data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Given that a 1-step requires 4 toothpicks, a 2-step requires 10 toothpicks, a 3-step requires 18 toothpicks, and a 4-step requires 28 toothpicks, determine the number of additional toothpicks needed to build a 6-step staircase. Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 36449,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If\n\\begin{align*}\na + b + c &= 1, \\\\\na^2 + b^2 + c^2 &= 2, \\\\\na^3 + b^3 + c^3 &= 3,\n\\end{align*}find $a^4 + b^4 + c^4.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{25}{6}"
} | {
"split": "train",
"index": 36450,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence $\\left\\{a_{n}\\right\\}$ consists of 9 terms, where $a_{1} = a_{9} = 1$, and for each $i \\in \\{1,2, \\cdots, 8\\}$, we have $\\frac{a_{i+1}}{a_{i}} \\in \\left\\{2,1,-\\frac{1}{2}\\right\\}$. Find the number of such sequences. Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "491"
} | {
"split": "train",
"index": 36451,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The ratio of the length to the width of a rectangle is $5$ : $2$. If the rectangle has a diagonal of length $13$ units, find the constant $k$ such that the area of the rectangle can be expressed as $kd^2$. Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "\\frac{10}{29}"
} | {
"split": "train",
"index": 36452,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the curve defined by the equation\n\\[r = 2.\\](A) Line\n(B) Circle\n(C) Parabola\n(D) Ellipse\n(E) Hyperbola\n\nEnter the letter of the correct option. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\text{(B)}"
} | {
"split": "train",
"index": 36453,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the equation $\\frac{1}{m} + \\frac{1}{n} = \\frac{1}{4}$, where $m$ and $n$ are positive integers, determine the sum of all possible values for $n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "51"
} | {
"split": "train",
"index": 36454,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle PQR$, $PR=15$, $QR=20$, and $PQ=25$. Points $A$ and $B$ lie on $\\overline{PQ}$, points $C$ and $D$ lie on $\\overline{QR}$, and points $E$ and $F$ lie on $\\overline{PR}$, with $PA=QB=QC=RD=RE=PF=5$. Find the area of hexagon $ABCDEF$. Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 36455,
"solution": "1. **Identify the type of triangle $PQR$**:\n Given $PR=15$, $QR=20$, and $PQ=25$, we check if $\\triangle PQR$ is a right triangle using the Pythagorean theorem:\n \\[\n PR^2 + QR^2 = 15^2 + 20^2 = 225 + 400 = 625 = 25^2 = PQ^2.\n \\]\n Since $PR^2 + QR^... |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given $\\overrightarrow{m}=(\\sin C,\\sin B\\cos A)$ and $\\overrightarrow{n}=(b,2c)$ with $\\overrightarrow{m}\\cdot \\overrightarrow{n}=0$.\n\n(1) Find angle $A$;\n(2) If ... | math | {
"style": "rule",
"ground_truth": "\\sqrt {3}"
} | {
"split": "train",
"index": 36456,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the maximum and minimum values of the function $f(x)=\\frac{1}{3}x^3-4x$ on the interval $\\left[-3,3\\right]$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{16}{3}"
} | {
"split": "train",
"index": 36457,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Explore the pattern, observe the following equations, and answer the questions.\n$1=1^{2}$;\n$1+3=4=2^{2}$;\n$1+3+5=9=3^{2}$;\n$1+3+5+7=16=4^{2}$;\n$1+3+5+7+9=25=5^{2}$;\n$\\ldots $\n$(1)$ Please guess $1+3+5+7+9+\\ldots +21=$______;\n$(2)$ Please guess $1+3+5+7+9+\\ldots +(2n-... | math | {
"style": "rule",
"ground_truth": "12500"
} | {
"split": "train",
"index": 36458,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is $2343_6+15325_6$? Express your answer in base $6$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22112_6"
} | {
"split": "train",
"index": 36459,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If there are 150 seats in a row, calculate the fewest number of seats that must be occupied so the next person to be seated must sit next to someone. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "37"
} | {
"split": "train",
"index": 36460,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest prime number $p<1000$ for which there exists a complex number $z$ satisfying\nthe real and imaginary part of $z$ are both integers;\n$|z|=\\sqrt{p},$ and\nthere exists a triangle whose three side lengths are $p,$ the real part of $z^{3},$ and the imaginary part... | math | {
"style": "rule",
"ground_truth": "349"
} | {
"split": "train",
"index": 36461,
"solution": "Assume that $z=a+bi$. Then, \\[z^3=(a^3-3ab^2)+(3a^2b-b^3)i\\]Note that by the Triangle Inequality, \\[|(a^3-3ab^2)-(3a^2b-b^3)|<p\\implies |a^3+b^3-3ab^2-3a^2b|<a^2+b^2\\]Thus, we know \\[|a+b||a^2+b^2-4ab|<a^2+b^2\\]Without loss of generality, assume $a>b$ (as ot... |
deepscaler | [
{
"role": "user",
"content": "A certain quadratic polynomial is known to have the following properties: its leading coefficient is equal to one, it has integer roots, and its graph (parabola) intersects the line \\( y = 2017 \\) at two points with integer coordinates. Can the ordinate of the vertex of the p... | math | {
"style": "rule",
"ground_truth": "-1016064"
} | {
"split": "train",
"index": 36462,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An inverted frustum with a bottom diameter of 12 and height of 18, filled with water, is emptied into another cylindrical container with a bottom diameter of 24. Assuming the cylindrical container is sufficiently tall, what will be the height of the water level in the cylindric... | math | {
"style": "rule",
"ground_truth": "1.5"
} | {
"split": "train",
"index": 36463,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given 8 teams, of which 3 are weak teams, they are randomly divided into two groups, A and B, with 4 teams in each group. Find:\n\n(1) The probability that one of the groups A or B has exactly 2 weak teams.\n(2) The probability that group A has at least 2 weak teams. Let's thin... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 36464,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all \\( a_{0} \\in \\mathbb{R} \\) such that the sequence defined by\n\\[ a_{n+1} = 2^{n} - 3a_{n}, \\quad n = 0, 1, 2, \\cdots \\]\nis increasing. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"split": "train",
"index": 36465,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two vectors $a$ and $b$ in a plane that are orthogonal to each other, with $|a|=2$ and $|b|=1$. Let $k$ and $t$ be real numbers that are not simultaneously zero.\n\n(1) If $x=a+(t-3)b$ and $y=-ka+tb$ are orthogonal, find the functional relationship $k=f(t)$.\n\n(2) Find t... | math | {
"style": "rule",
"ground_truth": "-\\frac{9}{16}"
} | {
"split": "train",
"index": 36466,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $X$ is a random variable that takes on only nonnegative integer values, with $E\\left[ X \\right] = 1$, $E\\left[ X^2 \\right] = 2$, and $E \\left[ X^3 \\right] = 5$. Determine the smallest possible value of the probability of the event $X=0$. Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 36467,
"solution": "The answer is $\\frac{1}{3}$. Let $a_n = P(X=n)$; we want the minimum value for $a_0$. If we write $S_k = \\sum_{n=1}^\\infty n^k a_n$, then the given expectation values imply that $S_1 = 1$, $S_2 = 2$, $S_3 = 5$. Now define $f(n) = 11n-6n^2+n^3$, and note that $f(... |
deepscaler | [
{
"role": "user",
"content": "Let \\[g(x) =\n\\begin{cases}\n2x - 4 &\\text{if } x < 0, \\\\\n5 - 3x &\\text{if } x \\geq 0.\n\\end{cases}\\]\nFind $g(-2)$ and $g(3)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 36468,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the gradient of the function \\( z = x^2 - xy + y^3 \\) at the point \\( A(1, -1) \\) and the derivative in the direction of the vector \\( \\bar{a} = 3\\bar{i} - 4\\bar{j} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"split": "train",
"index": 36469,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a,$ $b,$ and $c$ be nonnegative real numbers such that $a + b + c = 1.$ Find the maximum value of\n\\[a + \\sqrt{ab} + \\sqrt[3]{abc}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"split": "train",
"index": 36470,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\dbinom{6}{3}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 36471,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ok, let's solve it :\n\nWe know that $f^2(1)+f(1)$ divides $4$ and is greater than $1$, so that it is $2$ or $4$. Solving the quadratic equations in $f(1)$ we easily find that $f(1)=1.$\nIt follows that for each prime $p$ the number $1+f(p-1)$ divides $p^2$ and is greater than ... | math | {
"style": "rule",
"ground_truth": "f(n) = n"
} | {
"split": "train",
"index": 36472,
"solution": "\nLet us find a function \\( f \\) such that the conditions given in the problem statement are satisfied, starting from given hints and systematically addressing each part of the problem.\n\nFirst, we analyze the condition \\( f^2(1) + f(1) \\mid 4 \\) and \\( f^2(... |
deepscaler | [
{
"role": "user",
"content": "Let $\\omega$ be a complex number such that $\\omega^7 = 1$ and $\\omega \\ne 1.$ Compute\n\\[\\omega^{16} + \\omega^{18} + \\omega^{20} + \\dots + \\omega^{54}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 36473,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p,$ $q,$ $r,$ $s$ be distinct real numbers such that the roots of $x^2 - 12px - 13q = 0$ are $r$ and $s,$ and the roots of $x^2 - 12rx - 13s = 0$ are $p$ and $q.$ Find the value of $p + q + r + s.$ Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "2028"
} | {
"split": "train",
"index": 36474,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the greatest solution on the interval $[0 ; 10 \\pi]$ of the equation $|2 \\sin x - 1| + |2 \\cos 2x - 1| = 0$. Round the answer to three significant digits according to rounding rules and enter it in the provided field. Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "27.7"
} | {
"split": "train",
"index": 36475,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A company has 200 employees randomly assigned numbers from 1 to 200, and they are evenly divided into 40 groups. If employee number 22 is selected from the 5th group, find the number of the employee to be selected from the 10th group. Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "47"
} | {
"split": "train",
"index": 36476,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the tetrahedron \\(ABCD\\), \\(CD \\perp BC\\), \\(AB \\perp BC\\), \\(CD = AC\\), \\(AB = BC = 1\\). The dihedral angle between the planes \\(BCD\\) and \\(ABC\\) is \\(45^\\circ\\). Find the distance from point \\(B\\) to the plane \\(ACD\\). Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 36477,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of the function $y=\\sin 2x-\\sqrt{3}\\cos 2x$ can be obtained by shifting the graph of the function $y=\\sin 2x+\\sqrt{3}\\cos 2x$ to the right by $\\frac{\\pi}{3}$ units. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"split": "train",
"index": 36478,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\( f(x) = x^{5} + a_{1} x^{4} + a_{2} x^{3} + a_{3} x^{2} + a_{4} x + a_{5} \\), and \\( f(m) = 2017m \\) for \\( m = 1, 2, 3, 4 \\), find \\( f(10) - f(-5) \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "75615"
} | {
"split": "train",
"index": 36479,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On an island of Liars and Knights, a circular arrangement is called correct if each person in the circle can say that among their two neighbors, there is at least one member of their tribe. One day, 2019 natives formed a correct circle. A liar approached them and said, \"Now we... | math | {
"style": "rule",
"ground_truth": "1346"
} | {
"split": "train",
"index": 36480,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the surface integrals of the first kind:\na) \\(\\iint_{\\sigma}|x| dS\\), where \\(\\sigma\\) is defined by \\(x^2 + y^2 + z^2 = 1\\), \\(z \\geqslant 0\\).\n\nb) \\(\\iint_{\\sigma} (x^2 + y^2) dS\\), where \\(\\sigma\\) is defined by \\(x^2 + y^2 = 2z\\), \\(z = 1\... | math | {
"style": "rule",
"ground_truth": "3\\sqrt{2} \\pi"
} | {
"split": "train",
"index": 36481,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $f(x)= \\frac{a\\cdot 2^{x}+a-2}{2^{x}+1}$ ($x\\in\\mathbb{R}$), if $f(x)$ satisfies $f(-x)+f(x)=0$,\n(1) find the value of the real number $a$ and $f(3)$;\n(2) determine the monotonicity of the function and provide a proof. Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{9}"
} | {
"split": "train",
"index": 36482,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $z_1$, $z_2$, $z_3$, $\\dots$, $z_{16}$ be the 16 zeros of the polynomial $z^{16} - 16^{4}$. For each $j$, let $w_j$ be one of $z_j$ or $iz_j$. Find the maximum possible value of the real part of\n\\[\\sum_{j = 1}^{16} w_j.\\] Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 36483,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Matrices $A$ , $B$ are given as follows.\n\\[A=\\begin{pmatrix} 2 & 1 & 0 1 & 2 & 0 0 & 0 & 3 \\end{pmatrix}, \\quad B = \\begin{pmatrix} 4 & 2 & 0 2 & 4 & 0 0 & 0 & 12\\end{pmatrix}\\]\nFind volume of $V=\\{\\mathbf{x}\\in\\mathbb{R}^3 : \\mathbf{x}\\cdot A\\mathbf{x} ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"split": "train",
"index": 36484,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose\n\\[\\frac{1}{x^3-7x^2+11x+15} = \\frac{A}{x-5} + \\frac{B}{x+3} + \\frac{C}{(x+3)^2}\\]\nwhere $A$, $B$, and $C$ are real constants. What is $A$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{64}"
} | {
"split": "train",
"index": 36485,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The total number of toothpicks used to build a rectangular grid 15 toothpicks high and 12 toothpicks wide, with internal diagonal toothpicks, is calculated by finding the sum of the toothpicks. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "567"
} | {
"split": "train",
"index": 36486,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The expressions \\[A=1 \\times 2 + 3 \\times 4 + 5 \\times 6 + \\cdots + 37 \\times 38 + 39\\]and \\[B = 1 + 2 \\times 3 + 4 \\times 5 + \\cdots + 36 \\times 37 + 38 \\times 39\\]are obtained by writing multiplication and addition operators in an alternating pattern between suc... | math | {
"style": "rule",
"ground_truth": "722"
} | {
"split": "train",
"index": 36487,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let P_{1}, P_{2}, \\ldots, P_{6} be points in the complex plane, which are also roots of the equation x^{6}+6 x^{3}-216=0. Given that P_{1} P_{2} P_{3} P_{4} P_{5} P_{6} is a convex hexagon, determine the area of this hexagon. Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "9 \\sqrt{3}"
} | {
"split": "train",
"index": 36488,
"solution": "Factor x^{6}+6 x^{3}-216=\\left(x^{3}-12\\right)\\left(x^{3}+18\\right). This gives us 6 points equally spaced in terms of their angles from the origin, alternating in magnitude between \\sqrt[3]{12} and \\sqrt[3]{18}. This means our hexagon is composed of 6 triang... |
deepscaler | [
{
"role": "user",
"content": "In the trapezoid $ABCD$, $CD$ is three times the length of $AB$. Given that the area of the trapezoid is $30$ square units, determine the area of $\\triangle ABC$.\n\n[asy]\ndraw((0,0)--(1,4)--(7,4)--(12,0)--cycle);\ndraw((1,4)--(0,0));\nlabel(\"$A$\",(1,4),NW);\nlabel(\"$B$\",... | math | {
"style": "rule",
"ground_truth": "7.5"
} | {
"split": "train",
"index": 36489,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Amy rolls six fair 8-sided dice, each numbered from 1 to 8. What is the probability that exactly three of the dice show a prime number and at least one die shows an 8? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2899900}{16777216}"
} | {
"split": "train",
"index": 36490,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "2000 people are sitting around a round table. Each one of them is either a truth-sayer (who always tells the truth) or a liar (who always lies). Each person said: \"At least two of the three people next to me to the right are liars\". How many truth-sayers are there in the circ... | math | {
"style": "rule",
"ground_truth": "666"
} | {
"split": "train",
"index": 36491,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Out of the 200 natural numbers between 1 and 200, how many numbers must be selected to ensure that there are at least 2 numbers among them whose product equals 238? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "198"
} | {
"split": "train",
"index": 36492,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( a_1, a_2, \\ldots, a_{2020} \\) be the roots of the polynomial\n\\[ x^{2020} + x^{2019} + \\cdots + x^2 + x - 2022 = 0. \\]\nCompute\n\\[ \\sum_{n = 1}^{2020} \\frac{1}{1 - a_n}. \\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2041210"
} | {
"split": "train",
"index": 36493,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of the digits of the number $\\underbrace{44 \\ldots 4}_{2012 \\text { times}} \\cdot \\underbrace{99 \\ldots 9}_{2012 \\text { times}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18108"
} | {
"split": "train",
"index": 36494,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Seven cards numbered $1$ through $7$ are to be lined up in a row. Find the number of arrangements of these seven cards where one of the cards can be removed, leaving the remaining six cards in either ascending or descending order. Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "74"
} | {
"split": "train",
"index": 36495,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the angle between vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ is $60^{\\circ}$, $\\overrightarrow{a}=(2,)$, $|\\overrightarrow{b}|=1$, calculate $|\\overrightarrow{a}+2\\overrightarrow{b}|$. Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"split": "train",
"index": 36496,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When Joyce counts the pennies in her bank by fives, she has one left over. When she counts them by threes, there are two left over. What is the least possible number of pennies in the bank? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 36497,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the tangent line of the curve $y=\\ln x$ at point $P(x_{1}, y_{1})$ is tangent to the curve $y=e^{x}$ at point $Q(x_{2}, y_{2})$, then $\\frac{2}{{x_1}-1}+x_{2}=$____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 36498,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Select 4 out of 6 sprinters to participate in a 4×100 relay race. If neither A nor B runs the first leg, then there are $\\boxed{\\text{different}}$ possible team compositions. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 36499,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of the digits of the greatest prime number that is a divisor of $16,385$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 36500,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n>5$ be an integer. There are $n$ points in the plane, no three of them collinear. Each day, Tom erases one of the points, until there are three points left. On the $i$-th day, for $1<i<n-3$, before erasing that day's point, Tom writes down the positive integer $v(i)$ such... | math | {
"style": "rule",
"ground_truth": "2n - 8"
} | {
"split": "train",
"index": 36501,
"solution": "\nGiven an integer \\( n > 5 \\), there are \\( n \\) points in the plane with no three collinear. Tom sequentially erases a point each day until only three points remain. On the \\( i \\)-th day (\\( 1 < i < n-3 \\)), he notes a positive integer \\( v(i) \\) repre... |
deepscaler | [
{
"role": "user",
"content": "Given three composite numbers \\( A, B, \\) and \\( C \\) which are pairwise coprime, and \\( A \\times B \\times C = 11011 \\times 28 \\). What is the maximum value of \\( A + B + C \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1626"
} | {
"split": "train",
"index": 36502,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "each of the squares in a 2 x 2018 grid of squares is to be coloured black or white such that in any 2 x 2 block , at least one of the 4 squares is white. let P be the number of ways of colouring the grid. find the largest k so that $3^k$ divides P. Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "1009"
} | {
"split": "train",
"index": 36503,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest $n$ for which the numbers $1,2, \\ldots, 14$ can be colored in red and blue so that for any number $k=1,2, \\ldots, n$, there are a pair of blue numbers and a pair of red numbers, each pair having a difference equal to $k$? Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 36504,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many different routes can Samantha take by biking on streets to the southwest corner of City Park, then taking a diagonal path through the park to the northeast corner, and then biking on streets to school? Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "400"
} | {
"split": "train",
"index": 36505,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Qiang drives $15$ miles at an average speed of $30$ miles per hour. How many additional miles will he have to drive at $55$ miles per hour to average $50$ miles per hour for the entire trip? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "110"
} | {
"split": "train",
"index": 36506,
"solution": "1. **Determine the time for the first part of the trip**: Qiang drives 15 miles at an average speed of 30 miles per hour. The time taken for this part of the trip is calculated by the formula:\n \\[\n \\text{Time} = \\frac{\\text{Distance}}{\\text{Speed}} = \\f... |
deepscaler | [
{
"role": "user",
"content": "A point \\(A_{1}\\) is taken on the side \\(AC\\) of triangle \\(ABC\\), and a point \\(C_{1}\\) is taken on the extension of side \\(BC\\) beyond point \\(C\\). The length of segment \\(A_{1}C\\) is 85% of the length of side \\(AC\\), and the length of segment \\(BC_{1}\\) is ... | math | {
"style": "rule",
"ground_truth": "102"
} | {
"split": "train",
"index": 36507,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a group with the numbers $-3, 0, 5, 8, 11, 13$, and the following rules: the largest isn't first, and it must be within the first four places, the smallest isn't last, and it must be within the last four places, and the median isn't in the first or last position, determin... | math | {
"style": "rule",
"ground_truth": "5.5"
} | {
"split": "train",
"index": 36508,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate plane $xOy$, the parametric equations of the curve $C_1$ are given by $$\\begin{cases} x=2\\cos\\phi \\\\ y=2\\sin\\phi \\end{cases}$$ where $\\phi$ is the parameter. By shrinking the abscissa of points on curve $C_1$ to $\\frac{1}{2}$ of the origina... | math | {
"style": "rule",
"ground_truth": "\\frac{60}{19}"
} | {
"split": "train",
"index": 36509,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of the function $f(x)=3 \\sqrt {2}\\cos (x+\\varphi)+\\sin x$, where $x\\in \\mathbb{R}$ and $\\varphi\\in\\left(- \\frac {\\pi}{2}, \\frac {\\pi}{2}\\right)$, passes through the point $\\left( \\frac {\\pi}{2},4\\right)$. Find the minimum value of $f(x)$. Let's think... | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"split": "train",
"index": 36510,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find a unit vector that is orthogonal to both $\\begin{pmatrix} 1 \\\\ 1 \\\\ 0 \\end{pmatrix}$ and $\\begin{pmatrix} 1 \\\\ 0 \\\\ 2 \\end{pmatrix}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} -2/3 \\\\ 2/3 \\\\ 1/3 \\end{pmatrix}"
} | {
"split": "train",
"index": 36511,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ is isosceles, with $AB=AC$ and altitude $AM=11.$ Suppose that there is a point $D$ on $\\overline{AM}$ with $AD=10$ and $\\angle BDC=3\\angle BAC.$ Then the perimeter of $\\triangle ABC$ may be written in the form $a+\\sqrt{b},$ where $a$ and $b$ are integers. Fi... | math | {
"style": "rule",
"ground_truth": "616"
} | {
"split": "train",
"index": 36512,
"solution": "Let $x=\\angle CAM$, so $3x=\\angle CDM$. Then, $\\frac{\\tan 3x}{\\tan x}=\\frac{CM/1}{CM/11}=11$. Expanding $\\tan 3x$ using the angle sum identity gives \\[\\tan 3x=\\tan(2x+x)=\\frac{3\\tan x-\\tan^3x}{1-3\\tan^2x}.\\] Thus, $\\frac{3-\\tan^2x}{1-3\\tan^2x}=11$... |
deepscaler | [
{
"role": "user",
"content": "Say that a complex number $z$ is three-presentable if there is a complex number $w$ of absolute value $3$ such that $z = w - \\frac{1}{w}$. Let $T$ be the set of all three-presentable complex numbers. The set $T$ forms a closed curve in the complex plane. What is the area in... | math | {
"style": "rule",
"ground_truth": "\\frac{80}{9}\\pi"
} | {
"split": "train",
"index": 36513,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ 2^{1110} \\equiv n \\bmod{1111} $ with $ 0 \\leq n < 1111 $ . Compute $ n $ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1024"
} | {
"split": "train",
"index": 36514,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The tower function of twos is defined recursively as follows: $T(1) = 2$ and $T(n + 1) = 2^{T(n)}$ for $n\\ge1$. Let $A = (T(2009))^{T(2009)}$ and $B = (T(2009))^A$. What is the largest integer $k$ for which $\\underbrace{\\log_2\\log_2\\log_2\\ldots\\log_2B}_{k\\text{ times}}$... | math | {
"style": "rule",
"ground_truth": "2010"
} | {
"split": "train",
"index": 36515,
"solution": "To solve this problem, we need to understand the recursive definition of the tower function $T(n)$ and how logarithms interact with powers and products. We start by analyzing the expression for $B$ and then determine how many times we can apply the logarithm base 2... |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $b=2$.\n\n(1) If angles $A$, $B$, $C$ form an arithmetic progression, find the radius of the circumcircle of $\\triangle ABC$.\n(2) If sides $a$, $b$, $c$ form an arithmetic prog... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 36516,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A grocer creates a display of cans where the top row contains two cans and each subsequent lower row has three more cans than the row preceding it. If the total number of cans used in the display is 120, how many rows are there in the display? Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "n = 9"
} | {
"split": "train",
"index": 36517,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest integer \\( n \\) such that the expanded form of \\( (xy - 7x - 3y + 21)^n \\) has 2012 terms. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 36518,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Warehouse A and Warehouse B originally stored whole bags of grain. If 90 bags are transferred from Warehouse A to Warehouse B, then the grain in Warehouse B will be twice that in Warehouse A. If a certain number of bags are transferred from Warehouse B to Warehouse A, then the ... | math | {
"style": "rule",
"ground_truth": "153"
} | {
"split": "train",
"index": 36519,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For all composite integers $n$, what is the largest integer that always divides into the difference between $n^4 - n^2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 36520,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three real numbers $a, b,$ and $c$ have a sum of 114 and a product of 46656. If $b=ar$ and $c=ar^2$ for some real number $r$, what is the value of $a+c$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "78"
} | {
"split": "train",
"index": 36521,
"solution": "Since $b=ar, c=ar^2$, and the product of $a, b,$ and $c$ is 46656, then $a(ar)(ar^2)=46656$ or $a^3r^3=46656$ or $(ar)^3=46656$ or $ar=\\sqrt[3]{46656}=36$. Therefore, $b=ar=36$. Since the sum of $a, b,$ and $c$ is 114, then $a+c=114-b=114-36=78$."
} |
deepscaler | [
{
"role": "user",
"content": "On a blackboard lies $50$ magnets in a line numbered from $1$ to $50$ , with different magnets containing different numbers. David walks up to the blackboard and rearranges the magnets into some arbitrary order. He then writes underneath each pair of consecutive magnets ... | math | {
"style": "rule",
"ground_truth": "4925"
} | {
"split": "train",
"index": 36522,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A trapezoid \\(ABCD\\) is inscribed in a circle, with bases \\(AB = 1\\) and \\(DC = 2\\). Let \\(F\\) denote the intersection point of the diagonals of this trapezoid. Find the ratio of the sum of the areas of triangles \\(ABF\\) and \\(CDF\\) to the sum of the areas of triang... | math | {
"style": "rule",
"ground_truth": "5/4"
} | {
"split": "train",
"index": 36523,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a regular quadrilateral pyramid $S-ABCD$ with lateral edge length of $4$ and $\\angle ASB = 30^\\circ$, points $E$, $F$, and $G$ are taken on lateral edges $SB$, $SC$, and $SD$ respectively. Find the minimum value of the perimeter of the spatial quadrilateral $AEFG$. Let'... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{3}"
} | {
"split": "train",
"index": 36524,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the product of the roots of the equation \\[(2x^3 + x^2 - 8x + 20)(5x^3 - 25x^2 + 19) = 0.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "38"
} | {
"split": "train",
"index": 36525,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integers $k$ are there such that $$\\frac{k}{2013}(a+b)=\\operatorname{lcm}(a, b)$$ has a solution in positive integers $(a, b)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1006"
} | {
"split": "train",
"index": 36526,
"solution": "First, we can let $h=\\operatorname{gcd}(a, b)$ so that $(a, b)=(h A, h B)$ where $\\operatorname{gcd}(A, B)=1$. Making these substitutions yields $\\frac{k}{2013}(h A+h B)=h A B$, so $k=\\frac{2013 A B}{A+B}$. Because $A$ and $B$ are relatively prime, $A+B$ shares... |
deepscaler | [
{
"role": "user",
"content": "If $a*b=a^2+ab-b^2$, find $3*2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 36527,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Tetrahedron $A B C D$ with volume 1 is inscribed in circumsphere $\\omega$ such that $A B=A C=A D=2$ and $B C \\cdot C D \\cdot D B=16$. Find the radius of $\\omega$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{3}"
} | {
"split": "train",
"index": 36528,
"solution": "Let $X$ be the foot of the perpendicular from $A$ to $\\triangle B C D$. Since $A B=A C=A D$, it follows that $X$ is the circumcenter of $\\triangle B C D$. Denote $X B=X C=X D=r$. By the Pythagorean Theorem on $\\triangle A B X$, we have $A X=\\sqrt{4-r^{2}}$. Now... |
deepscaler | [
{
"role": "user",
"content": "The volumes of a regular tetrahedron and a regular octahedron have equal edge lengths. Find the ratio of their volumes without calculating the volume of each polyhedron. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 36529,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\\times 5$ square array of dots? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "225"
} | {
"split": "train",
"index": 36530,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the volume of tetrahedron $ABCD$ with edge lengths $AB = 2$, $AC = 3$, $AD = 4$, $BC = \\sqrt{13}$, $BD = 2\\sqrt{5}$, and $CD = 5$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 36531,
"solution": "1. **Positioning the Tetrahedron in the Coordinate System**:\n We place tetrahedron $ABCD$ in the $xyz$-plane. Assume $A=(0,0,0)$, $C=(3,0,0)$, and $D=(0,4,0)$. This positioning is based on the edge lengths $AC = 3$ and $AD = 4$. We also know that $CD = 5$, and b... |
deepscaler | [
{
"role": "user",
"content": "Given positive integers $n$ and $k$, $n > k^2 >4.$ In a $n \\times n$ grid, a $k$[i]-group[/i] is a set of $k$ unit squares lying in different rows and different columns.\nDetermine the maximal possible $N$, such that one can choose $N$ unit squares in the grid and color them, ... | math | {
"style": "rule",
"ground_truth": "n(k-1)^2"
} | {
"split": "train",
"index": 36532,
"solution": "\nGiven positive integers \\( n \\) and \\( k \\) such that \\( n > k^2 > 4 \\), we aim to determine the maximal possible \\( N \\) such that one can choose \\( N \\) unit squares in an \\( n \\times n \\) grid and color them, with the condition that in any \\( k \... |
deepscaler | [
{
"role": "user",
"content": "In a given area, there are 10 famous tourist attractions, of which 8 are for daytime visits and 2 are for nighttime visits. A tour group wants to select 5 from these 10 spots for a two-day tour. The itinerary is arranged with one spot in the morning, one in the afternoon, and o... | math | {
"style": "rule",
"ground_truth": "2352"
} | {
"split": "train",
"index": 36533,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A box contains 6 cards numbered 1, 2, ..., 6. A card is randomly drawn from the box, and its number is denoted as $a$. The box is then adjusted to retain only the cards with numbers greater than $a$. A second draw is made, and the probability that the first draw is an odd numbe... | math | {
"style": "rule",
"ground_truth": "\\frac{17}{45}"
} | {
"split": "train",
"index": 36534,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The line $y = \\frac{5}{3} x - \\frac{17}{3}$ is to be parameterized using vectors. Which of the following options are valid parameterizations?\n\n(A) $\\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\begin{pmatrix} 4 \\\\ 1 \\end{pmatrix} + t \\begin{pmatrix} -3 \\\\ -5 \\end{pma... | math | {
"style": "rule",
"ground_truth": "\\text{A,C}"
} | {
"split": "train",
"index": 36535,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the base 4 representation of the base 2 number $11011000_2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3120_4"
} | {
"split": "train",
"index": 36536,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The height of a right triangle $ABC$ dropped to the hypotenuse is 9.6. From the vertex $C$ of the right angle, a perpendicular $CM$ is raised to the plane of the triangle $ABC$, with $CM = 28$. Find the distance from point $M$ to the hypotenuse $AB$. Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "29.6"
} | {
"split": "train",
"index": 36537,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The first and thirteenth terms of an arithmetic sequence are 5 and 29, respectively. What is the fiftieth term? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "103"
} | {
"split": "train",
"index": 36538,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "During a break between voyages, a sailor turned 20 years old. All six crew members gathered in the cabin to celebrate. \"I am twice the age of the cabin boy and 6 years older than the engineer,\" said the helmsman. \"And I am as much older than the cabin boy as I am younger tha... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 36539,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sum of the first n terms of a geometric sequence {a_n} (where all terms are real numbers) is S_n, if S_10=10 and S_30=70, determine the value of S_40. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 36540,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At a conference there are $n$ mathematicians. Each of them knows exactly $k$ fellow mathematicians. Find the smallest value of $k$ such that there are at least three mathematicians that are acquainted each with the other two.\n\n[color=#BF0000]Rewording of the last line for cla... | math | {
"style": "rule",
"ground_truth": "\\left\\lfloor \\frac{n}{2} \\right\\rfloor +1"
} | {
"split": "train",
"index": 36541,
"solution": "\nLet \\( n \\) be the number of mathematicians at the conference. We are tasked with finding the smallest value of \\( k \\) such that there are at least three mathematicians, say \\( X, Y, \\) and \\( Z \\), who are all acquainted with each other (i.e., each know... |
deepscaler | [
{
"role": "user",
"content": "Compute $\\dbinom{8}{4}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "70"
} | {
"split": "train",
"index": 36542,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\sin\\alpha + \\sin\\beta = \\frac{1}{3}$, find the maximum and minimum values of $y = \\sin\\beta - \\cos^2\\alpha$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{11}{12}"
} | {
"split": "train",
"index": 36543,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An arithmetic sequence {a_n} has a sum of the first n terms as S_n, and S_6/S_3 = 4. Find the value of S_9/S_6. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{9}{4}"
} | {
"split": "train",
"index": 36544,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $x$ such that $\\log_{12}3x=2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 36545,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two players, A and B, take turns shooting baskets. The probability of A making a basket on each shot is $\\frac{1}{2}$, while the probability of B making a basket is $\\frac{1}{3}$. The rules are as follows: A goes first, and if A makes a basket, A continues to shoot; otherwise... | math | {
"style": "rule",
"ground_truth": "\\frac{247}{432}"
} | {
"split": "train",
"index": 36546,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$ S$ is a non-empty subset of the set $ \\{ 1, 2, \\cdots, 108 \\}$, satisfying:\r\n\r\n(1) For any two numbers $ a,b \\in S$ ( may not distinct), there exists $ c \\in S$, such that $ \\gcd(a,c)\\equal{}\\gcd(b,c)\\equal{}1$.\r\n\r\n(2) For any two numbers $ a,b \\in S$ ( may ... | math | {
"style": "rule",
"ground_truth": "79"
} | {
"split": "train",
"index": 36547,
"solution": "\nLet \\( S \\) be a non-empty subset of the set \\( \\{ 1, 2, \\ldots, 108 \\} \\) satisfying the following conditions:\n\n1. For any two numbers \\( a, b \\in S \\) (not necessarily distinct), there exists \\( c \\in S \\) such that \\( \\gcd(a, c) = \\gcd(b, c) ... |
deepscaler | [
{
"role": "user",
"content": "In right triangle \\( ABC \\), a point \\( D \\) is on hypotenuse \\( AC \\) such that \\( BD \\perp AC \\). Let \\(\\omega\\) be a circle with center \\( O \\), passing through \\( C \\) and \\( D \\) and tangent to line \\( AB \\) at a point other than \\( B \\). Point \\( X ... | math | {
"style": "rule",
"ground_truth": "8041"
} | {
"split": "train",
"index": 36548,
"solution": ""
} |
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