data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Let \\(a,\\) \\(b,\\) \\(c\\) be distinct real numbers such that\n\\[\\frac{a}{1 + b} = \\frac{b}{1 + c} = \\frac{c}{1 + a} = k.\\] Find the product of all possible values of \\(k.\\) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 30349,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, $AB$ is congruent to $AC$, the measure of angle $ABC$ is $72^{\\circ}$ and segment $BD$ bisects angle $ABC$ with point $D$ on side $AC$. If point $E$ is on side $BC$ such that segment $DE$ is parallel to side $AB$, and point $F$ is on side $AC$ such that segm... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 30350,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has a width of $6$ meters, and it takes her $36$ seconds longer to walk around the outside edge of the track than around th... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"split": "train",
"index": 30351,
"solution": "\nLet's denote Keiko's speed as $s$ meters per second. The track consists of two straight sections and two semicircular ends. The width of the track is $6$ meters, which means the difference in radius between the inner and outer semicircles is $6$ meters.\n\n#### S... |
deepscaler | [
{
"role": "user",
"content": "There are $12$ small balls in a bag, which are red, black, and yellow respectively (these balls are the same in other aspects except for color). The probability of getting a red ball when randomly drawing one ball is $\\frac{1}{3}$, and the probability of getting a black ball i... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 30352,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Adam has $15$ of a certain type of rare coin and is interested in knowing how much this collection is worth. He discovers that $5$ of these coins are worth $12$ dollars in total. Assuming that the value of each coin is the same, how many dollars is his entire collection worth?... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 30353,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the value $-2\\frac{1}{2}$, calculate its absolute value. 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": 30354,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The base of isosceles $\\triangle ABC$ is $24$ and its area is $60$. What is the length of one of the congruent sides? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 30355,
"solution": "1. **Identify the given information**: \n - Base of isosceles triangle $\\triangle ABC$, $AB = 24$.\n - Area of $\\triangle ABC = 60$.\n\n2. **Calculate the height using the area formula**:\n The area of a triangle is given by the formula:\n \\[\n \\text{... |
deepscaler | [
{
"role": "user",
"content": "If the line $(m+2)x+3y+3=0$ is parallel to the line $x+(2m-1)y+m=0$, then the real number $m=$ \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{5}{2}"
} | {
"split": "train",
"index": 30356,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Petya and Vasya calculated that if they walk at a speed of 4 km per hour to the neighboring village, which is 4 kilometers away, they will be 10 minutes late for the football match held there for the district championship. How should they proceed to arrive at the match on time ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 30357,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For some positive integers $a$ and $b$, the product \\[\\log_a(a+1) \\cdot \\log_{a+1} (a+2) \\dotsm \\log_{b-2} (b-1) \\cdot\\log_{b-1} b\\]contains exactly $870$ terms, and its value is $2.$ Compute $a+b.$ Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "930"
} | {
"split": "train",
"index": 30358,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the distance between the points $(2,1,-4)$ and $(5,8,-3).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{59}"
} | {
"split": "train",
"index": 30359,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 7 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{40}"
} | {
"split": "train",
"index": 30360,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each vertex of the parallelogram $ABCD$ lies on the same side of the plane $S$ such that the distances of the vertices $A, B$, and $C$ from the plane $S$ are 4 cm, 6 cm, and 8 cm, respectively. The area of the projection of the parallelogram onto the plane $S$, which forms the ... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 30361,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest number of consecutive integers whose sum is $45?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 30362,
"solution": "To find the greatest number of consecutive integers whose sum is $45$, we need to consider sequences of integers, both positive and negative.\n\n1. **Understanding the sum of consecutive integers**:\n The sum of $N$ consecutive integers starting from $a$ can be e... |
deepscaler | [
{
"role": "user",
"content": "A cube is painted so that one side is blue, two sides are red, and three sides are green. How many different such cubes can be painted? Two cubes are considered the same if one cube can be rotated in any way to match the second cube. Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 30363,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the parabola $C: x^{2}=2py\\left(p \\gt 0\\right)$ with focus $F$, and the minimum distance between $F$ and a point on the circle $M: x^{2}+\\left(y+4\\right)^{2}=1$ is $4$. \n$(1)$ Find $p$; \n$(2)$ If point $P$ lies on $M$, $PA$ and $PB$ are two tangents to $C$ with p... | math | {
"style": "rule",
"ground_truth": "20\\sqrt{5}"
} | {
"split": "train",
"index": 30364,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a box, there are 4 cards each with a function defined on the domain \\\\( R \\\\): \\\\( f_{1}(x)=x \\\\), \\\\( f_{2}(x)=|x| \\\\), \\\\( f_{3}(x)=\\sin x \\\\), \\\\( f_{4}(x)=\\cos x \\\\). Now, two cards are randomly selected from the box, and the functions written on th... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 30365,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $x$ is an integer that satisfies the following congruences:\n\\begin{align*}\n2+x &\\equiv 3^2 \\pmod{2^4}, \\\\\n3+x &\\equiv 2^3 \\pmod{3^4}, \\\\\n4+x &\\equiv 3^3 \\pmod{2^3}.\n\\end{align*}\nWhat is the remainder when $x$ is divided by $24$? Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 30366,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is $2^{-1} + 2^{-2} + 2^{-3} + 2^{-4} + 2^{-5} + 2^{-6} \\pmod{13}$?\n\nExpress your answer as an integer from $0$ to $12$, inclusive. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 30367,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In right triangle $DEF$, we have $\\angle D = 25^\\circ$, $\\angle E = 90^\\circ$, and $EF = 9$. Find $DE$ to the nearest tenth. You may use a calculator for this problem. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19.3"
} | {
"split": "train",
"index": 30368,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that Mr. A initially owns a home worth $\\$15,000$, he sells it to Mr. B at a $20\\%$ profit, then Mr. B sells it back to Mr. A at a $15\\%$ loss, then Mr. A sells it again to Mr. B at a $10\\%$ profit, and finally Mr. B sells it back to Mr. A at a $5\\%$ loss, calculate ... | math | {
"style": "rule",
"ground_truth": "3541.50"
} | {
"split": "train",
"index": 30369,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The lengths of the sides of a non-degenerate triangle are $x$, 13 and 37 units. How many integer values of $x$ are possible? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 30370,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the decimal representation of an even number \\( M \\), only the digits \\( 0, 2, 4, 5, 7, \\) and \\( 9 \\) are used, and the digits may repeat. It is known that the sum of the digits of the number \\( 2M \\) equals 39, and the sum of the digits of the number \\( M / 2 \\) ... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 30371,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle intersects the sides $AC$ and $CB$ of an isosceles triangle $ACB$ at points $P$ and $Q$ respectively, and is circumscribed around triangle $ABQ$. The segments $AQ$ and $BP$ intersect at point $D$ such that $AQ: AD = 4:3$. Find the area of triangle $DQB$ if the area of ... | math | {
"style": "rule",
"ground_truth": "9/2"
} | {
"split": "train",
"index": 30372,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A pair of dice is rolled twice. What is the probability that the sum of the numbers facing up is 5?\n\nA) $\\frac{1}{9}$\nB) $\\frac{1}{4}$\nC) $\\frac{1}{36}$\nD) 97 Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{9}"
} | {
"split": "train",
"index": 30373,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a rectangle $ABCD$ with all vertices on a sphere centered at $O$, where $AB = \\sqrt{3}$, $BC = 3$, and the volume of the pyramid $O-ABCD$ is $4\\sqrt{3}$, find the surface area of the sphere $O$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "76\\pi"
} | {
"split": "train",
"index": 30374,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $m$ be the smallest positive, three-digit integer congruent to 6 (mod 13). Let $n$ be the smallest positive, four-digit integer congruent to 7 (mod 17). What is $n-m$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "900"
} | {
"split": "train",
"index": 30375,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five students from a certain class participated in a speech competition and the order of appearance was determined by drawing lots, under the premise that student A must appear before student B. Calculate the probability of students A and B appearing adjacent to each other. Let... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"split": "train",
"index": 30376,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the product of all real numbers that are tripled when added to their reciprocals? 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": 30377,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a right-angled triangle \\(ABC\\) (with right angle at \\(A\\)), the bisectors of the acute angles intersect at point \\(P\\). The distance from \\(P\\) to the hypotenuse is \\(\\sqrt{80000}\\). What is the distance from \\(P\\) to \\(A\\)? Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "400"
} | {
"split": "train",
"index": 30378,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of all two-digit positive integers whose squares end with the digits 25? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "495"
} | {
"split": "train",
"index": 30379,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two integers have a sum of $26$. When two more integers are added to the first two, the sum is $41$. Finally, when two more integers are added to the sum of the previous $4$ integers, the sum is $57$. What is the minimum number of even integers among the $6$ integers? Let's thi... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 30380,
"solution": "1. **Identify the sums at each stage:**\n - Let the first two integers be $x$ and $y$. We know $x + y = 26$.\n - Let the next two integers added be $a$ and $b$. Then, $x + y + a + b = 41$.\n - Let the final two integers added be $m$ and $n$. Then, $x + y + a ... |
deepscaler | [
{
"role": "user",
"content": "Consider the function $ f: \\mathbb{N}_0\\to\\mathbb{N}_0$, where $ \\mathbb{N}_0$ is the set of all non-negative\nintegers, defined by the following conditions :\n\n $ (i)$ $ f(0) \\equal{} 0$; $ (ii)$ $ f(2n) \\equal{} 2f(n)$ and $ (iii)$ $ f(2n \\plus{} 1) \\equal{} n \\p... | math | {
"style": "rule",
"ground_truth": "2^k - 1"
} | {
"split": "train",
"index": 30381,
"solution": "\n### Part (a)\n\nWe have the function \\( f: \\mathbb{N}_0 \\rightarrow \\mathbb{N}_0 \\) defined by:\n- \\( f(0) = 0 \\)\n- \\( f(2n) = 2f(n) \\)\n- \\( f(2n + 1) = n + 2f(n) \\)\n\nWe need to determine the sets:\n- \\( L = \\{ n \\mid f(n) < f(n + 1) \\} \\)\n- ... |
deepscaler | [
{
"role": "user",
"content": "Suppose that the roots of $x^3+3x^2+4x-11=0$ are $a$, $b$, and $c$, and that the roots of $x^3+rx^2+sx+t=0$ are $a+b$, $b+c$, and $c+a$. Find $t$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 30382,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In America, temperature is measured in degrees Fahrenheit. This is a linear scale where the freezing point of water is $32^{\\circ} \\mathrm{F}$ and the boiling point is $212^{\\circ} \\mathrm{F}$.\n\nSomeone provides the temperature rounded to whole degrees Fahrenheit, which w... | math | {
"style": "rule",
"ground_truth": "13/18"
} | {
"split": "train",
"index": 30383,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways can the integers from $-7$ to $7$ inclusive be arranged in a sequence such that the absolute value of the numbers in the sequence does not decrease? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "128"
} | {
"split": "train",
"index": 30384,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sequence of squares is made of identical square tiles. The edge of each square is one tile length longer than the edge of the previous square. The first three squares are shown. How many more tiles does the seventh square require than the sixth? Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 30385,
"solution": "\n1. **Identify the pattern in the sequence of squares:**\n Each square in the sequence has a side length that is one tile longer than the side of the previous square. Therefore, the side length of the $n$-th square is $n$ tiles.\n\n2. **Calculate the number of t... |
deepscaler | [
{
"role": "user",
"content": "Given that $x$ is a multiple of $12600$, what is the greatest common divisor of $g(x) = (5x + 7)(11x + 3)(17x + 8)(4x + 5)$ and $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "840"
} | {
"split": "train",
"index": 30386,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A modified octahedron consists of two pyramids, each with a pentagonal base, glued together along their pentagonal bases, forming a polyhedron with twelve faces. An ant starts at the top vertex and walks randomly to one of the five adjacent vertices in the middle ring. From thi... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"split": "train",
"index": 30387,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "You have a number of gold coins that you were going to distribute equally among your 11 best friends. However, after dividing your gold coins into 11 equal piles, you realize that if you give away all your gold coins, 2 people will receive an extra gold coin. You have less than... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 30388,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An open box is constructed by starting with a rectangular sheet of metal 10 in. by 14 in. and cutting a square of side $x$ inches from each corner. The resulting projections are folded up and the seams welded. The volume of the resulting box is: Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "140x - 48x^2 + 4x^3"
} | {
"split": "train",
"index": 30389,
"solution": "To find the volume of the box formed by cutting squares from each corner of a rectangular sheet and folding the sides up, we follow these steps:\n\n1. **Identify the dimensions of the original rectangle:**\n The original metal sheet is a rectangle with dimensions... |
deepscaler | [
{
"role": "user",
"content": "The slope angle of the tangent line to the curve y=\\frac{1}{3}x^{3} at x=1 is what value? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"split": "train",
"index": 30390,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the remainder when $x^4 +x + 2$ is divided by $x-3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "86"
} | {
"split": "train",
"index": 30391,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x$ be a real number selected uniformly at random between 100 and 300. If $\\lfloor \\sqrt{x} \\rfloor = 14$, find the probability that $\\lfloor \\sqrt{100x} \\rfloor = 140$.\nA) $\\frac{281}{2900}$\nB) $\\frac{4}{29}$\nC) $\\frac{1}{10}$\nD) $\\frac{96}{625}$\nE) $\\frac{... | math | {
"style": "rule",
"ground_truth": "\\frac{281}{2900}"
} | {
"split": "train",
"index": 30392,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The distance from the center \\( O \\) of a sphere with radius 12, which is circumscribed around a regular quadrangular pyramid, to a lateral edge is \\( 4 \\sqrt{2} \\). Find: \n1) the height of the pyramid;\n2) the distance from point \\( O \\) to the lateral face of the pyra... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{3}\\left(2 \\sqrt{2} - 1\\right)"
} | {
"split": "train",
"index": 30393,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Distinct points $A$, $B$, $C$, and $D$ lie on a line, with $AB=BC=CD=1$. Points $E$ and $F$ lie on a second line, parallel to the first, with $EF=1$. A triangle with positive area has three of the six points as its vertices. How many possible values are there for the area of th... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 30394,
"solution": "1. **Identify Possible Bases and Heights**: \n - The points $A$, $B$, $C$, and $D$ are collinear with equal distances between consecutive points, i.e., $AB = BC = CD = 1$.\n - Points $E$ and $F$ are also collinear on a different line parallel to the line contai... |
deepscaler | [
{
"role": "user",
"content": "If $Q = 5+2i$, $E = i$, and $D = 5-2i$, find $Q\\cdot E \\cdot D$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "29i"
} | {
"split": "train",
"index": 30395,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Nine delegates, three each from three different countries, randomly select chairs at a round table that seats nine people. Let the probability that each delegate sits next to at least one delegate from another country be $\\frac{m}{n}$, where $m$ and $n$ are relatively prime po... | math | {
"style": "rule",
"ground_truth": "097"
} | {
"split": "train",
"index": 30396,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given points $P(-2,7)$ and $Q(4,y)$ in a coordinate plane, for what value of $y$ is the slope of the line through $P$ and $Q$ equal to $\\frac{-3}{2}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 30397,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x_1,$ $x_2,$ $\\dots,$ $x_n$ be nonnegative real numbers such that $x_1 + x_2 + \\dots + x_n = 1$ and\n\\[x_1^2 + x_2^2 + \\dots + x_n^2 \\le \\frac{1}{100}.\\]Find the smallest possible value of $n.$ Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 30398,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( A \\subseteq \\{0, 1, 2, \\cdots, 29\\} \\) such that for any integers \\( k \\) and any numbers \\( a \\) and \\( b \\) (possibly \\( a = b \\)), the expression \\( a + b + 30k \\) is not equal to the product of two consecutive integers. Determine the maximum possible ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 30399,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For any positive integer $M$, the notation $M!$ denotes the product of the integers $1$ through $M$. What is the largest integer $n$ for which $5^n$ is a factor of the sum $98! + 99! + 100!$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 30400,
"solution": "1. **Factor out $98!$ from the sum**: \n \\[\n 98! + 99! + 100! = 98!(1 + 99 + 100 \\times 99)\n \\]\n Simplifying the expression inside the parentheses:\n \\[\n 1 + 99 + 9900 = 10000\n \\]\n Therefore, the expression becomes:\n \\[\n 98! \\time... |
deepscaler | [
{
"role": "user",
"content": "Given that $\\frac{a}{25-a}+\\frac{b}{65-b}+\\frac{c}{60-c}=7$, evaluate $\\frac{5}{25-a}+\\frac{13}{65-b}+\\frac{12}{60-c}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 30401,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many binary strings of length $10$ do not contain the substrings $101$ or $010$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "178"
} | {
"split": "train",
"index": 30402,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Supposed that $x$ and $y$ are nonzero real numbers such that $\\frac{3x+y}{x-3y}=-2$. What is the value of $\\frac{x+3y}{3x-y}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 30403,
"solution": "1. Start with the given equation:\n \\[\n \\frac{3x+y}{x-3y} = -2\n \\]\n\n2. Cross-multiply to eliminate the fraction:\n \\[\n 3x + y = -2(x - 3y)\n \\]\n\n3. Distribute the -2 on the right-hand side:\n \\[\n 3x + y = -2x + 6y\n \\]\n\n4. Rearran... |
deepscaler | [
{
"role": "user",
"content": "Calculate the lengths of the arcs of curves defined by the equations in polar coordinates.\n\n$$\n\\rho=5(1-\\cos \\varphi),-\\frac{\\pi}{3} \\leq \\varphi \\leq 0\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20 \\left(1 - \\frac{\\sqrt{3}}{2}\\right)"
} | {
"split": "train",
"index": 30404,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of two numbers is $10$; their product is $20$. The sum of their reciprocals 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": 30405,
"solution": "Given the equations:\n1. \\(x + y = 10\\)\n2. \\(xy = 20\\)\n\nWe are asked to find the sum of the reciprocals of \\(x\\) and \\(y\\), which is \\(\\frac{1}{x} + \\frac{1}{y}\\).\n\nUsing the identity for the sum of reciprocals:\n\\[\n\\frac{1}{x} + \\frac{1}{y} = ... |
deepscaler | [
{
"role": "user",
"content": "A library has six identical copies of a certain book. At any given time, some of these copies are at the library and some are checked out. How many different ways are there for some of the books to be in the library and the rest to be checked out if at least one book is in the ... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 30406,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Aaron takes a square sheet of paper, with one corner labeled $A$ . Point $P$ is chosen at random inside of the square and Aaron folds the paper so that points $A$ and $P$ coincide. He cuts the sheet along the crease and discards the piece containing $A$ . Let $p$ be t... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 30407,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The area of the smallest equilateral triangle with one vertex on each of the sides of the right triangle with side lengths $2\\sqrt{3},~5,$ and $\\sqrt{37},$ as shown, is $\\frac{m\\sqrt{p}}{n},$ where $m,~n,$ and $p$ are positive integers, $m$ and $n$ are relatively prime, and... | math | {
"style": "rule",
"ground_truth": "145"
} | {
"split": "train",
"index": 30408,
"solution": "Let $AB=2\\sqrt{3}, BC=5$, $D$ lies on $BC$, $F$ lies on $AB$ and $E$ lies on $AC$\n Set $D$ as the origin, $BD=a,BF=b$, $F$ can be expressed as $-a+bi$ in argand plane, the distance of $CD$ is $5-a$\n We know that $(-a+bi)\\cdot cis(-\\frac{\\pi}{3})=(-\\frac{a+\\... |
deepscaler | [
{
"role": "user",
"content": "Find $y$ so that the vectors $\\begin{pmatrix} 1 \\\\ -3 \\\\ -4 \\end{pmatrix}$ and $\\begin{pmatrix} -2 \\\\ y \\\\ -1 \\end{pmatrix}$ are orthogonal. 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": 30409,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute\n\\[\n\\begin{vmatrix} \\cos 1 & \\cos 2 & \\cos 3 \\\\ \\cos 4 & \\cos 5 & \\cos 6 \\\\ \\cos 7 & \\cos 8 & \\cos 9 \\end{vmatrix}\n.\\]All the angles are in radians. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 30410,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $r_{1}, r_{2}, r_{3}, r_{4}$ be the four roots of the polynomial $x^{4}-4 x^{3}+8 x^{2}-7 x+3$. Find the value of $\\frac{r_{1}^{2}}{r_{2}^{2}+r_{3}^{2}+r_{4}^{2}}+\\frac{r_{2}^{2}}{r_{1}^{2}+r_{3}^{2}+r_{4}^{2}}+\\frac{r_{3}^{2}}{r_{1}^{2}+r_{2}^{2}+r_{4}^{2}}+\\frac{r_{4}... | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 30411,
"solution": "Add 1 to each fraction to get $$\\frac{r_{1}^{2}+r_{2}^{2}+r_{3}^{2}+r_{4}^{2}}{r_{2}^{2}+r_{3}^{2}+r_{4}^{2}}+\\frac{r_{1}^{2}+r_{2}^{2}+r_{3}^{2}+r_{4}^{2}}{r_{1}^{2}+r_{3}^{2}+r_{4}^{2}}+\\frac{r_{1}^{2}+r_{2}^{2}+r_{3}^{2}+r_{4}^{2}}{r_{1}^{2}+r_{2}^{2}+r_{4}^{... |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, we have $AC=BC=7$ and $AB=2$. Suppose that $D$ is a point on line $AB$ such that $B$ lies between $A$ and $D$ and $CD=8$. What is $BD$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 30412,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the complex plane, let $A$ be the set of solutions to $z^{3}-8=0$ and let $B$ be the set of solutions to $z^{3}-8z^{2}-8z+64=0.$ What is the greatest distance between a point of $A$ and a point of $B?$ Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{21}"
} | {
"split": "train",
"index": 30413,
"solution": "1. **Solve for Set A:**\n The equation given is $z^3 - 8 = 0$. We can rewrite this as:\n \\[\n z^3 = 8\n \\]\n Recognizing that $8 = 2^3$, we can express $z$ in polar form as $z = r(\\cos \\theta + i \\sin \\theta)$, where $r^3 = 8$ and hence $r = 2$. Usi... |
deepscaler | [
{
"role": "user",
"content": "Solve the equation \\(\\sqrt{8x+5} + 2 \\{x\\} = 2x + 2\\). Here, \\(\\{x\\}\\) denotes the fractional part of \\(x\\), i.e., \\(\\{x\\} = x - \\lfloor x \\rfloor\\). Write down the sum of all solutions. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "0.75"
} | {
"split": "train",
"index": 30414,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = a\\sin(\\pi x + \\alpha) + b\\cos(\\pi x + \\beta)$, and it is known that $f(2001) = 3$, find the value of $f(2012)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 30415,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the set $\\{a, \\frac{b}{a}, 1\\} = \\{a^2, a+b, 0\\}$, find the value of $a^{2015} + b^{2016}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 30416,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "My school's math club has 6 boys and 8 girls. I need to select a team to send to the state math competition. We want 6 people on the team. In how many ways can I select the team without restrictions? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "3003"
} | {
"split": "train",
"index": 30417,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Niall's four children have different integer ages under 18. The product of their ages is 882. What is the sum of their ages? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 30418,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The bar graph shows the grades in a mathematics class for the last grading period. If A, B, C, and D are satisfactory grades, what fraction of the grades shown in the graph are satisfactory? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 30419,
"solution": "1. **Identify Satisfactory Grades**: According to the problem, grades A, B, C, and D are considered satisfactory.\n\n2. **Count the Number of Satisfactory Grades**:\n - Number of students with grade A = 5\n - Number of students with grade B = 4\n - Number of ... |
deepscaler | [
{
"role": "user",
"content": "In a dark room, a drawer contains 120 red socks, 100 green socks, 70 blue socks, and 50 black socks. A person selects socks one by one from the drawer without being able to see their color. What is the minimum number of socks that must be selected to ensure that at least 15 pai... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 30420,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Reduced to lowest terms, $\\frac{a^{2}-b^{2}}{ab} - \\frac{ab-b^{2}}{ab-a^{2}}$ is equal to: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{a}{b}"
} | {
"split": "train",
"index": 30421,
"solution": "1. **Factorize and simplify the expressions:**\n - The first term is already simplified: \\(\\frac{a^2-b^2}{ab}\\).\n - Factorize the numerator and denominator of the second term:\n \\[\n -\\frac{ab-b^2}{ab-a^2} = -\\frac{b(a-b)}{a(b-a)}.\n \\]\n ... |
deepscaler | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ that satisfies $a_1=1$ and $a_{n+1}=2S_n+1$, where $S_n$ is the sum of the first $n$ terms of $\\{a_n\\}$, and $n \\in \\mathbb{N}^*$.\n\n$(1)$ Find $a_n$.\n\n$(2)$ If the sequence $\\{b_n\\}$ satisfies $b_n=\\frac{1}{(1+\\log_3{a_n})(3+\\log_3{a_n}... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 30422,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the expression $\\frac{3^2 + 3^0 + 3^{-1} + 3^{-2}}{3^{-1} + 3^{-2} + 3^{-3} + 3^{-4}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3807}{180}"
} | {
"split": "train",
"index": 30423,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $$\\sqrt{6+4\\sqrt2}+\\sqrt{6-4\\sqrt2}.$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 30424,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A deck of cards now contains 54 cards, including two jokers, one being a red joker and the other a black joker, along with the standard 52 cards. In how many ways can we pick two different cards such that at least one of them is a joker? (Order matters.) Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "210"
} | {
"split": "train",
"index": 30425,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $d$ be a number such that when 229 is divided by $d$, the remainder is 4. Compute the sum of all possible two-digit values of $d$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "135"
} | {
"split": "train",
"index": 30426,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence $\\{a_n\\}$ with the sum of its first $n$ terms $S_n = 6n - n^2$, find the sum of the first $20$ terms of the sequence $\\left\\{ \\frac{1}{a_na_{n+1}}\\right\\}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{4}{35}"
} | {
"split": "train",
"index": 30427,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given sets \\( A = \\{ x \\mid 5x - a \\leq 0 \\} \\) and \\( B = \\{ x \\mid 6x - b > 0 \\} \\), where \\( a, b \\in \\mathbf{N} \\), and \\( A \\cap B \\cap \\mathbf{N} = \\{ 2, 3, 4 \\} \\), the number of integer pairs \\((a, b)\\) is: Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 30428,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The 5-digit number $52\\,28\\square$ is a multiple of 6. Which digit is represented by $\\square$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 30429,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Petya has seven cards with the digits 2, 2, 3, 4, 5, 6, 8. He wants to use all the cards to form the largest natural number that is divisible by 12. What number should he get? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8654232"
} | {
"split": "train",
"index": 30430,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the set \\( A \\) formed by exponential functions, there are 10 odd functions, 8 increasing functions defined on \\((-\\infty, \\infty)\\), and 12 functions whose graphs pass through the origin. Determine the minimum number of elements in set \\( A \\). Let's think step b... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 30431,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Amy, Beth, and Claire each have some sweets. Amy gives one third of her sweets to Beth. Beth gives one third of all the sweets she now has to Claire. Then Claire gives one third of all the sweets she now has to Amy. All the girls end up having the same number of sweets.\n\nClai... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 30432,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Danica drove her new car on a trip for a whole number of hours, averaging 55 miles per hour. At the beginning of the trip, $abc$ miles was displayed on the odometer, where $abc$ is a 3-digit number with $a \\geq 1$ and $a+b+c \\leq 7$. At the end of the trip, the odometer showe... | math | {
"style": "rule",
"ground_truth": "37"
} | {
"split": "train",
"index": 30433,
"solution": "1. **Understanding the Problem:**\n Danica drove her car for a whole number of hours at an average speed of 55 miles per hour. The odometer readings at the start and end of the trip are given as $abc$ and $cba$, respectively, where $abc$ and $cba$ are three-digit... |
deepscaler | [
{
"role": "user",
"content": "The sum\\[\\sum_{x=2}^{44} 2\\sin{x}\\sin{1}[1 + \\sec (x-1) \\sec (x+1)]\\]can be written in the form $\\sum_{n=1}^{4} (-1)^n \\frac{\\Phi(\\theta_n)}{\\Psi(\\theta_n)}$, where $\\Phi,\\, \\Psi$ are trigonometric functions and $\\theta_1,\\, \\theta_2, \\, \\theta_3, \\, \\the... | math | {
"style": "rule",
"ground_truth": "92"
} | {
"split": "train",
"index": 30434,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rugby team scored 24 points, 17 points, and 25 points in the seventh, eighth, and ninth games of their season. Their mean points-per-game was higher after 9 games than it was after their first 6 games. What is the smallest number of points that they could score in their 10th ... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 30435,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $n\\in N^{*}$, select $k(k\\in N, k\\geqslant 2)$ numbers $j\\_1$, $j\\_2$, $...$, $j\\_k$ from the set ${1,2,3,...,n}$ such that they simultaneously satisfy the following two conditions: $①1\\leqslant j\\_1 < j\\_2 < ...j\\_k\\leqslant n$; $②j_{i+1}-j_{i}\\geqslant ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 30436,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The equation $x^2+12x=73$ has two solutions. The positive solution has the form $\\sqrt{a}-b$ for positive natural numbers $a$ and $b$. What is $a+b$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "115"
} | {
"split": "train",
"index": 30437,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The value of $(\\sqrt{1+\\sqrt{1+\\sqrt{1}}})^{4}$ is:\n(a) $\\sqrt{2}+\\sqrt{3}$;\n(b) $\\frac{1}{2}(7+3 \\sqrt{5})$;\n(c) $1+2 \\sqrt{3}$;\n(d) 3 ;\n(e) $3+2 \\sqrt{2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3 + 2 \\sqrt{2}"
} | {
"split": "train",
"index": 30438,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose the point $(1,2)$ is on the graph of $y=\\frac{f(x)}2$. Then there is one point which must be on the graph of $y=\\frac{f^{-1}(x)}{2}$. What is the sum of that point's coordinates? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac 92"
} | {
"split": "train",
"index": 30439,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Trapezoid $EFGH$ has sides $EF=105$, $FG=45$, $GH=21$, and $HE=80$, with $EF$ parallel to $GH$. A circle with center $Q$ on $EF$ is drawn tangent to $FG$ and $HE$. Find the exact length of $EQ$ using fractions. Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "\\frac{336}{5}"
} | {
"split": "train",
"index": 30440,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mike and Harry play a game on an 8x8 board. Mike chooses \\( k \\) squares and writes an \\( M \\) in each of them. Harry then chooses \\( k+1 \\) squares and writes an \\( H \\) in each of them. After Harry is done, Mike wins if there is a sequence of letters forming \" \\( H ... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 30441,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the largest solution to the equation \\[\\log_{5x^3} 5 + \\log_{25x^4} 5 = -1.\\] Find the value of \\( x^{10} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.0000001024"
} | {
"split": "train",
"index": 30442,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be a parallelogram with $\\angle BAD < 90^{\\circ}$. A circle tangent to sides $\\overline{DA}$, $\\overline{AB}$, and $\\overline{BC}$ intersects diagonal $\\overline{AC}$ at points $P$ and $Q$ with $AP < AQ$, as shown. Suppose that $AP = 3$, $PQ = 9$, and $QC = 16$... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 30443,
"solution": "Let $\\omega$ be the circle, let $r$ be the radius of $\\omega$, and let the points at which $\\omega$ is tangent to $AB$, $BC$, and $AD$ be $H$, $K$, and $T$, respectively. PoP on $A$ and $C$ with respect to $\\omega$ yields \\[AT=6, CK=20.\\]\n Let $TG = AC, CG||... |
deepscaler | [
{
"role": "user",
"content": "Geoff walks on the number line for 40 minutes, starting at the point 0. On the $n$th minute, he flips a fair coin. If it comes up heads he walks $\\frac{1}{n}$ in the positive direction and if it comes up tails he walks $\\frac{1}{n}$ in the negative direction. Let $p$ be the p... | math | {
"style": "rule",
"ground_truth": "8101"
} | {
"split": "train",
"index": 30444,
"solution": "To estimate it by hand, we'll do casework on the most likely ways that Geoff will go past +2, and double the answer. If Geoff starts with one of the three sequences below, he will be past 2 or very close to 2: $$(+,+,+,+),(+,+,+,-,+,+),(+,+,-,+,+,+)$$ The probabili... |
deepscaler | [
{
"role": "user",
"content": "Let $b_n$ be the number obtained by writing the integers $1$ to $n$ from left to right and then subtracting $n$ from the resulting number. For example, $b_4 = 1234 - 4 = 1230$ and $b_{12} = 123456789101112 - 12 = 123456789101100$. For $1 \\le k \\le 100$, how many $b_k$ are div... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 30445,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Flights are arranged between 13 countries. For $ k\\ge 2$ , the sequence $ A_{1} ,A_{2} ,\\ldots A_{k}$ is said to a cycle if there exist a flight from $ A_{1}$ to $ A_{2}$ , from $ A_{2}$ to $ A_{3}$ , $ \\ldots$ , from $ A_{k \\minus{} 1}$ to $ A_{k}$ , and from ... | math | {
"style": "rule",
"ground_truth": "79"
} | {
"split": "train",
"index": 30446,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A binary operation $\\diamondsuit$ has the properties that $a\\,\\diamondsuit\\, (b\\,\\diamondsuit \\,c) = (a\\,\\diamondsuit \\,b)\\cdot c$ and that $a\\,\\diamondsuit \\,a=1$ for all nonzero real numbers $a, b,$ and $c$. (Here $\\cdot$ represents multiplication). Find the so... | math | {
"style": "rule",
"ground_truth": "\\frac{25}{84}"
} | {
"split": "train",
"index": 30447,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Convert $6532_8$ to base 5. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "102313_5"
} | {
"split": "train",
"index": 30448,
"solution": ""
} |
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