data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xoy$, the parametric equation of line $l$ is $\\begin{cases} x=1- \\frac { \\sqrt {3}}{2}t \\\\ y= \\frac {1}{2}t\\end{cases}$ (where $t$ is the parameter), and in the polar coordinate system with the origin $O$ as the pole and the positive h... | math | {
"style": "rule",
"ground_truth": "2 \\sqrt {3}"
} | {
"split": "train",
"index": 26149,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A line \\( y = -\\frac{2}{3}x + 6 \\) crosses the \\( x \\)-axis at \\( P \\) and the \\( y \\)-axis at \\( Q \\). Point \\( T(r,s) \\) is on the line segment \\( PQ \\). If the area of \\( \\triangle POQ \\) is four times the area of \\( \\triangle TOP \\), what is the value o... | math | {
"style": "rule",
"ground_truth": "8.25"
} | {
"split": "train",
"index": 26150,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Scenario: In a math activity class, the teacher presented a set of questions and asked the students to explore the pattern by reading the following solution process:\n\n$\\sqrt{1+\\frac{5}{4}}=\\sqrt{\\frac{9}{4}}=\\sqrt{{(\\frac{3}{2})}^{2}}=\\frac{3}{2}$;\n$\\sqrt{1+\\frac{7}... | math | {
"style": "rule",
"ground_truth": "\\frac{1012}{1011}"
} | {
"split": "train",
"index": 26151,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that point $P$ is a moving point on circle $C_{1}$: $\\left(x-1\\right)^{2}+y^{2}=1$, point $Q$ is a moving point on circle $C_{2}$: $\\left(x-4\\right)^{2}+\\left(y-1\\right)^{2}=4$, and point $R$ moves on the line $l: x-y+1=0$, find the minimum value of $|PR|+|QR|$. Let... | math | {
"style": "rule",
"ground_truth": "\\sqrt{26}-3"
} | {
"split": "train",
"index": 26152,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a sequence $\\left\\{ a_n \\right\\}$ satisfying $a_1=4$ and $a_1+a_2+\\cdots +a_n=a_{n+1}$, and $b_n=\\log_{2}a_n$, calculate the value of $\\frac{1}{b_1b_2}+\\frac{1}{b_2b_3}+\\cdots +\\frac{1}{b_{2017}b_{2018}}$. Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "\\frac{3025}{4036}"
} | {
"split": "train",
"index": 26153,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve the congruence $15x + 3 \\equiv 9 \\pmod{21}$ for $x$, and express the solution as $x \\equiv a \\pmod{m}$, where $a < m$ and find $a + m$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 26154,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, \\( Z \\) lies on \\( XY \\) and the three circles have diameters \\( XZ \\), \\( ZY \\), and \\( XY \\). If \\( XZ = 12 \\) and \\( ZY = 8 \\), then the ratio of the area of the shaded region to the area of the unshaded region is Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "12:13"
} | {
"split": "train",
"index": 26155,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two ants, one starting at $(-1,1)$ and the other at $(1,1)$, walk to the right along the parabola $y=x^{2}$ such that their midpoint moves along the line $y=1$ with constant speed 1. When the left ant first hits the line $y=\\frac{1}{2}$, what is its speed? Let's think step by ... | math | {
"style": "rule",
"ground_truth": "3 \\sqrt{3} - 3"
} | {
"split": "train",
"index": 26156,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the greatest common factor of 8! and 9!. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40320"
} | {
"split": "train",
"index": 26157,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of all real numbers $x$ such that $5 x^{4}-10 x^{3}+10 x^{2}-5 x-11=0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 26158,
"solution": "Rearrange the equation to $x^{5}+(1-x)^{5}-12=0$. It's easy to see this has two real roots, and that $r$ is a root if and only if $1-r$ is a root, so the answer must be 1."
} |
deepscaler | [
{
"role": "user",
"content": "Given the assumption that smoking is unrelated to lung disease, calculate the confidence level that can be concluded from the chi-square statistic $K^2=5.231$, with $P(K^2 \\geq 3.841) = 0.05$ and $P(K^2 \\geq 6.635) = 0.01$. Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "95\\%"
} | {
"split": "train",
"index": 26159,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A positive two-digit number is even and is a multiple of 11. The product of its digits is a perfect cube. What is this two-digit number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "88"
} | {
"split": "train",
"index": 26160,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular piece of paper with vertices $A B C D$ is being cut by a pair of scissors. The pair of scissors starts at vertex $A$, and then cuts along the angle bisector of $D A B$ until it reaches another edge of the paper. One of the two resulting pieces of paper has 4 times... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}"
} | {
"split": "train",
"index": 26161,
"solution": "Without loss of generality, let $A B>A D$, and let $x=A D, y=A B$. Let the cut along the angle bisector of $\\angle D A B$ meet $C D$ at $E$. Note that $A D E$ is a $45-45-90$ triangle, so $D E=A D=x$, and $E C=y-x$. Now, $[A D E]=\\frac{x^{2}}{2}$, and $[A E C B]=... |
deepscaler | [
{
"role": "user",
"content": "Compute $\\cos \\left( \\arcsin \\frac{5}{13} \\right).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{12}{13}"
} | {
"split": "train",
"index": 26162,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An electric car is charged 3 times per week for 52 weeks. The cost to charge the car each time is $0.78. What is the total cost to charge the car over these 52 weeks? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\$121.68"
} | {
"split": "train",
"index": 26163,
"solution": "Since the car is charged 3 times per week for 52 weeks, it is charged \\(3 \\times 52 = 156\\) times. Since the cost per charge is $0.78, then the total cost is \\(156 \\times 0.78 = 121.68\\)."
} |
deepscaler | [
{
"role": "user",
"content": "How many positive rational numbers less than \\(\\pi\\) have denominator at most 7 when written in lowest terms? (Integers have denominator 1.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 26164,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A coin is tossed 10 times. Find the probability that no two heads appear consecutively. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9/64"
} | {
"split": "train",
"index": 26165,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a rectangular pan of brownies measuring 15 inches by 24 inches, cut into triangular pieces with a base of 3 inches and a height of 4 inches, determine the number of triangular pieces that can be cut from the pan. Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 26166,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $\\log_{10}{16} + 3\\log_{5}{25} + 4\\log_{10}{2} + \\log_{10}{64} - \\log_{10}{8}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9.311"
} | {
"split": "train",
"index": 26167,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We rolled ten fair six-sided dice at once, and the product of the numbers rolled was 7776. We know that the highest number rolled appeared only once. What is the sum of the numbers rolled? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 26168,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the measure of an angle, in degrees, if its supplement is six times its complement? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "72^{\\circ}"
} | {
"split": "train",
"index": 26169,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest number \\( n > 1980 \\) such that the number\n\n$$\n\\frac{x_{1} + x_{2} + x_{3} + \\ldots + x_{n}}{5}\n$$\n\nis an integer for any given integer values \\( x_{1}, x_{2}, x_{3}, \\ldots, x_{n} \\), none of which is divisible by 5. Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "1985"
} | {
"split": "train",
"index": 26170,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A counter begins at 0 . Then, every second, the counter either increases by 1 or resets back to 0 with equal probability. The expected value of the counter after ten seconds can be written as $\\frac{m}{n}$, where $m, n$ are positive integers and $\\operatorname{gcd}(m, n)=1$. ... | math | {
"style": "rule",
"ground_truth": "103324"
} | {
"split": "train",
"index": 26171,
"solution": "The probability that the counter is equal to $k$ corresponds to the last $k$ seconds all being increases by 1 and the second before that being a reset to 0 , which happens with probability $2^{-k-1}$. The only contradiction to this is when $k=10$ and the counter ge... |
deepscaler | [
{
"role": "user",
"content": "The maximum and minimum values of the function $f(x) = -x^2 + 2x + 3$ in the interval $[-2, 3]$ are ___ and ___, respectively. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"split": "train",
"index": 26172,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "8. Shortening. There is a sequence of 2015 digits. All digits are chosen randomly from the set {0, 9} independently of each other. The following operation is performed on the resulting sequence. If several identical digits go in a row, they are replaced by one such digit. For e... | math | {
"style": "rule",
"ground_truth": "1813.6"
} | {
"split": "train",
"index": 26173,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the distance between the vertices of the hyperbola given by the equation $4x^2 + 16x - 9y^2 + 18y - 23 = 0.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{30}"
} | {
"split": "train",
"index": 26174,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, $a$, $b$, $c$ are the opposite sides of the internal angles $A$, $B$, $C$, respectively, and $\\sin ^{2}A+\\sin A\\sin C+\\sin ^{2}C+\\cos ^{2}B=1$. \n$(1)$ Find the measure of angle $B$; \n$(2)$ If $a=5$, $b=7$, find $\\sin C$. Let's think step ... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{3}}{14}"
} | {
"split": "train",
"index": 26175,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $PQR$, angle $R$ is a right angle, $PR=15$ units, and $Q$ is on a circle centered at $R$. Squares $PQRS$ and $PRUT$ are formed on sides $PQ$ and $PR$, respectively. What is the number of square units in the sum of the areas of the two squares $PQRS$ and $PRUT$? Let'... | math | {
"style": "rule",
"ground_truth": "450"
} | {
"split": "train",
"index": 26176,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve for $x$: $x = \\dfrac{35}{6-\\frac{2}{5}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{25}{4}"
} | {
"split": "train",
"index": 26177,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $A=30^{\\circ}$, $2 \\overrightarrow{AB}\\cdot \\overrightarrow{AC}=3 \\overrightarrow{BC}^{2}$, find the cosine value of the largest angle in $\\triangle ABC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"split": "train",
"index": 26178,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find constants $A,$ $B,$ and $C$ so that\n\\[\\frac{x^2 - 7}{(x - 2)(x - 3)(x - 5)} = \\frac{A}{x - 2} + \\frac{B}{x - 3} + \\frac{C}{x - 5}.\\]Enter the ordered triple $(A,B,C).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(-1,-1,3)"
} | {
"split": "train",
"index": 26179,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that there are 5 cards of the same size and shape, each marked with the numbers 1, 2, 3, 4, and 5 respectively. If two cards are drawn at random, the probability that the larger number on these two cards is 3 is ______. Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "\\dfrac {1}{5}"
} | {
"split": "train",
"index": 26180,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A table $110\\times 110$ is given, we define the distance between two cells $A$ and $B$ as the least quantity of moves to move a chess king from the cell $A$ to cell $B$ . We marked $n$ cells on the table $110\\times 110$ such that the distance between any two cell... | math | {
"style": "rule",
"ground_truth": "6050"
} | {
"split": "train",
"index": 26181,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In Class 5A, a survey was conducted to find out which fruits the students like. It turned out that:\n\n- 13 students like apples,\n- 11 students like plums,\n- 15 students like peaches, and\n- 6 students like melons.\n\nA student can like more than one fruit. Each student who l... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 26182,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of all four-digit numbers in which the digits $0, 4, 5, 9$ are absent. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6479352"
} | {
"split": "train",
"index": 26183,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider three coins where two are fair and a third coin lands on heads with a probability of $\\frac{3}{5}$. Alice flips the three coins, and then Bob flips the same three coins. Let $\\frac{p}{q}$ be the probability that Alice and Bob get the same number of heads, where $p$ a... | math | {
"style": "rule",
"ground_truth": "263"
} | {
"split": "train",
"index": 26184,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let\n\\[f(x) = \\left\\{\n\\begin{array}{cl}\nx + 3 & \\text{if $x < 20$}, \\\\\n2x - 2 & \\text{if $x \\ge 20$}.\n\\end{array}\n\\right.\\]Find $f^{-1}(7) + f^{-1}(46).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 26185,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From the $7$ integers from $2$ to $8$, randomly select $2$ different numbers, and calculate the probability that these $2$ numbers are coprime. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 26186,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The year 2009 has the property that rearranging its digits never results in a smaller four-digit number (numbers do not start with zero). In which year will this property first repeat? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2022"
} | {
"split": "train",
"index": 26187,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the lateral surface of a cone, when unrolled, forms a semicircle with a radius of $2\\sqrt{3}$, and the vertex and the base circle of the cone are on the surface of a sphere O, calculate the volume of sphere O. Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "\\frac{32}{3}\\pi"
} | {
"split": "train",
"index": 26188,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $f(x)= \\begin{cases} \\sin \\frac{\\pi}{3}x, & x\\leqslant 2011, \\\\ f(x-4), & x > 2011, \\end{cases}$ find $f(2012)$. 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": 26189,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points \\( M \\) and \\( N \\) are the midpoints of the sides \\( AC \\) and \\( CB \\) of the isosceles triangle \\( ACB \\). Point \\( L \\) lies on the median \\( BM \\) such that \\( BL : BM = 4 : 9 \\). A circle with center at point \\( L \\) is tangent to the line \\( MN ... | math | {
"style": "rule",
"ground_truth": "2(2 + \\sqrt{13})"
} | {
"split": "train",
"index": 26190,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A clock chimes 2 times at 2 o'clock, taking 2 seconds to complete. How many seconds does it take to chime 12 times at 12 o'clock? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 26191,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A modified deck of cards contains 60 cards consisting of 15 ranks (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King, Prince, and Princess) and 4 suits ($\\spadesuit$, $\\heartsuit$, $\\diamondsuit$, $\\clubsuit$). Hence, each suit now has 15 cards. Four suits are still divide... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{20}"
} | {
"split": "train",
"index": 26192,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We wrote letters to ten of our friends and randomly placed the letters into addressed envelopes. What is the probability that exactly 5 letters will end up with their intended recipients? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.0031"
} | {
"split": "train",
"index": 26193,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Below is a portion of the graph of a function, $y=f(x)$:\n\n[asy]\nimport graph; size(8cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-3.25,xmax=5.25,ymin=-3.25,ymax=4.25;\n\npen cqcqcq=rgb(0.75,0.75,0.75);\n\n/*grid*/ pen gs=li... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 26194,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $P(z)$, $Q(z)$, and $R(z)$ are polynomials with real coefficients, having degrees $2$, $3$, and $6$, respectively, and constant terms $1$, $2$, and $3$, respectively. Let $N$ be the number of distinct complex numbers $z$ that satisfy the equation $P(z) \\cdot Q(z) ... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 26195,
"solution": "1. **Understanding the Problem:**\n Given polynomials $P(z)$, $Q(z)$, and $R(z)$ with specified degrees and constant terms, we need to find the number of distinct complex solutions to the equation $P(z) \\cdot Q(z) = R(z)$.\n\n2. **Analyzing the Degrees:**\n - ... |
deepscaler | [
{
"role": "user",
"content": "In square $ABCD$, point $M$ is the midpoint of side $AB$ and point $N$ is the midpoint of side $BC$. What is the ratio of the area of triangle $AMN$ to the area of square $ABCD$? Express your answer as a common fraction. Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"split": "train",
"index": 26196,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given circle $C$: $(x-2)^2 + y^2 = 4$, and line $l$: $x - \\sqrt{3}y = 0$, the probability that the distance from point $A$ on circle $C$ to line $l$ is not greater than $1$ is $\\_\\_\\_\\_\\_\\_\\_.$ Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 26197,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simply the expression\n\\[\\frac{(\\sqrt{2} - 1)^{1 - \\sqrt{3}}}{(\\sqrt{2} + 1)^{1 + \\sqrt{3}}},\\]writing your answer as $a - b \\sqrt{c},$ where $a,$ $b,$ and $c$ are positive integers, and $c$ is not divisible by the square of a prime. Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "3 - 2 \\sqrt{2}"
} | {
"split": "train",
"index": 26198,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x$ is real and $4y^2+4xy+x+6=0$, then the complete set of values of $x$ for which $y$ is real, is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "x \\le -2$ or $x \\ge 3"
} | {
"split": "train",
"index": 26199,
"solution": "1. **Identify the type of equation**: The given equation $4y^2 + 4xy + x + 6 = 0$ is a quadratic equation in terms of $y$. The standard form of a quadratic equation is $ay^2 + by + c = 0$.\n\n2. **Coefficients of the quadratic equation**: Here, $a = 4$, $b = 4x$, a... |
deepscaler | [
{
"role": "user",
"content": "Given that a full circle is 800 clerts on Venus and is 360 degrees, calculate the number of clerts in an angle of 60 degrees. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "133.\\overline{3}"
} | {
"split": "train",
"index": 26200,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many integers $n$ are there such that $3 \\leq n \\leq 10$ and $121_n$ (the number written as $121$ in base $n$) is a perfect square? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 26201,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Azar and Carl play a game of tic-tac-toe. Azar places an \\(X\\) in one of the boxes in a \\(3\\)-by-\\(3\\) array of boxes, then Carl places an \\(O\\) in one of the remaining boxes. After that, Azar places an \\(X\\) in one of the remaining boxes, and so on until all boxes ar... | math | {
"style": "rule",
"ground_truth": "148"
} | {
"split": "train",
"index": 26202,
"solution": "To solve this problem, we need to count the number of ways the tic-tac-toe board can be filled such that Carl wins by placing his third $O$ in a winning position, and there are exactly 3 $X$s and 3 $O$s on the board. We will consider two cases based on the arrangem... |
deepscaler | [
{
"role": "user",
"content": "The area of the triangle formed by the tangent to the curve $y=\\ln(x)-2x$ at the point $(1, -2)$ and the coordinate axes. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 26203,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\sin \\alpha$ is a root of the equation $5x^2 - 7x - 6 = 0$, find the value of $\\frac{\\sin \\left(\\alpha - \\frac{\\pi}{2}\\right)\\sin^2\\left(-\\alpha + \\pi\\right)\\sin \\left( \\frac{3\\pi}{2}-\\alpha\\right)\\tan \\left(\\alpha - \\pi\\right)}{\\sin \\left... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"split": "train",
"index": 26204,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Xiao Ming and Xiao Hua are counting picture cards in a box together. Xiao Ming is faster, being able to count 6 cards in the same time it takes Xiao Hua to count 4 cards. When Xiao Hua reached 48 cards, he forgot how many cards he had counted and had to start over. When he coun... | math | {
"style": "rule",
"ground_truth": "169"
} | {
"split": "train",
"index": 26205,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of the function $y=\\sin (2x+\\varphi) (0 < \\varphi < \\pi)$ is shifted to the right by $\\frac{\\pi}{8}$ and then is symmetric about the $y$-axis. Determine the possible value(s) of $\\varphi$. Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\pi}{4}"
} | {
"split": "train",
"index": 26206,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "David has a unit triangular array of 10 points, 4 on each side. A looping path is a sequence $A_{1}, A_{2}, \\ldots, A_{10}$ containing each of the 10 points exactly once, such that $A_{i}$ and $A_{i+1}$ are adjacent (exactly 1 unit apart) for $i=1,2, \\ldots, 10$. (Here $A_{11... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 26207,
"solution": "There are $10 \\cdot 2$ times as many loop sequences as loops. To count the number of loops, first focus on the three corners of the array: their edges are uniquely determined. It's now easy to see there are 3 loops (they form \" $V$-shapes\"), so the answer is $10... |
deepscaler | [
{
"role": "user",
"content": "A polynomial $P$ of degree 2015 satisfies the equation $P(n)=\\frac{1}{n^{2}}$ for $n=1,2, \\ldots, 2016$. Find \\lfloor 2017 P(2017)\\rfloor. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-9"
} | {
"split": "train",
"index": 26208,
"solution": "Let $Q(x)=x^{2} P(x)-1$. Then $Q(n)=n^{2} P(n)-1=0$ for $n=1,2, \\ldots, 2016$, and $Q$ has degree 2017 . Thus we may write $$Q(x)=x^{2} P(x)-1=(x-1)(x-2) \\ldots(x-2016) L(x)$$ where $L(x)$ is some linear polynomial. Then $Q(0)=-1=(-1)(-2) \\ldots(-2016) L(0)$, so... |
deepscaler | [
{
"role": "user",
"content": "Equilateral $\\triangle ABC$ has side length $2$, and shapes $ABDE$, $BCHT$, $CAFG$ are formed outside the triangle such that $ABDE$ and $CAFG$ are squares, and $BCHT$ is an equilateral triangle. What is the area of the geometric shape formed by $DEFGHT$?\nA) $3\\sqrt{3} - 1$\n... | math | {
"style": "rule",
"ground_truth": "3\\sqrt{3} - 2"
} | {
"split": "train",
"index": 26209,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A survey of $120$ teachers determined the following:\n\n$70$ had high blood pressure\n\n$40$ had heart trouble\n\n$20$ had both high blood pressure and heart trouble\n\nWhat percent of the teachers surveyed had neither high blood pressure nor heart trouble? Let's think step by ... | math | {
"style": "rule",
"ground_truth": "25\\%"
} | {
"split": "train",
"index": 26210,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the numbers $1, 2, \\cdots, 20$, calculate the probability that three randomly selected numbers form an arithmetic sequence. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{38}"
} | {
"split": "train",
"index": 26211,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graphs of $2y + x + 3 = 0$ and $3y + ax + 2 = 0$ are perpendicular. Solve for $a.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"split": "train",
"index": 26212,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the equation of the ellipse $\\frac{x^2}{a^2}+\\frac{y^2}{b^2}=1$ with $a > b > 0$ and eccentricity $e = \\frac{\\sqrt{5}-1}{2}$, find the product of the slopes of lines $PA$ and $PB$ for a point $P$ on the ellipse that is not the left or right vertex. Let's think step by... | math | {
"style": "rule",
"ground_truth": "\\frac{1-\\sqrt{5}}{2}"
} | {
"split": "train",
"index": 26213,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the constraints \\(x + 2y \\leq 5\\), \\(2x + y \\leq 4\\), \\(x \\geq 0\\), and \\(y \\geq 0\\), find the coordinates \\((x, y)\\) where \\(3x + 4y\\) achieves its maximum value, and determine that maximum value. Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 26214,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many numbers between $1$ and $2500$ are integer multiples of $4$ or $5$ but not $15$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "959"
} | {
"split": "train",
"index": 26215,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $\\frac{{1+\\cos{20}°}}{{2\\sin{20}°}}-\\sin{10°}\\left(\\frac{1}{{\\tan{5°}}}-\\tan{5°}\\right)=\\_\\_\\_\\_\\_\\_$. 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": 26216,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sarah places four ounces of coffee into an eight-ounce cup and four ounces of cream into a second cup of the same size. She then pours half the coffee from the first cup to the second and, after stirring thoroughly, pours half the liquid in the second cup back to the first. Wha... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"split": "train",
"index": 26217,
"solution": "We will analyze the problem step by step to determine the fraction of cream in the first cup after the described operations.\n\n#### Initial Setup:\n- **Cup 1:** 4 ounces of coffee.\n- **Cup 2:** 4 ounces of cream.\n\n#### Step 1: Pour half the coffee from Cup 1 to... |
deepscaler | [
{
"role": "user",
"content": "Consider the following data from a new season graph, showing the number of home runs hit in April by the top hitters in the baseball league:\n- 5 players hit 6 home runs each.\n- 6 players hit 8 home runs each.\n- 4 players hit 10 home runs each.\n\nCalculate the mean number of... | math | {
"style": "rule",
"ground_truth": "\\frac{118}{15}"
} | {
"split": "train",
"index": 26218,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that Jessica uses 150 grams of lemon juice and 100 grams of sugar, and there are 30 calories in 100 grams of lemon juice and 386 calories in 100 grams of sugar, and water contains no calories, compute the total number of calories in 300 grams of her lemonade. Let's think ... | math | {
"style": "rule",
"ground_truth": "152.1"
} | {
"split": "train",
"index": 26219,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that point $M(x\\_0, y\\_0)$ moves on the circle $x^{2}+y^{2}=4$, $N(4,0)$, and point $P(x,y)$ is the midpoint of segment $MN$.\n\n(1) Find the trajectory equation of point $P(x,y)$;\n\n(2) Find the maximum and minimum distances from point $P(x,y)$ to the line $3x+4y-86=0... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 26220,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two tangents are drawn to a circle from an exterior point $A$; they touch the circle at points $B$ and $C$ respectively. \nA third tangent intersects segment $AB$ in $P$ and $AC$ in $R$, and touches the circle at $Q$. If $AB=20$, then the perimeter of $\\triangle APR$ is Let's ... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 26221,
"solution": "1. **Identify Tangent Properties**: From the problem, we know that $AB$ and $AC$ are tangents from point $A$ to the circle, touching the circle at points $B$ and $C$ respectively. By the tangent properties, $AB = AC$.\n\n2. **Tangent Lengths**: Since $AB = 20$, we ... |
deepscaler | [
{
"role": "user",
"content": "An eight-sided die numbered from 1 to 8 is rolled, and $Q$ is the product of the seven numbers that are visible. What is the largest number that is certain to divide $Q$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "192"
} | {
"split": "train",
"index": 26222,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nEach segment whose ends are vertices of a regular 100-sided polygon is colored - in red if there are an even number of vertices between its ends, and in blue otherwise (in particular, all sides of the 100-sided polygon are red). Numbers are placed at the vertices, the sum of ... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 26223,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The opposite of the real number $2023$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2023"
} | {
"split": "train",
"index": 26224,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABC$ be an equilateral triangle, and let $D$ and $F$ be points on sides $BC$ and $AB$, respectively, with $FA = 5$ and $CD = 2$. Point $E$ lies on side $CA$ such that angle $DEF = 60^{\\circ}$. The area of triangle $DEF$ is $14\\sqrt{3}$. The two possible values of the len... | math | {
"style": "rule",
"ground_truth": "989"
} | {
"split": "train",
"index": 26225,
"solution": "First of all, assume $EC=x,BD=m, ED=a, EF=b$, then we can find $BF=m-3, AE=2+m-x$ It is not hard to find $ab*sin60^{\\circ}*\\frac{1}{2}=14\\sqrt{3}, ab=56$, we apply LOC on $\\triangle{DEF}, \\triangle{BFD}$, getting that $(m-3)^2+m^2-m(m-3)=a^2+b^2-ab$, leads to ... |
deepscaler | [
{
"role": "user",
"content": "How many triangles with positive area can be formed where each vertex is at point $(i,j)$ in the coordinate grid, with integers $i$ and $j$ ranging from $1$ to $4$ inclusive? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "516"
} | {
"split": "train",
"index": 26226,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a board game played with dice, our piece is four spaces away from the finish line. If we roll at least a four, we reach the finish line. If we roll a three, we are guaranteed to finish in the next roll.\n\nWhat is the probability that we will reach the finish line in more th... | math | {
"style": "rule",
"ground_truth": "1/12"
} | {
"split": "train",
"index": 26227,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three circles have the same center O. Point X divides segment OP in the ratio 1:3, with X being closer to O. Calculate the ratio of the area of the circle with radius OX to the area of the circle with radius OP. Next, find the area ratio of a third circle with radius 2*OX to th... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 26228,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a circle, an inscribed hexagon has three consecutive sides each of length 3, and the other three sides each of length 5. A chord of the circle splits the hexagon into two quadrilaterals: one quadrilateral has three sides each of length 3, and the other quadrilateral has thre... | math | {
"style": "rule",
"ground_truth": "409"
} | {
"split": "train",
"index": 26229,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the six whole numbers 10-15, compute the largest possible value for the sum, S, of the three numbers on each side of the triangle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "39"
} | {
"split": "train",
"index": 26230,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many non- empty subsets $S$ of $\\{1,2,3,\\ldots ,15\\}$ have the following two properties?\n$(1)$ No two consecutive integers belong to $S$.\n$(2)$ If $S$ contains $k$ elements, then $S$ contains no number less than $k$.\n$\\mathrm{(A) \\ } 277\\qquad \\mathrm{(B) \\ } 311... | math | {
"style": "rule",
"ground_truth": "405"
} | {
"split": "train",
"index": 26231,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The decimal representation of $m/n,$ where $m$ and $n$ are relatively prime positive integers and $m < n,$ contains the digits $2, 5$, and $1$ consecutively, and in that order. Find the smallest value of $n$ for which this is possible. Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "127"
} | {
"split": "train",
"index": 26232,
"solution": "To find the smallest value of $n$, we consider when the first three digits after the decimal point are $0.251\\ldots$.\n Otherwise, suppose the number is in the form of $\\frac{m}{n} = 0.X251 \\ldots$, where $X$ is a string of $k$ digits and $n$ is small as possibl... |
deepscaler | [
{
"role": "user",
"content": "Compute $\\arccos (\\cos 3).$ All functions are in radians. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3 - 2\\pi"
} | {
"split": "train",
"index": 26233,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $W,X,Y$ and $Z$ are four different digits selected from the set ${1,2,3,4,5,6,7,8,9}$, find the value of $\\frac{W}{X} + \\frac{Y}{Z}$ when it is as small as possible. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{25}{72}"
} | {
"split": "train",
"index": 26234,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cuckoo clock chimes \"cuckoo\" on the hour, with the number of \"cuckoo\" calls equal to the hour indicated by the hour hand (e.g., at 19:00, it chimes 7 times). One morning, Maxim approached the clock when it showed 9:05. He started turning the minute hand with his finger un... | math | {
"style": "rule",
"ground_truth": "43"
} | {
"split": "train",
"index": 26235,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $4a = \\sqrt{5}c$ and $\\cos C = \\frac{3}{5}$. \n$(Ⅰ)$ Find the value of $\\sin A$. \n$(Ⅱ)$ If $b = 11$, find the area of $\\triangle ABC$. Let's... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 26236,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of ordered quadruples \\((a,b,c,d)\\) of nonnegative real numbers such that\n\\[\na^2 + b^2 + c^2 + d^2 = 9,\n\\]\n\\[\n(a + b + c + d)(a^3 + b^3 + c^3 + d^3) = 81.\n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 26237,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A line contains the points $(6,8)$, $(-2, k)$ and $(-10, 4)$. What is the value of $k$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 26238,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the result of $46_8 - 63_8$ and express your answer in base 10. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-13"
} | {
"split": "train",
"index": 26239,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $3+\\triangle=5$ and $\\triangle+\\square=7$, what is the value of $\\triangle+\\Delta+\\Delta+\\square+\\square$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 26240,
"solution": "Since $3+\\triangle=5$, then $\\triangle=5-3=2$. Since $\\triangle+\\square=7$ and $\\triangle=2$, then $\\square=5$. Thus, $\\triangle+\\Delta+\\Delta+\\square+\\square=3 \\times 2+2 \\times 5=6+10=16$."
} |
deepscaler | [
{
"role": "user",
"content": "The set of points $(x,y,z)$ that are equidistant to $(1,2,-5)$ and point $P$ satisfy an equation of the form\n\\[10x - 4y + 24z = 55.\\]Find the point $P.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(6,0,7)"
} | {
"split": "train",
"index": 26241,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\( DE = 9 \\), \\( EB = 6 \\), \\( AB = \\frac{1}{3}AD \\), and \\( FD = \\frac{3}{4}AD \\), find \\( FC \\). Assume triangle \\( ADE \\) is similar to triangle \\( AFB \\) and triangle \\( AFC \\). Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "16.875"
} | {
"split": "train",
"index": 26242,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four players stand at distinct vertices of a square. They each independently choose a vertex of the square (which might be the vertex they are standing on). Then, they each, at the same time, begin running in a straight line to their chosen vertex at 10 mph, stopping when they ... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 26243,
"solution": "Observe that no two players can choose the same vertex, and no two players can choose each others vertices. Thus, if two players choose their own vertices, then the remaining two also must choose their own vertices (because they can't choose each others vertices), ... |
deepscaler | [
{
"role": "user",
"content": "Calculate: ${2}^{0}-|-3|+(-\\frac{1}{2})=\\_\\_\\_\\_\\_\\_$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2\\frac{1}{2}"
} | {
"split": "train",
"index": 26244,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of all values of $a + b$ , where $(a, b)$ is an ordered pair of positive integers and $a^2+\\sqrt{2017-b^2}$ is a perfect square. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "67"
} | {
"split": "train",
"index": 26245,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of \\(\\cos ^{5} \\frac{\\pi}{9}+\\cos ^{5} \\frac{5 \\pi}{9}+\\cos ^{5} \\frac{7 \\pi}{9}\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{15}{32}"
} | {
"split": "train",
"index": 26246,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x) = x^4 + 20x^3 + 150x^2 + 500x + 625$. Let $z_1, z_2, z_3, z_4$ be the four roots of $f$. Find the smallest possible value of $|z_a^2 + z_b^2 + z_cz_d|$ where $\\{a, b, c, d\\} = \\{1, 2, 3, 4\\}$. Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "1875"
} | {
"split": "train",
"index": 26247,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Lord Moneybag said to his grandson, \"Bill, listen carefully! Christmas is almost here. I have taken an amount between 300 and 500 pounds, which is a multiple of 6. You will receive 5 pounds in 1-pound coins. When I give you each pound, the remaining amount will first be divisi... | math | {
"style": "rule",
"ground_truth": "426"
} | {
"split": "train",
"index": 26248,
"solution": ""
} |
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