data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "\nA set \\( A \\) consists of 40 elements chosen from \\(\\{1, 2, \\ldots, 50\\}\\), and \\( S \\) is the sum of all elements in the set \\( A \\). How many distinct values can \\( S \\) take? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "401"
} | {
"split": "train",
"index": 32449,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\mathbf{a} = \\begin{pmatrix} -7 \\\\ 0 \\\\ 1 \\end{pmatrix}$ and $\\mathbf{b} = \\begin{pmatrix} 4 \\\\ 2 \\\\ -1 \\end{pmatrix},$ find $\\mathbf{a} - 3 \\mathbf{b}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} -19 \\\\ -6 \\\\ 4 \\end{pmatrix}"
} | {
"split": "train",
"index": 32450,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all positive integers $x$ such that the product of all digits of $x$ is given by $x^2 - 10 \\cdot x - 22.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 32451,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "24 people participate in a training competition consisting of 12 rounds. After each round, every participant receives a certain score \\( a_k \\) based on their ranking \\( k \\) in that round, where \\( a_{k} \\in \\mathbb{N}_+, k = 1, 2, \\ldots, n, a_1 > a_2 > \\cdots > a_n ... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 32452,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the adjoining figure the five circles are tangent to one another consecutively and to the lines $L_1$ and $L_2$. If the radius of the largest circle is $18$ and that of the smallest one is $8$, then the radius of the middle circle is Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 32453,
"solution": "1. **Identify the Configuration**: We are given five circles tangent to each other and to two lines $L_1$ and $L_2$. The radii of the largest and smallest circles are given as $18$ and $8$, respectively.\n\n2. **Understanding the Geometry**: The centers of three co... |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is given that $c=2$, $b=\\sqrt{2}a$. The maximum area of $\\triangle ABC$ is ______. Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"split": "train",
"index": 32454,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the product of two positive integers $a$ and $b$ is $161$ after reversing the digits of the two-digit number $a$, find the correct value of the product of $a$ and $b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "224"
} | {
"split": "train",
"index": 32455,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "I have a bag with only red, blue, and green marbles. The ratio of red marbles to blue marbles to green marbles is $1:5:3$. There are 27 green marbles in the bag. How many marbles are there in the bag? Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "81"
} | {
"split": "train",
"index": 32456,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest prime which is not the difference (in some order) of a power of $2$ and a power of $3$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "41"
} | {
"split": "train",
"index": 32457,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $a$ is a multiple of $456$, find the greatest common divisor of $3a^3+a^2+4a+57$ and $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "57"
} | {
"split": "train",
"index": 32458,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For every non-empty subset of the natural number set $N^*$, we define the \"alternating sum\" as follows: arrange the elements of the subset in descending order, then start with the largest number and alternately add and subtract each number. For example, the alternating sum of... | math | {
"style": "rule",
"ground_truth": "448"
} | {
"split": "train",
"index": 32459,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a certain area, there are 100,000 households, among which there are 99,000 ordinary households and 1,000 high-income households. A simple random sampling method is used to select 990 households from the ordinary households and 100 households from the high-income households f... | math | {
"style": "rule",
"ground_truth": "4.8\\%"
} | {
"split": "train",
"index": 32460,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Quadrilateral $ABCD$ has $\\angle BCD=\\angle DAB=90^{\\circ}$. The perimeter of $ABCD$ is 224 and its area is 2205. One side of $ABCD$ has length 7. The remaining three sides have integer lengths. What is the integer formed by the rightmost two digits of the sum of the squares... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 32461,
"solution": "Suppose that $AB=x, BC=y, CD=z$, and $DA=7$. Since the perimeter of $ABCD$ is 224, we have $x+y+z+7=224$ or $x+y+z=217$. Join $B$ to $D$. The area of $ABCD$ is equal to the sum of the areas of $\\triangle DAB$ and $\\triangle BCD$. Since these triangles are right-a... |
deepscaler | [
{
"role": "user",
"content": "Reimu and Sanae play a game using 4 fair coins. Initially both sides of each coin are white. Starting with Reimu, they take turns to color one of the white sides either red or green. After all sides are colored, the 4 coins are tossed. If there are more red sides showing up, th... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{16}"
} | {
"split": "train",
"index": 32462,
"solution": "Clearly Reimu will always color a side red and Sanae will always color a side green, because their situation is never worse off when a side of a coin changes to their own color. Since the number of red-only coins is always equal to the number of green-only coins, n... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = \\ln x - ax$, where $a \\in \\mathbb{R}$.\n(1) If the line $y = 3x - 1$ is a tangent line to the graph of the function $f(x)$, find the value of the real number $a$.\n(2) If the maximum value of the function $f(x)$ on the interval $[1, e^2]$ is $1 - a... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{e}"
} | {
"split": "train",
"index": 32463,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A$ be the area of the largest semicircle that can be inscribed in a quarter-circle of radius 1. Compute $\\frac{120 A}{\\pi}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 32464,
"solution": "The optimal configuration is when the two ends $X$ and $Y$ of the semicircle lie on the arc of the quarter circle. Let $O$ and $P$ be the centers of the quarter circle and semicircle, respectively. Also, let $M$ and $N$ be the points where the semicircle is tangent... |
deepscaler | [
{
"role": "user",
"content": "\nFor the four-digit number $\\overline{abcd}$ (where $1 \\leq a \\leq 9$ and $0 \\leq b, c, d \\leq 9$):\n- If $a > b$, $b < c$, and $c > d$, then $\\overline{abcd}$ is called a $P$-type number;\n- If $a < b$, $b > c$, and $c < d$, then $\\overline{abcd}$ is called a $Q$-type ... | math | {
"style": "rule",
"ground_truth": "285"
} | {
"split": "train",
"index": 32465,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Loki, Moe, Nick and Ott are good friends. Ott had no money, but the others did. Moe gave Ott one-fifth of his money, Loki gave Ott one-fourth of his money and Nick gave Ott one-third of his money. Each gave Ott the same amount of money. What fractional part of the group's money... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 32466,
"solution": "1. **Assumption of Money Given**: Assume Moe, Loki, and Nick each give Ott $1. This assumption is based on the problem statement that each gave Ott the same amount of money.\n\n2. **Calculation of Original Money**:\n - Since Moe gave Ott one-fifth of his money, a... |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ has $BC=20.$ The incircle of the triangle evenly trisects the median $AD.$ If the area of the triangle is $m \\sqrt{n}$ where $m$ and $n$ are integers and $n$ is not divisible by the square of a prime, find $m+n.$\n Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "38"
} | {
"split": "train",
"index": 32467,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that\n\\begin{align*}x_{1}&=211,\\\\ x_{2}&=375,\\\\ x_{3}&=420,\\\\ x_{4}&=523,\\ \\text{and}\\\\ x_{n}&=x_{n-1}-x_{n-2}+x_{n-3}-x_{n-4}\\ \\text{when}\\ n\\geq5, \\end{align*}\n find the value of $x_{531}+x_{753}+x_{975}$. Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "898"
} | {
"split": "train",
"index": 32468,
"solution": "Calculate the first few terms:\n\\[211,375,420,523,267,-211,-375,-420,-523,\\dots\\]\n At this point it is pretty clear that the sequence is periodic with period 10 (one may prove it quite easily like in solution 1) so our answer is obviously $211+420+267=\\boxed{8... |
deepscaler | [
{
"role": "user",
"content": "Petya considered moments in his life to be happy when his digital clock showed the number of hours to be six times the number of minutes, or vice versa. Petya fell asleep and woke up at a happy moment in his life, without missing any such moment during his sleep. What is the ma... | math | {
"style": "rule",
"ground_truth": "361"
} | {
"split": "train",
"index": 32469,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the addition shown, each of \\( P \\), \\( Q \\), and \\( R \\) is a digit.\n\n\\[\n\\begin{array}{r}\nP Q R \\\\\n+\\quad Q R \\\\\n\\hline 1012\n\\end{array}\n\\]\n\nWhat is the value of \\( P + Q + R \\)? Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 32470,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the minimum possible value of $\\sqrt{58-42 x}+\\sqrt{149-140 \\sqrt{1-x^{2}}}$ where $-1 \\leq x \\leq 1$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{109}"
} | {
"split": "train",
"index": 32471,
"solution": "Substitute $x=\\cos \\theta$ and $\\sqrt{1-x^{2}}=\\sin \\theta$, and notice that $58=3^{2}+7^{2}, 42=2 \\cdot 3 \\cdot 7,149=7^{2}+10^{2}$, and $140=2 \\cdot 7 \\cdot 10$. Therefore the first term is an application of Law of Cosines on a triangle that has two side... |
deepscaler | [
{
"role": "user",
"content": "If \\( n \\) is a positive integer such that \\( n^{6} + 206 \\) is divisible by \\( n^{2} + 2 \\), find the sum of all possible values of \\( n \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 32472,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$(1)$ Calculate: $tan60°×{({-2})^{-1}}-({\\sqrt{\\frac{3}{4}}-\\sqrt[3]{8}})+|{-\\frac{1}{2}\\sqrt{12}}|$;<br/>$(2)$ Simplify and find the value: $({\\frac{{x+2}}{{{x^2}-2x}}-\\frac{{x-1}}{{{x^2}-4x+4}}})÷\\frac{{x-4}}{x}$, where $x=\\sqrt{2}+2$. Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 32473,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that there are $3$ red cubes, $4$ blue cubes, $2$ green cubes, and $2$ yellow cubes, determine the total number of different towers with a height of $10$ cubes that can be built, with the condition that the tower must always have a yellow cube at the top. Let's think step... | math | {
"style": "rule",
"ground_truth": "1,260"
} | {
"split": "train",
"index": 32474,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=2\\cos \\left(2x+ \\frac{2\\pi}{3}\\right)+ \\sqrt{3}\\sin 2x$.\n\n$(1)$ Find the smallest positive period and the maximum value of the function $f(x)$;\n\n$(2)$ Let $\\triangle ABC$ have internal angles $A$, $B$, and $C$ respectively. If $f\\left(\\fra... | math | {
"style": "rule",
"ground_truth": "\\frac{3 \\sqrt{21}}{14}"
} | {
"split": "train",
"index": 32475,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a_{0}, a_{1}, a_{2}, \\ldots$ be an infinite sequence where each term is independently and uniformly random in the set $\\{1,2,3,4\\}$. Define an infinite sequence $b_{0}, b_{1}, b_{2}, \\ldots$ recursively by $b_{0}=1$ and $b_{i+1}=a_{i}^{b_{i}}$. Compute the expected val... | math | {
"style": "rule",
"ground_truth": "\\frac{35}{16}"
} | {
"split": "train",
"index": 32476,
"solution": "Do casework on what $a_{0}$ is. If $a_{0}=1$ then $k=1$. If $a_{0}=4$ then $k=2$. If $a_{0}=3$ then - if $a_{1}=1$, then $k=2$ - if $a_{1}=2$ or 4 , then $k=3$ - if $a_{1}=3$, then you make no progress. so in expectation it requires $E=(2+3+(E+1)+3) / 4 \\Longright... |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC^{}_{}$ has $AB=9^{}_{}$ and $BC: AC=40: 41^{}_{}$. What's the largest area that this triangle can have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "820"
} | {
"split": "train",
"index": 32477,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square sheet of paper that measures $18$ cm on a side has corners labeled $A$ , $B$ , $C$ , and $D$ in clockwise order. Point $B$ is folded over to a point $E$ on $\\overline{AD}$ with $DE=6$ cm and the paper is creased. When the paper is unfolded, the crease i... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 32478,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the acute angle \\(\\alpha\\) satisfies \\(\\frac{1}{\\sqrt{\\tan \\frac{\\alpha}{2}}}=\\sqrt{2 \\sqrt{3}} \\sqrt{\\tan 10^{\\circ}}+\\sqrt{\\tan \\frac{\\alpha}{2}}\\), then the measure of the angle \\(\\alpha\\) in degrees is \\(\\qquad\\) Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 32479,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a dodecahedron, which is made up of 12 pentagonal faces. An ant starts at one of the top vertices and walks to one of the three adjacent vertices (vertex A). From vertex A, the ant walks again to one of its adjacent vertices (vertex B). What is the probability that ver... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 32480,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the shadow length $l$ is equal to the product of the table height $h$ and the tangent value of the solar zenith angle $\\theta$, and $\\tan(\\alpha-\\beta)=\\frac{1}{3}$, if the shadow length in the first measurement is three times the table height, determine the shadow l... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"split": "train",
"index": 32481,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(A\\) and \\(B\\) be two moving points on the ellipse \\(x^2 + 3y^2 = 1\\), and \\(OA \\perp OB\\) (where \\(O\\) is the origin). Find the product of the maximum and minimum values of \\( |AB| \\). Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "\\frac{2 \\sqrt{3}}{3}"
} | {
"split": "train",
"index": 32482,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A lateral face of a regular triangular pyramid $SABC$ is inclined to the base plane $ABC$ at an angle $\\alpha = \\operatorname{arctg} \\frac{3}{4}$. Points $M, N, K$ are midpoints of the sides of the base $ABC$. The triangle $MNK$ serves as the lower base of a rectangular pris... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 32483,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the set $\\{a, b, c\\} = \\{0, 1, 2\\}$, and the following three conditions: ① $a \\neq 2$; ② $b = 2$; ③ $c \\neq 0$ are correct for only one of them, then $10a + 2b + c$ equals $\\boxed{?}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 32484,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A collector has \\( N \\) precious stones. If he takes away the three heaviest stones, then the total weight of the stones decreases by \\( 35\\% \\). From the remaining stones, if he takes away the three lightest stones, the total weight further decreases by \\( \\frac{5}{13} ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 32485,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A jar of peanut butter which is 3 inches in diameter and 4 inches high sells for $\\$$0.60. At the same rate, what would be the price for a jar that is 6 inches in diameter and 6 inches high? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\$3.60"
} | {
"split": "train",
"index": 32486,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Remove all positive integers that are divisible by 7, and arrange the remaining numbers in ascending order to form a sequence $\\{a_n\\}$. Calculate the value of $a_{100}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "116"
} | {
"split": "train",
"index": 32487,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $35^{1723} - 16^{1723}$ modulo 6. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 32488,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the common ratio of the infinite geometric series: $$\\frac{-4}{7}+\\frac{14}{3}+\\frac{-98}{9} + \\dots$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{49}{6}"
} | {
"split": "train",
"index": 32489,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Every day, from Monday to Friday, an old man went to the blue sea and cast his net into the sea. Each day, he caught no more fish than the previous day. In total, he caught exactly 100 fish over the five days. What is the minimum total number of fish he could have caught on Mon... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 32490,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given 6 people $A$, $B$, $C$, $D$, $E$, $F$, they are randomly arranged in a line. The probability of the event \"$A$ is adjacent to $B$ and $A$ is not adjacent to $C$\" is $\\_\\_\\_\\_\\_\\_$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{15}"
} | {
"split": "train",
"index": 32491,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the area of the region enclosed by the graph of the equation $x^2+y^2=|x|+|y|+1?$\nA) $\\frac{\\pi}{2} + 2$\nB) $\\frac{3\\pi}{2}$\nC) $\\frac{3\\pi}{2} + 2$\nD) $2\\pi + 2$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3\\pi}{2} + 2"
} | {
"split": "train",
"index": 32492,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the numbers from $000$ to $999$, calculate how many have three digits in non-decreasing or non-increasing order, including cases where digits can repeat. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "430"
} | {
"split": "train",
"index": 32493,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 196 students numbered from 1 to 196 arranged in a line. Students at odd-numbered positions (1, 3, 5, ...) leave the line. The remaining students are renumbered starting from 1 in order. Then, again, students at odd-numbered positions leave the line. This process repea... | math | {
"style": "rule",
"ground_truth": "128"
} | {
"split": "train",
"index": 32494,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Heather compares the price of a new computer at two different stores. Store $A$ offers $15\\%$ off the sticker price followed by a $\\$90$ rebate, and store $B$ offers $25\\%$ off the same sticker price with no rebate. Heather saves $\\$15$ by buying the computer at store $A$ i... | math | {
"style": "rule",
"ground_truth": "750"
} | {
"split": "train",
"index": 32495,
"solution": "\nLet the sticker price of the computer be denoted as \\( x \\).\n\n**Store A Calculation:**\n- Store A offers a 15% discount, so the price after the discount is \\( 0.85x \\).\n- After applying the additional $90 rebate, the final price at store A becomes \\( 0.85... |
deepscaler | [
{
"role": "user",
"content": "Dean scored a total of 252 points in 28 basketball games. Ruth played 10 fewer games than Dean. Her scoring average was 0.5 points per game higher than Dean's scoring average. How many points, in total, did Ruth score? Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "171"
} | {
"split": "train",
"index": 32496,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $a$ and $b$ are digits, not both nine and not both zero, and the repeating decimal $0.\\overline{ab}$ is expressed as a fraction in lowest terms. How many different denominators are possible? Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 32497,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the total number of different selections possible for five donuts when choosing from four types of donuts (glazed, chocolate, powdered, and jelly), with the additional constraint of purchasing at least one jelly donut. Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 32498,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle inscribed in an isosceles trapezoid divides its lateral side into segments equal to 4 and 9. Find the area of the trapezoid. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "156"
} | {
"split": "train",
"index": 32499,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence ${a_n}$ satisfying $a_1=1$, $a_2=2$, $a_3=3$, $a_{n+3}=a_n$ ($n\\in\\mathbb{N}^*$). If $a_n=A\\sin(\\omega n+\\varphi)+c$ $(ω>0,|\\varphi|<\\frac{\\pi}{2})$, find the real number $A$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "-\\frac{2\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 32500,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The ratio of the area of the rectangle to the area of the decagon can be calculated given that a regular decagon $ABCDEFGHIJ$ contains a rectangle $AEFJ$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"split": "train",
"index": 32501,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the volume of a cube whose surface area is twice that of a cube with volume 1? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"split": "train",
"index": 32502,
"solution": "1. **Understanding the problem**: We need to find the volume of a cube whose surface area is twice that of another cube with a volume of 1.\n\n2. **Calculating the surface area of the first cube**:\n - The volume of the first cube is given as 1 cubic unit.\n - ... |
deepscaler | [
{
"role": "user",
"content": "Given that $f(x)$ is an odd function defined on $\\mathbb{R}$ with a minimal positive period of $3$, and $f(x)=\\log_2(1-x)$ when $x \\in \\left(-\\frac{3}{2}, 0\\right)$, find the value of $f(2014) + f(2016)$. Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 32503,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ten identical crates each of dimensions $3\\mathrm{ft}\\times 4\\mathrm{ft}\\times 6\\mathrm{ft}$. The first crate is placed flat on the floor. Each of the remaining nine crates is placed, in turn, flat on top of the previous crate, and the orientation of each crate is chosen a... | math | {
"style": "rule",
"ground_truth": "190"
} | {
"split": "train",
"index": 32504,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $({11011_{(2)}} - {101_{(2)}} = )$\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_$(.$ (represented in binary) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10110_{(2)}"
} | {
"split": "train",
"index": 32505,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose a sequence starts with 1254, 2547, 5478, and ends with 4781. Let $T$ be the sum of all terms in this sequence. Find the largest prime factor that always divides $T$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "101"
} | {
"split": "train",
"index": 32506,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number of elderly employees in the sampled group is what fraction of the number of young employees in the sampled group if the sampled group consists of the employees of a certain unit, which has a total of 430 employees, among which there are 160 young employees, and the n... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 32507,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the value of the following product as a common fraction:\n\\[ \\left(2 \\cdot \\left(1-\\frac{1}{2}\\right)\\right) \\cdot \\left(1-\\frac{1}{3}\\right) \\cdot \\left(1-\\frac{1}{4}\\right) \\dotsm \\left(1-\\frac{1}{50}\\right) \\] Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{50}"
} | {
"split": "train",
"index": 32508,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, we have $\\angle C = 90^\\circ$, $AB = 26$, and $BC = 10$. What is $\\sin A$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{13}"
} | {
"split": "train",
"index": 32509,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In base \\( R_1 \\), the fractional expansion of \\( F_1 \\) is \\( 0.373737 \\cdots \\), and the fractional expansion of \\( F_2 \\) is \\( 0.737373 \\cdots \\). In base \\( R_2 \\), the fractional expansion of \\( F_1 \\) is \\( 0.252525 \\cdots \\), and the fractional expans... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 32510,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $r(x)$ be a monic quartic polynomial such that $r(1) = 0,$ $r(2) = 3,$ $r(3) = 8,$ and $r(4) = 15$. Find $r(5)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 32511,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We seek the true statement. There are one hundred statements written in a notebook:\n\n1) There is exactly one false statement in this notebook.\n2) There are exactly two false statements in this notebook.\n...\n100) There are exactly one hundred false statements in this notebo... | math | {
"style": "rule",
"ground_truth": "99"
} | {
"split": "train",
"index": 32512,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A positive integer $N$ greater than $1$ is described as special if in its base- $8$ and base- $9$ representations, both the leading and ending digit of $N$ are equal to $1$ . What is the smallest special integer in decimal representation?\n\n*Proposed by Michael Ren* L... | math | {
"style": "rule",
"ground_truth": "793"
} | {
"split": "train",
"index": 32513,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An athlete's target heart rate, in beats per minute, is $80\\%$ of the theoretical maximum heart rate. The maximum heart rate is found by subtracting the athlete's age, in years, from $220$. To the nearest whole number, what is the target heart rate of an athlete who is $26$ ye... | math | {
"style": "rule",
"ground_truth": "134"
} | {
"split": "train",
"index": 32514,
"solution": "1. **Calculate the Maximum Heart Rate**: The maximum heart rate for an athlete is calculated by subtracting the athlete's age from 220. For an athlete who is 26 years old, the calculation is:\n \\[\n \\text{Maximum Heart Rate} = 220 - 26 = 194 \\text{ beats per... |
deepscaler | [
{
"role": "user",
"content": "Given that in $\\triangle ABC$, $\\sin A + 2 \\sin B \\cos C = 0$, find the maximum value of $\\tan A$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 32515,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sum of the first n terms of a geometric sequence {a_n} is denoted as S_n, if S_4 = 2 and S_8 = 6, calculate the value of S_{12}. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 32516,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the least positive integer that is divisible by the next three distinct primes larger than 5? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1001"
} | {
"split": "train",
"index": 32517,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the set \\( A = \\{1, 2, \\cdots, 2016\\} \\). For any 1008-element subset \\( X \\) of \\( A \\), if there exist \\( x \\) and \\( y \\in X \\) such that \\( x < y \\) and \\( x \\mid y \\), then \\( X \\) is called a \"good set\". Find the largest positive integer \\( a \... | math | {
"style": "rule",
"ground_truth": "1008"
} | {
"split": "train",
"index": 32518,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Which triplet of numbers has a sum NOT equal to 1? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1.1 + (-2.1) + 1.0"
} | {
"split": "train",
"index": 32519,
"solution": "To find which triplet of numbers has a sum NOT equal to 1, we will add the numbers in each triplet and check the result.\n\n1. **Triplet (A):** $(1/2, 1/3, 1/6)$\n \\[\n \\frac{1}{2} + \\frac{1}{3} + \\frac{1}{6} = \\frac{3}{6} + \\frac{2}{6} + \\frac{1}{6} = \... |
deepscaler | [
{
"role": "user",
"content": "The arithmetic mean of 12 scores is 82. When the highest and lowest scores are removed, the new mean becomes 84. If the highest of the 12 scores is 98, what is the lowest score? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "46"
} | {
"split": "train",
"index": 32520,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define a $\\it{good\\ word}$ as a sequence of letters that consists only of the letters $A$, $B$, $C$, and $D$ --- some of these letters may not appear in the sequence --- and in which $A$ is never immediately followed by $B$, $B$ is never immediately followed by $C$, $C$ is ne... | math | {
"style": "rule",
"ground_truth": "8748"
} | {
"split": "train",
"index": 32521,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The lengths of a pair of corresponding medians of two similar triangles are 10cm and 4cm, respectively, and the sum of their perimeters is 140cm. The perimeters of these two triangles are , and the ratio of their areas is . Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "25:4"
} | {
"split": "train",
"index": 32522,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be the number of ordered quadruples $(x_1,x_2,x_3,x_4)$ of positive odd integers that satisfy $\\sum_{i = 1}^4 x_i = 98.$ Find $\\frac n{100}.$\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "196"
} | {
"split": "train",
"index": 32523,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In 2010, the sum of the year's digits is $3$. What is the first year after 2010 in which the sum of the digits is 3 again? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2100"
} | {
"split": "train",
"index": 32524,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two real numbers $a$ and $b$ are chosen independently and uniformly at random from the interval $(0, 75)$. Let $O$ and $P$ be two points on the plane with $OP = 200$. Let $Q$ and $R$ be on the same side of line $OP$ such that the degree measures of $\\angle POQ$ and $\\angle PO... | math | {
"style": "rule",
"ground_truth": "41"
} | {
"split": "train",
"index": 32525,
"solution": "Noting that $\\angle OQP$ and $\\angle ORP$ are right angles, we realize that we can draw a semicircle with diameter $\\overline{OP}$ and points $Q$ and $R$ on the semicircle. Since the radius of the semicircle is $100$, if $\\overline{QR} \\leq 100$, then $\\overa... |
deepscaler | [
{
"role": "user",
"content": "Let GCF(a, b) be the abbreviation for the greatest common factor of a and b, and let LCM(c, d) be the abbreviation for the least common multiple of c and d. What is GCF(LCM(8, 14), LCM(7, 12))? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 32526,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The measure of angle $ACB$ is 70 degrees. If ray $CA$ is rotated 600 degrees about point $C$ in a clockwise direction, what will be the positive measure of the new obtuse angle $ACB$, in degrees? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "170"
} | {
"split": "train",
"index": 32527,
"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": "100"
} | {
"split": "train",
"index": 32528,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A standard deck of 52 cards has 13 ranks (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King) and 4 suits ($\\spadesuit$, $\\heartsuit$, $\\diamondsuit$, and $\\clubsuit$), such that there is exactly one card for any given rank and suit. Two of the suits ($\\spadesuit$ and $\\... | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{13}"
} | {
"split": "train",
"index": 32529,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We inscribe a cone around a sphere of unit radius. What is the minimum surface area of the cone? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8\\pi"
} | {
"split": "train",
"index": 32530,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What percent of the positive integers less than or equal to $120$ have no remainders when divided by $6$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16.67\\%"
} | {
"split": "train",
"index": 32531,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Neznaika is drawing closed paths inside a $5 \\times 8$ rectangle, traveling along the diagonals of $1 \\times 2$ rectangles. In the illustration, an example of a path passing through 12 such diagonals is shown. Help Neznaika draw the longest possible path. Let's think step by ... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 32532,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ satisfy $|\\overrightarrow{a}|=\\sqrt{2}$, $|\\overrightarrow{b}|=2$, and $\\overrightarrow{a}\\bot (\\overrightarrow{a}-\\overrightarrow{b})$, calculate the angle between $\\overrightarrow{a}$ and $\\overrighta... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"split": "train",
"index": 32533,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number of sets of two or more consecutive positive integers whose sum is 100 is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 32534,
"solution": "1. **Formulate the sum of consecutive integers**: Let's consider a sequence of $n$ consecutive integers starting from $a$. The integers are $a, a+1, a+2, \\ldots, a+n-1$. The sum $S$ of these integers can be calculated using the formula for the sum of an arithmetic... |
deepscaler | [
{
"role": "user",
"content": "The four circles in the diagram intersect to divide the interior into 8 parts. Fill these 8 parts with the numbers 1 through 8 such that the sum of the 3 numbers within each circle is equal. Calculate the maximum possible sum and provide one possible configuration. Let's think ... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 32535,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular garden measuring 60 feet by 20 feet is enclosed by a fence. In a redesign to maximize the area using the same amount of fencing, its shape is changed to a circle. How many square feet larger or smaller is the new garden compared to the old one? Let's think step by... | math | {
"style": "rule",
"ground_truth": "837.62"
} | {
"split": "train",
"index": 32536,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are \\( n \\) players participating in a round-robin chess tournament, where each player competes exactly once against every other player. The winner of a match earns 3 points, a draw gives each player 1 point, and the loser earns 0 points. If the total sum of all players... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 32537,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x$ and $y$ be distinct real numbers such that\n\\[\n\\begin{vmatrix} 2 & 3 & 7 \\\\ 4 & x & y \\\\ 4 & y & x+1 \\end{vmatrix}\n= 0.\\]Find $x + y.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 32538,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate:<br/>$(1)-3+\\left(-9\\right)+10-\\left(-18\\right)$;<br/>$(2)12÷2×(-\\frac{1}{2})-75÷(-5)$;<br/>$(3)-4^3-2×(-5)^2+6÷(-\\frac{1}{3})^2$;<br/>$(4)(-1\\frac{1}{4}-1\\frac{5}{6}+2\\frac{8}{9})÷(-\\frac{1}{6})^2$. Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"split": "train",
"index": 32539,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A parallelogram has side lengths of 10, $12x-2$, $5y+5$, and $4$. What is the value of $x+y$?\n\n[asy]draw((0,0)--(24,0)--(30,20)--(6,20)--cycle);\nlabel(\"$12x-2$\",(15,0),S);\nlabel(\"10\",(3,10),W);\nlabel(\"$5y+5$\",(18,20),N);\nlabel(\"4\",(27,10),E);\n[/asy] Let's think s... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{5}"
} | {
"split": "train",
"index": 32540,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of $\\frac{5x+9y}{45xy}$, given $x = \\frac{3}{5}$ and $y = \\frac{7}{9}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{10}{21}"
} | {
"split": "train",
"index": 32541,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the digits of the decimal representation of $2^{2005} \\times 5^{2007} \\times 3$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 32542,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "As shown in the diagram, A and B are the endpoints of the diameter of a circular track. Three miniature robots, labeled as J, Y, and B, start simultaneously and move uniformly along the circular track. Robots J and Y start from point A, while robot B starts from point B. Robot ... | math | {
"style": "rule",
"ground_truth": "56"
} | {
"split": "train",
"index": 32543,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the complex number z = $$\\frac{a^2i}{2-i} + \\frac{1-2ai}{5}$$ (where a ∈ R, i is the imaginary unit), find the value(s) of a if z is a purely imaginary number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 32544,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive four-digit integers of the form $\\_\\_35$ are divisible by 35? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 32545,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Among the integers from 1 to 100, how many integers can be divided by exactly two of the following four numbers: 2, 3, 5, 7? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 32546,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=2x^{3}-ax^{2}+1$ $(a\\in\\mathbb{R})$ has exactly one zero in the interval $(0,+\\infty)$, find the sum of the maximum and minimum values of $f(x)$ on the interval $[-1,1]$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 32547,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the area enclosed by the graph of $|x| + |3y| = 9$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 32548,
"solution": ""
} |
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