
} | {
"split": "train",
"index": 7863,
"solution": "This is a case of conditional probability; the answer is the probability that the first ball is red and the second ball is black, divided by the probability that the second ball is black. First, we compute the numerator. If the first ball is drawn from Urn A, we hav... |
deepscaler | [
{
"role": "user",
"content": "What is the remainder of $8^6 + 7^7 + 6^8$ divided by 5? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 7864,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a natural number $n$, if $n+6$ divides $n^3+1996$, then $n$ is called a lucky number of 1996. For example, since $4+6$ divides $4^3+1996$, 4 is a lucky number of 1996. Find the sum of all lucky numbers of 1996. Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "3720"
} | {
"split": "train",
"index": 7865,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the slant height of a cone is 2, and its net is a semicircle, what is the area of the cross section of the axis of the cone? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 7866,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In trapezoid $ABCD$ with $\\overline{BC}\\parallel\\overline{AD}$, let $BC = 800$ and $AD = 1800$. Let $\\angle A = 45^\\circ$, $\\angle D = 45^\\circ$, and $P$ and $Q$ be the midpoints of $\\overline{BC}$ and $\\overline{AD}$, respectively. Find the length $PQ$. Let's think st... | math | {
"style": "rule",
"ground_truth": "500"
} | {
"split": "train",
"index": 7867,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of all the two-digit primes that are greater than 20 but less than 80 and are still prime when their two digits are interchanged? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "291"
} | {
"split": "train",
"index": 7868,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular triangular prism has a triangle $ABC$ with side $a$ as its base. Points $A_{1}, B_{1}$, and $C_{1}$ are taken on the lateral edges and are located at distances of $a / 2, a, 3a / 2$ from the base plane, respectively. Find the angle between the planes $ABC$ and $A_{1}B... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"split": "train",
"index": 7869,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a circle with center $O$ and radius $r$, chord $AB$ is drawn with length equal to $r$ (units). From $O$, a perpendicular to $AB$ meets $AB$ at $M$. From $M$ a perpendicular to $OA$ meets $OA$ at $D$. In terms of $r$ the area of triangle $MDA$, in appropriate square units, is... | math | {
"style": "rule",
"ground_truth": "\\frac{r^2\\sqrt{3}}{32}"
} | {
"split": "train",
"index": 7870,
"solution": "1. **Identify the Geometry of Triangle $AOB$:**\n Since $AB = r$ and $AO = OB = r$ (radii of the circle), triangle $AOB$ is isosceles with $AO = OB = AB$. Additionally, since all sides are equal, $\\triangle AOB$ is an equilateral triangle. Therefore, each angle i... |
deepscaler | [
{
"role": "user",
"content": "Given a positive integer $n$, let $M(n)$ be the largest integer $m$ such that \\[ \\binom{m}{n-1} > \\binom{m-1}{n}. \\] Evaluate \\[ \\lim_{n \\to \\infty} \\frac{M(n)}{n}. \\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3+\\sqrt{5}}{2}"
} | {
"split": "train",
"index": 7871,
"solution": "The answer is $\\frac{3+\\sqrt{5}}{2}$. Note that for $m > n+1$, both binomial coefficients are nonzero and their ratio is \\[ {m\\choose n-1}/{m-1\\choose n} = \\frac{m!n!(m-n-1)!}{(m-1)!(n-1)!(m-n+1)!} = \\frac{mn}{(m-n+1)(m-n)}. \\] Thus the condition ${m\\choose... |
deepscaler | [
{
"role": "user",
"content": "A circle with a radius of 6 cm is tangent to three sides of a rectangle. The area of the circle is half the area of the rectangle. What is the length of the longer side of the rectangle, in centimeters? Express your answer in terms of $\\pi$. Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "6\\pi"
} | {
"split": "train",
"index": 7872,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In 1991 the population of a town was a perfect square. Ten years later, after an increase of 150 people, the population was 9 more than a perfect square. Now, in 2011, with an increase of another 150 people, the population is once again a perfect square. What is the percent gro... | math | {
"style": "rule",
"ground_truth": "62"
} | {
"split": "train",
"index": 7873,
"solution": "1. **Define Variables:**\n Let the population of the town in 1991 be $p^2$. In 2001, after an increase of 150 people, the population is $p^2 + 150$. According to the problem, this new population is 9 more than a perfect square, so we can write it as $q^2 + 9$. Thu... |
deepscaler | [
{
"role": "user",
"content": "Find the area of triangle \\(ABC\\), if \\(AC = 3\\), \\(BC = 4\\), and the medians \\(AK\\) and \\(BL\\) are mutually perpendicular. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{11}"
} | {
"split": "train",
"index": 7874,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $tan({θ+\\frac{π}{4}})=2tanθ-7$, determine the value of $\\sin 2\\theta$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{5}"
} | {
"split": "train",
"index": 7875,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider arrangements of the $9$ numbers $1, 2, 3, \\dots, 9$ in a $3 \\times 3$ array. For each such arrangement, let $a_1$, $a_2$, and $a_3$ be the medians of the numbers in rows $1$, $2$, and $3$ respectively, and let $m$ be the median of $\\{a_1, a_2, a_3\\}$. Let $Q$ be th... | math | {
"style": "rule",
"ground_truth": "360"
} | {
"split": "train",
"index": 7876,
"solution": "We see that if one of the medians is 5, then there are two remaining numbers greater than 5 and two less than 5, so it follows that $m=5$. There are 3 ways to choose which row to have 5 in, $4\\cdot 4=16$ ways to choose the other two numbers in that row, $3!=6$ ways... |
deepscaler | [
{
"role": "user",
"content": "If 5 points are placed in the plane at lattice points (i.e. points $(x, y)$ where $x$ and $y$ are both integers) such that no three are collinear, then there are 10 triangles whose vertices are among these points. What is the minimum possible number of these triangles that have... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 7877,
"solution": "By the pigeonhole principle, the 5 points cannot all be distinct modulo 2, so two of them must have a midpoint that is also a lattice point. This midpoint is not one of the 5 since no 3 are collinear. Pick's theorem states that the area of a polygon whose vertices a... |
deepscaler | [
{
"role": "user",
"content": "Luka is making lemonade to sell at a school fundraiser. His recipe requires $4$ times as much water as sugar and twice as much sugar as lemon juice. He uses $3$ cups of lemon juice. How many cups of water does he need? Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 7878,
"solution": "1. **Identify the ratios**: According to the problem, the recipe requires:\n - 4 times as much water as sugar.\n - Twice as much sugar as lemon juice.\n\n Let's denote the amount of lemon juice used as $L$, the amount of sugar as $S$, and the amount of water a... |
deepscaler | [
{
"role": "user",
"content": "Two hunters, $A$ and $B$, went duck hunting. Assume that each of them hits a duck as often as they miss it. Hunter $A$ encountered 50 ducks during the hunt, while hunter $B$ encountered 51 ducks. What is the probability that hunter $B$'s catch exceeds hunter $A$'s catch? Let's ... | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 7879,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "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 | {
"style": "rule",
"ground_truth": "56"
} | {
"split": "train",
"index": 7880,
"solution": "We will just bash. Let $z=a+bi$ where $a,b\\in\\mathbb{R}$. We see that $\\frac{z_3-z_1}{z_2-z_1}=\\frac{-4+15i}{11}$ after doing some calculations. We also see that $\\frac{[(a-18)+(b-39)i][(a-78)-(b-99)i]}{\\text{some real stuff}}.$ We note that $[(a-18)+(b-39)i][... |
deepscaler | [
{
"role": "user",
"content": "Given the function \\( f(x) = \\sin^4 x \\),\n1. Let \\( g(x) = f(x) + f\\left(\\frac{\\pi}{2} - x\\right) \\). Find the maximum and minimum values of \\( g(x) \\) in the interval \\(\\left[\\frac{\\pi}{6}, \\frac{3\\pi}{8}\\right]\\).\n2. Find the value of \\(\\sum_{k=1}^{89} ... | math | {
"style": "rule",
"ground_truth": "\\frac{133}{4}"
} | {
"split": "train",
"index": 7881,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the function $f(x)=a\\ln x-x- \\frac {1}{2}x^{2}$. \n(I) For $a=2$, find the extreme values of the function $f(x)$. \n(II) Discuss the monotonicity of the function $f(x)$. 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": 7882,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The plane angle at the vertex of a regular triangular pyramid is $90^{\\circ}$. Find the ratio of the lateral surface area of the pyramid to the area of its base. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 7883,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five years ago, Tim was three times as old as his sister Sarah, and three years before that, Tim was five times as old as Sarah. Determine the number of years it will take for the ratio of their ages to be 3 : 2. Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 7884,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the double integrals and change the order of integration.\n\n$$\n\\text{a) } \\int_{0}^{2} dx \\int_{0}^{3}\\left(x^{2} + 2xy\\right) dy\n$$\n\nb) $\\int_{-2}^{0} dy \\int_{0}^{y^{2}}(x+2y) dx$\n\nc) $\\int_{0}^{5} dx \\int_{0}^{5-x} \\sqrt{4 + x + y} dy$ Let's think ... | math | {
"style": "rule",
"ground_truth": "\\frac{506}{3}"
} | {
"split": "train",
"index": 7885,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the times between $8$ and $9$ o'clock, correct to the nearest minute, when the hands of a clock will form an angle of $120^{\\circ}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8:22"
} | {
"split": "train",
"index": 7886,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the sum of the first $n$ terms of an arithmetic sequence $\\{a_n\\}$ be $S_n$. If $a_5=3$ and $S_{10}=40$, then the minimum value of $nS_n$ is . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-32"
} | {
"split": "train",
"index": 7887,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\\(ABCD\\) is a parallelogram with \\(AB = 7\\), \\(BC = 2\\), and \\(\\angle DAB = 120^\\circ\\). Parallelogram \\(ECFA\\) is contained within \\(ABCD\\) and is similar to it. Find the ratio of the area of \\(ECFA\\) to the area of \\(ABCD\\). Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "39/67"
} | {
"split": "train",
"index": 7888,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the volume of the solid formed by rotating around the $O Y$ axis the curvilinear trapezoid which is bounded by the hyperbola $x y=2$ and the lines $y_{1}=1, y_{2}=4$, and $y_{3}=0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3\\pi"
} | {
"split": "train",
"index": 7889,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$ with $\\overline{AB}=\\overline{AC}=3.6$ , a point $D$ is taken on $AB$ at a distance $1.2$ from $A$ . Point $D$ is joined to $E$ in the prolongation of $AC$ so that triangle $AED$ is equal in area to $ABC$ . Then $\\overline{AE}$ is: Le... | math | {
"style": "rule",
"ground_truth": "10.8"
} | {
"split": "train",
"index": 7890,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $0 \\le x_0 < 1$, let\n\\[x_n = \\begin{cases} 2x_{n-1} & \\text{ if } 2x_{n-1} < 1 \\\\ 2x_{n-1} - 1 & \\text{ if } 2x_{n-1} \\ge 1 \\end{cases}\\]for all integers $n > 0$. For how many $x_0$ is it true that $x_0 = x_5$? Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 7891,
"solution": "1. **Understanding the Sequence**: The sequence defined by $x_n$ is a binary sequence where each term is generated by doubling the previous term and subtracting 1 if the result is at least 1. This can be interpreted as a shift and truncate operation in binary repres... |
deepscaler | [
{
"role": "user",
"content": "Arrange 3 volunteer teachers to 4 schools, with at most 2 people per school. How many different distribution plans are there? (Answer with a number) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 7892,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Kolya traveled on an electric scooter to a store in a neighboring village at a speed of 10 km/h. After covering exactly one-third of the total distance, he realized that if he continued at the same speed, he would arrive exactly at the store's closing time. He then doubled his ... | math | {
"style": "rule",
"ground_truth": "6.666666666666667"
} | {
"split": "train",
"index": 7893,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many degrees are in the sum of the measures of the six numbered angles pictured? [asy]\ndraw((3,8)--(10,4)--(1,0)--cycle,linewidth(1));\ndraw((7,8)--(9,0)--(0,4)--cycle,linewidth(1));\nlabel(\"1\",(3,8),SSE);\nlabel(\"2\",(7,8),SSW);\nlabel(\"3\",(10,4),2W);\nlabel(\"4\",(9... | math | {
"style": "rule",
"ground_truth": "360^\\circ"
} | {
"split": "train",
"index": 7894,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function $f$ is graphed below. Each small box has width and height 1.\n\n[asy]\nsize(150);\nreal ticklen=3;\nreal tickspace=2;\n\nreal ticklength=0.1cm;\nreal axisarrowsize=0.14cm;\npen axispen=black+1.3bp;\nreal vectorarrowsize=0.2cm;\nreal tickdown=-0.5;\nreal tickdownle... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 7895,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\sin \\alpha $ and $\\cos \\alpha $ are the two roots of the quadratic equation $2x^{2}-x-m=0$ with respect to $x$, find $\\sin \\alpha +\\cos \\alpha =$______ and $m=\\_\\_\\_\\_\\_\\_$. Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 7896,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Robyn has 4 tasks to do and Sasha has 14 tasks to do. How many of Sasha's tasks should Robyn do in order for them to have the same number of tasks? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 7897,
"solution": "Between them, Robyn and Sasha have \\(4 + 14 = 18\\) tasks to do. If each does the same number of tasks, each must do \\(18 \\div 2 = 9\\) tasks. This means that Robyn must do \\(9 - 4 = 5\\) of Sasha's tasks."
} |
deepscaler | [
{
"role": "user",
"content": "Given that the chord common to circle C: x²+(y-4)²=18 and circle D: (x-1)²+(y-1)²=R² has a length of $6\\sqrt {2}$, find the radius of circle D. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt {7}"
} | {
"split": "train",
"index": 7898,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sequence consists of 2010 terms. Each term after the first is 1 larger than the previous term. The sum of the 2010 terms is 5307. When every second term is added up, starting with the first term and ending with the second last term, what is the sum? Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "2151"
} | {
"split": "train",
"index": 7899,
"solution": "We label the terms $x_{1}, x_{2}, x_{3}, \\ldots, x_{2009}, x_{2010}$. Suppose that $S$ is the sum of the odd-numbered terms in the sequence; that is, $S=x_{1}+x_{3}+x_{5}+\\cdots+x_{2007}+x_{2009}$. We know that the sum of all of the terms is 5307; that is, $x_{1}+... |
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