data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "A nine-joint bamboo tube has rice capacities of 4.5 *Sheng* in the lower three joints and 3.8 *Sheng* in the upper four joints. Find the capacity of the middle two joints. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2.5"
} | {
"split": "train",
"index": 38649,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x) = 3x-8$ and $g(f(x)) = 2x^2 + 5x - 3.$ Find $g(-5).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 38650,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\triangle{ABC}$ be a triangle with $AB = 10$ and $AC = 11$ . Let $I$ be the center of the inscribed circle of $\\triangle{ABC}$ . If $M$ is the midpoint of $AI$ such that $BM = BC$ and $CM = 7$ , then $BC$ can be expressed in the form $\\frac{\\sqrt{a}-b... | math | {
"style": "rule",
"ground_truth": "622"
} | {
"split": "train",
"index": 38651,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In \\\\(\\Delta ABC\\\\), given that \\\\(a= \\sqrt{3}, b= \\sqrt{2}, B=45^{\\circ}\\\\), find \\\\(A, C\\\\) and \\\\(c\\\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}- \\sqrt{2}}{2}"
} | {
"split": "train",
"index": 38652,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two students are preparing to register for the independent admission tests of Zhejiang University, Fudan University, and Shanghai Jiao Tong University, with the requirement that each person can choose at most two schools. Calculate the number of different registration results. ... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 38653,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A right triangle has legs $a$ and $b$ and hypotenuse $c.$ Find the largest possible value of\n\\[\\frac{a + b}{c}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 38654,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "You use a lock with four dials, each of which is set to a number between 0 and 9 (inclusive). You can never remember your code, so normally you just leave the lock with each dial one higher than the correct value. Unfortunately, last night someone changed all the values to 5. A... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 38655,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a,b,c,d$ be positive integers such that $a+c=20$ and $\\frac{a}{b}+\\frac{c}{d}<1$ . Find the maximum possible value of $\\frac{a}{b}+\\frac{c}{d}$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20/21"
} | {
"split": "train",
"index": 38656,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A long thin strip of paper is $1024$ units in length, $1$ unit in width, and is divided into $1024$ unit squares. The paper is folded in half repeatedly. For the first fold, the right end of the paper is folded over to coincide with and lie on top of the left end. The result is... | math | {
"style": "rule",
"ground_truth": "593"
} | {
"split": "train",
"index": 38657,
"solution": "We can keep track of the position of the square labeled 942 in each step. We use an $(x,y)$ coordinate system, so originally the 942 square is in the position $(942,1)$. In general, suppose that we've folded the strip into an array $r=2^k$ squares wide and $c=1024/... |
deepscaler | [
{
"role": "user",
"content": "The points $P(3,-2), Q(3,1), R(7,1)$, and $S$ form a rectangle. What are the coordinates of $S$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(7,-2)"
} | {
"split": "train",
"index": 38658,
"solution": "Since $P$ and $Q$ have the same $x$-coordinate, then side $PQ$ of the rectangle is vertical. This means that side $SR$ must also be vertical, and so the $x$-coordinate of $S$ is the same as the $x$-coordinate of $R$, which is 7. Since $Q$ and $R$ have the same $y$-... |
deepscaler | [
{
"role": "user",
"content": "In parallelogram $ABCD$, $\\overline{DE}$ is the altitude to the base $\\overline{AB}$ and $\\overline{DF}$ is the altitude to the base $\\overline{BC}$. [Note: Both pictures represent the same parallelogram.] If $DC=12$, $EB=4$, and $DE=6$, then $DF=$ Let's think step by step... | math | {
"style": "rule",
"ground_truth": "6.4"
} | {
"split": "train",
"index": 38659,
"solution": "To solve for $DF$, we first need to find the length of $AB$ using the information given about $EB$ and the properties of a parallelogram.\n\n1. **Identify the length of $AB$**:\n Since $ABCD$ is a parallelogram, opposite sides are equal, so $AB = DC = 12$.\n\n2. ... |
deepscaler | [
{
"role": "user",
"content": "In the geometric sequence $\\{a_n\\}$, $S_4=1$, $S_8=3$, then the value of $a_{17}+a_{18}+a_{19}+a_{20}$ is. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 38660,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $\\{a_n\\}$ is an arithmetic sequence with $$\na_1+a_2+ \\cdots +a_{100}=100 \\quad \\text{and} \\quad\na_{101}+a_{102}+ \\cdots + a_{200}=200.\n$$What is the value of $a_2 - a_1$? Express your answer as a common fraction. Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{100}"
} | {
"split": "train",
"index": 38661,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the positive divisors of 1184? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2394"
} | {
"split": "train",
"index": 38662,
"solution": "We start by finding the prime factors of 1184: $1184=2 \\cdot 592=2^{2} \\cdot 296=2^{3} \\cdot 148=2^{4} \\cdot 74=2^{5} \\cdot 37$. The positive divisors of 1184 cannot contain prime factors other than 2 and 37, and cannot contain more than 5 factors of 2 or 1 fa... |
deepscaler | [
{
"role": "user",
"content": "A circle of radius 1 is surrounded by 4 circles of radius $r$ as shown. What is $r$?\n\n[asy]\nunitsize(0.6cm);\nfor(int i=0; i<2; ++i){\nfor(int j=0; j<2; ++j){\ndraw(Circle((-2.4+4.8i,-2.4+4.8j),2.4),linewidth(0.7));\ndraw((-2.4+4.8i,-2.4+4.8j)--(-0.7+4.8i,-0.7+4.8j));\nlabel... | math | {
"style": "rule",
"ground_truth": "1+\\sqrt{2}"
} | {
"split": "train",
"index": 38663,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain cafeteria serves ham and cheese sandwiches, ham and tomato sandwiches, and tomato and cheese sandwiches. It is common for one meal to include multiple types of sandwiches. On a certain day, it was found that 80 customers had meals which contained both ham and cheese; ... | math | {
"style": "rule",
"ground_truth": "230"
} | {
"split": "train",
"index": 38664,
"solution": "230. Everyone who ate just one sandwich is included in exactly one of the first three counts, while everyone who ate more than one sandwich is included in all four counts. Thus, to count each customer exactly once, we must add the first three figures and subtract t... |
deepscaler | [
{
"role": "user",
"content": "The polynomial\n\\[ax^4 + bx^3 + cx^2 + dx + e = 0\\]has coefficients that are all integers, and has roots $-2,$ $5,$ $9,$ and $-1/3.$ If $e$ is a positive integer, then find its smallest possible value. Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 38665,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two positive integers \\(x\\) and \\(y\\), \\(xy - (x + y) = \\operatorname{HCF}(x, y) + \\operatorname{LCM}(x, y)\\), where \\(\\operatorname{HCF}(x, y)\\) and \\(\\operatorname{LCM}(x, y)\\) are respectively the greatest common divisor and the least common multiple of \... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 38666,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Experts and Viewers play \"What? Where? When?\" until one side wins six rounds. The probability of Experts winning a single round is 0.6, and there are no ties. Currently, the Experts are losing with a score of 3 to 4. Find the probability that the Experts will eventually win. ... | math | {
"style": "rule",
"ground_truth": "0.4752"
} | {
"split": "train",
"index": 38667,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, $A=60^{\\circ}$, $a=\\sqrt{6}$, $b=2$. \n$(1)$ Find $\\angle B$; \n$(2)$ Find the area of $\\triangle ABC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3 + \\sqrt{3}}{2}"
} | {
"split": "train",
"index": 38668,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest natural number that consists of identical digits and is divisible by 18. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "666"
} | {
"split": "train",
"index": 38669,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of all integers from 80 through 90, inclusive? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "935"
} | {
"split": "train",
"index": 38670,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We define two sequences of vectors $(\\mathbf{v}_n)$ and $(\\mathbf{w}_n)$ as follows: First, $\\mathbf{v}_0 = \\begin{pmatrix} 1 \\\\ 3 \\end{pmatrix},$ $\\mathbf{w}_0 = \\begin{pmatrix} 4 \\\\ 0 \\end{pmatrix}.$ Then for all $n \\ge 1,$ $\\mathbf{v}_n$ is the projection of $... | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 4/9 \\\\ 4/3 \\end{pmatrix}"
} | {
"split": "train",
"index": 38671,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many parallelograms with sides 1 and 2, and angles \\(60^{\\circ}\\) and \\(120^{\\circ}\\), can be placed inside a regular hexagon with side length 3? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 38672,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( S = \\{1, 2, \\ldots, 98\\} \\). Find the smallest natural number \\( n \\) such that in any \\( n \\)-element subset of \\( S \\), it is always possible to select 10 numbers, and no matter how these 10 numbers are divided into two groups of five, there will always be a... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 38673,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that Sia and Kira count sequentially, where Sia skips every fifth number, find the 45th number said in this modified counting game. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 38674,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rosencrantz plays $n \\leq 2015$ games of question, and ends up with a win rate (i.e. $\\frac{\\# \\text { of games won }}{\\# \\text { of games played }}$ ) of $k$. Guildenstern has also played several games, and has a win rate less than $k$. He realizes that if, after playing... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2015}"
} | {
"split": "train",
"index": 38675,
"solution": "Write $k=\\frac{m}{n}$, for relatively prime integers $m, n$. For the property not to hold, there must exist integers $a$ and $b$ for which $$\\frac{a}{b}<\\frac{m}{n}<\\frac{a+1}{b+1}$$ (i.e. at some point, Guildenstern must \"jump over\" $k$ with a single win) $$... |
deepscaler | [
{
"role": "user",
"content": "Only nine out of the original thirteen colonies had to ratify the U.S. Constitution in order for it to take effect. What is this ratio, nine to thirteen, rounded to the nearest tenth? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.7"
} | {
"split": "train",
"index": 38676,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n=x-y^{x-y}$. Find $n$ when $x=2$ and $y=-2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-14"
} | {
"split": "train",
"index": 38677,
"solution": "1. **Substitute the values of $x$ and $y$ into the expression for $n$:**\n \\[ n = x - y^{x-y} \\]\n Given $x = 2$ and $y = -2$, substitute these values:\n \\[ n = 2 - (-2)^{2 - (-2)} \\]\n\n2. **Simplify the exponent:**\n \\[ 2 - (-2) = 2 + 2 = 4 \\]\n S... |
deepscaler | [
{
"role": "user",
"content": "If $A\\ \\clubsuit\\ B$ is defined as $A\\ \\clubsuit\\ B = 3A + 2B + 5$, what is the value of $A$ for which $A\\ \\clubsuit\\ 4 = 58$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 38678,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $m,n$ be natural numbers such that $\\hspace{2cm} m+3n-5=2LCM(m,n)-11GCD(m,n).$ Find the maximum possible value of $m+n$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "70"
} | {
"split": "train",
"index": 38679,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The Lucas numbers $L_n$ are given by the recursion $L_0 = 2, L_1 = 1$, and $L_n = L_{n-1} + L_{n-2}$. Find the units digit of $L_{L_{10}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 38680,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When simplified $\\sqrt{1+ \\left (\\frac{x^4-1}{2x^2} \\right )^2}$ equals: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{x^2}{2}+\\frac{1}{2x^2}"
} | {
"split": "train",
"index": 38681,
"solution": "1. **Start by rewriting the expression inside the square root:**\n \\[\n \\sqrt{1+ \\left(\\frac{x^4-1}{2x^2}\\right)^2}\n \\]\n We first simplify the squared term:\n \\[\n \\left(\\frac{x^4-1}{2x^2}\\right)^2 = \\frac{(x^4-1)^2}{(2x^2)^2} = \\frac{(x^4... |
deepscaler | [
{
"role": "user",
"content": "(Answer in numbers) \nFrom 5 different storybooks and 4 different math books, 4 books are to be selected and given to 4 students, one book per student. How many different ways are there to: \n(1) Select 2 storybooks and 2 math books? \n(2) Ensure one specific storybook and o... | math | {
"style": "rule",
"ground_truth": "1080"
} | {
"split": "train",
"index": 38682,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A 6 cm by 8 cm rectangle is inscribed in a circle. What is the number of centimeters in the circumference of the circle? Express your answer in terms of $\\pi$.\n\n[asy]import graph;\ndefaultpen(linewidth(0.7));\ndraw(Circle((0,0),20));\ndraw((-16,-12)--(16,-12)--(16,12)--(-16,... | math | {
"style": "rule",
"ground_truth": "10\\pi"
} | {
"split": "train",
"index": 38683,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A boss and two engineers are to meet at a park. Each of them arrives at a random time between 1:00 PM and 3:00 PM. The boss leaves instantly if not both engineers are present upon his arrival. Each engineer will wait for up to 1.5 hours for the other before leaving. What is the... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 38684,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "1. \\(\\lim _{x \\rightarrow 1}(1-x) \\operatorname{tg} \\frac{\\pi x}{2}\\)\n2. \\(\\lim _{x \\rightarrow \\frac{\\pi}{4}}\\left(\\frac{\\pi}{4}-x\\right) \\operatorname{cosec}\\left(\\frac{3}{4} \\pi+x\\right)\\)\n3. \\(\\lim _{x \\rightarrow+\\infty} x \\operatorname{arcctg}... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 38685,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, points \\(B\\), \\(C\\), and \\(D\\) lie on a line. Also, \\(\\angle ABC = 90^\\circ\\) and \\(\\angle ACD = 150^\\circ\\). The value of \\(x\\) is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 38686,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The equation of the line that passes through the points $(-3,5)$ and $(0,-4)$ can be expressed in the form $y=mx+b$. What is the value of $m+b$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"split": "train",
"index": 38687,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many whole numbers between 1 and 500 do not contain the digit 2? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "323"
} | {
"split": "train",
"index": 38688,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Add 74.6893 to 23.152 and round to the nearest hundredth. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "97.84"
} | {
"split": "train",
"index": 38689,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Divide the sides of a unit square \\(ABCD\\) into 5 equal parts. Let \\(D'\\) denote the second division point from \\(A\\) on side \\(AB\\), and similarly, let the second division points from \\(B\\) on side \\(BC\\), from \\(C\\) on side \\(CD\\), and from \\(D\\) on side \\(... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{29}"
} | {
"split": "train",
"index": 38690,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Many states are considering a new license-plate pattern that consists of a sequence of four letters followed by a sequence of four digits. Assuming that each four-letter four-digit arrangement is equally likely, what is the probability that such a license plate will contain at ... | math | {
"style": "rule",
"ground_truth": "2735"
} | {
"split": "train",
"index": 38691,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The last three digits of \\( 1978^n \\) and \\( 1978^m \\) are the same. Find the positive integers \\( m \\) and \\( n \\) such that \\( m+n \\) is minimized (here \\( n > m \\geq 1 \\)). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "106"
} | {
"split": "train",
"index": 38692,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a positive integer \\( A \\) whose prime factorization can be written as \\( A = 2^{\\alpha} \\times 3^{\\beta} \\times 5^{\\gamma} \\), where \\(\\alpha, \\beta, \\gamma\\) are natural numbers. If one-half of \\( A \\) is a perfect square, one-third of \\( A \\) is a per... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 38693,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Name the number that consists of 11 hundreds, 11 tens, and 11 units. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1221"
} | {
"split": "train",
"index": 38694,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of functions $f : \\mathbb{R} \\to \\mathbb{R}$ such that\n\\[f(xy) + f(xz) - f(x) f(yz) \\ge 1\\]for all real numbers $x,$ $y,$ and $z.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 38695,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $O$ be the origin, and let $(a,b,c)$ be a fixed point. A plane passes through $(a,b,c)$ and intersects the $x$-axis, $y$-axis, and $z$-axis at $A,$ $B,$ and $C,$ respectively, all distinct from $O.$ Let $(p,q,r)$ be the center of the sphere passing through $A,$ $B,$ $C,$ ... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 38696,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a rectangle, the perimeter of quadrilateral $PQRS$ is given. If the horizontal distance between adjacent dots in the same row is 1 and the vertical distance between adjacent dots in the same column is 1, what is the perimeter of quadrilateral $PQRS$? Let's think step by step... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 38697,
"solution": "The perimeter of quadrilateral $PQRS$ equals $PQ+QR+RS+SP$. Since the dots are spaced 1 unit apart horizontally and vertically, then $PQ=4, QR=4$, and $PS=1$. Thus, the perimeter equals $4+4+RS+1$ which equals $RS+9$. We need to determine the length of $RS$. If we ... |
deepscaler | [
{
"role": "user",
"content": "What is the least four-digit whole number that is both a perfect square and a perfect cube? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4096"
} | {
"split": "train",
"index": 38698,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given real numbers $x$ and $y$ satisfying $x^{2}+2y^{2}-2xy=4$, find the maximum value of $xy$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2} + 2"
} | {
"split": "train",
"index": 38699,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\tan \\alpha =2$, find the value of $\\frac{4\\sin^{3} \\alpha -2\\cos \\alpha }{5\\cos \\alpha +3\\sin \\alpha }$. 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": 38700,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Cindy wants to arrange her coins into $X$ piles, each consisting of the same number of coins, $Y$. Each pile will have more than one coin and no pile will have all the coins. If there are 16 possible values for $Y$ given all of the restrictions, what is the smallest number of c... | math | {
"style": "rule",
"ground_truth": "131072"
} | {
"split": "train",
"index": 38701,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $\\left(1 - \\frac{1}{2}\\right)\\left(1-\\frac{1}{3}\\right)\\left(1-\\frac{1}{4}\\right)\\left(1-\\frac{1}{5}\\right) \\dotsm \\left(1-\\frac{1}{50}\\right)$? Express your answer as a common fraction. Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{50}"
} | {
"split": "train",
"index": 38702,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A point is chosen at random within the square in the coordinate plane whose vertices are $(0, 0), (2020, 0), (2020, 2020),$ and $(0, 2020)$. The probability that the point is within $d$ units of a lattice point is $\\frac{1}{2}$. (A point $(x, y)$ is a lattice point if $x$ and ... | math | {
"style": "rule",
"ground_truth": "0.4"
} | {
"split": "train",
"index": 38703,
"solution": "\nWe are given a square with vertices at $(0, 0), (2020, 0), (2020, 2020),$ and $(0, 2020)$, and we need to find the radius $d$ such that the probability a randomly chosen point within the square is within $d$ units of a lattice point is $\\frac{1}{2}$. \n\n#### St... |
deepscaler | [
{
"role": "user",
"content": "What is the value of $c$ if the lines with equations $y = 8x + 2$ and $y = (2c)x - 4$ are parallel? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 38704,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $N$ be the smallest positive integer for which $$x^{2}+x+1 \\quad \\text { divides } \\quad 166-\\sum_{d \\mid N, d>0} x^{d}$$ Find the remainder when $N$ is divided by 1000. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "N \\equiv 672 \\pmod{1000}"
} | {
"split": "train",
"index": 38705,
"solution": "Let $\\omega=e^{2 \\pi i / 3}$. The condition is equivalent to $$166=\\sum_{d \\mid N, d>0} \\omega^{d}$$ Let's write $N=3^{d} n$ where $n$ is not divisible by 3. If all primes dividing $n$ are $1 \\bmod 3$, then $N$ has a positive number of factors that are $1 \\b... |
deepscaler | [
{
"role": "user",
"content": "If 700 were expressed as a sum of at least three distinct powers of 2, what would be the least possible sum of the exponents of these powers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 38706,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of the first 10 elements that appear both among the terms of the arithmetic progression $\\{4,7,10,13, \\ldots\\}$ and the geometric progression $\\{20,40,80,160, \\ldots\\}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13981000"
} | {
"split": "train",
"index": 38707,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A tetrahedron \\(ABCD\\) has edge lengths 7, 13, 18, 27, 36, 41, with \\(AB = 41\\). Determine the length of \\(CD\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 38708,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The equations of the asymptotes of the hyperbola $\\frac{x^2}{2}-y^2=1$ are ________, and its eccentricity is ________. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{2}"
} | {
"split": "train",
"index": 38709,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a set $A$ with 10 elements, find the number of consistent 2-configurations of $A$ of order 2 with exactly 2 cells. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "99144"
} | {
"split": "train",
"index": 38710,
"solution": "Notice that if we look only at the pairs contained within any fixed cell, each element of that cell still lies in 2 such pairs, since all the pairs it belongs to are contained within that cell. Thus we have an induced consistent 2-configuration of order 2 of each c... |
deepscaler | [
{
"role": "user",
"content": "Compute $26\\times33+67\\times26$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2600"
} | {
"split": "train",
"index": 38711,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\\\( \\{a_n\\} \\\\) be a sequence with the sum of the first \\\\( n \\\\) terms denoted as \\\\( S_n \\\\). If \\\\( S_2 = 4 \\\\) and \\\\( a_{n+1} = 2S_n + 1 \\\\) where \\\\( n \\in \\mathbb{N}^* \\\\), find the values of \\\\( a_1 \\\\) and \\\\( S_5 \\\\). Let's thin... | math | {
"style": "rule",
"ground_truth": "121"
} | {
"split": "train",
"index": 38712,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a circle, $15$ equally spaced points are drawn and arbitrary triangles are formed connecting $3$ of these points. How many non-congruent triangles can be drawn? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 38713,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A conference lasted for 2 days. On the first day, the sessions lasted for 7 hours and 15 minutes, and on the second day, they lasted for 8 hours and 45 minutes. Calculate the total number of minutes the conference sessions lasted. Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "960"
} | {
"split": "train",
"index": 38714,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a}=(\\sin \\theta, 2)$ and $\\overrightarrow{b}=(\\cos \\theta, 1)$, which are collinear, where $\\theta \\in (0, \\frac{\\pi}{2})$.\n1. Find the value of $\\tan (\\theta + \\frac{\\pi}{4})$.\n2. If $5\\cos (\\theta - \\phi)=3 \\sqrt{5}\\cos \\ph... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"split": "train",
"index": 38715,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $a$ and $b$ are positive numbers, and $a+b=1$, find the minimum value of $\\frac{1}{2a} + \\frac{1}{b}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2} + \\frac{3}{2}"
} | {
"split": "train",
"index": 38716,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the function $f(x) = e^x + \\frac{a}{e^x}$ has a derivative $y = f'(x)$ that is an odd function and the slope of a tangent line to the curve $y = f(x)$ is $\\frac{3}{2}$, determine the abscissa of the tangent point. Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "\\ln 2"
} | {
"split": "train",
"index": 38717,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The necessary and sufficient condition for the graph of the function $y=f(x)$ to be symmetric with respect to the origin is that the function $y=f(x)$ is an odd function. This can be extended as follows: The necessary and sufficient condition for the graph of the function $y=f(... | math | {
"style": "rule",
"ground_truth": "-8086"
} | {
"split": "train",
"index": 38718,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the isosceles triangle \\(ABC\\) with the sides \\(AB = BC\\), the angle \\(\\angle ABC\\) is \\(80^\\circ\\). Inside the triangle, a point \\(O\\) is taken such that \\(\\angle OAC = 10^\\circ\\) and \\(\\angle OCA = 30^\\circ\\). Find the angle \\(\\angle AOB\\). Let's thi... | math | {
"style": "rule",
"ground_truth": "70"
} | {
"split": "train",
"index": 38719,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all integers \\( z \\) for which exactly two of the following five statements are true, and three are false:\n\n1) \\( 2z > 130 \\)\n2) \\( z < 200 \\)\n3) \\( 3z > 50 \\)\n4) \\( z > 205 \\)\n5) \\( z > 15 \\) Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 38720,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x_1, x_2, \\ldots , x_n$ be a sequence of integers such that\n(i) $-1 \\le x_i \\le 2$ for $i = 1,2, \\ldots n$\n(ii) $x_1 + \\cdots + x_n = 19$; and\n(iii) $x_1^2 + x_2^2 + \\cdots + x_n^2 = 99$.\nLet $m$ and $M$ be the minimal and maximal possible values of $x_1^3 + \\cd... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 38721,
"solution": "1. **Identify the range and possible values of $x_i$:**\n Given that $-1 \\leq x_i \\leq 2$ for all $i$, and $x_i$ are integers, the possible values for each $x_i$ are $-1, 0, 1, 2$. However, since adding or removing zeros does not affect the sums or sums of squa... |
deepscaler | [
{
"role": "user",
"content": "The least common multiple of $a$ and $b$ is $20$, and the least common multiple of $b$ and $c$ is $21$. Find the least possible value of the least common multiple of $a$ and $c$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "420"
} | {
"split": "train",
"index": 38722,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If a number eight times as large as $x$ is increased by two, then one fourth of the result equals Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2x + \\frac{1}{2}"
} | {
"split": "train",
"index": 38723,
"solution": "1. **Identify the expression**: The problem states that a number eight times as large as $x$ is increased by two. This can be expressed mathematically as:\n \\[\n 8x + 2\n \\]\n\n2. **Calculate one fourth of the result**: We need to find one fourth of the exp... |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, with the origin $O$ as the pole and the non-negative half-axis of the $x$-axis as the polar axis, a polar coordinate system is established. It is known that the polar equation of curve $C$ is $\\rho^{2}= \\dfrac {16}{1+3\\sin ^{2}\\thet... | math | {
"style": "rule",
"ground_truth": "2 \\sqrt {2}+4"
} | {
"split": "train",
"index": 38724,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The digits 2, 3, 5 and 7 are arranged randomly to form a four-digit number. What is the probability that the number is odd? Express your answer as a common fraction. 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": 38725,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Regular octagon $A_1A_2A_3A_4A_5A_6A_7A_8$ is inscribed in a circle of area $1.$ Point $P$ lies inside the circle so that the region bounded by $\\overline{PA_1},\\overline{PA_2},$ and the minor arc $\\widehat{A_1A_2}$ of the circle has area $\\tfrac{1}{7},$ while the region bo... | math | {
"style": "rule",
"ground_truth": "504"
} | {
"split": "train",
"index": 38726,
"solution": "The actual size of the diagram doesn't matter. To make calculation easier, we discard the original area of the circle, $1$, and assume the side length of the octagon is $2$. Let $r$ denote the radius of the circle, $O$ be the center of the circle. Then $r^2= 1^2 + ... |
deepscaler | [
{
"role": "user",
"content": "Given square $PQRS$ with side length $12$ feet, a circle is drawn through vertices $P$ and $S$, and tangent to side $QR$. If the point of tangency divides $QR$ into segments of $3$ feet and $9$ feet, calculate the radius of the circle. Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{(6 - 3\\sqrt{2})^2 + 9^2}"
} | {
"split": "train",
"index": 38727,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many lines pass through exactly two points in the following hexagonal grid? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 38728,
"solution": "First solution. From a total of 19 points, there are $\\binom{19}{2}=171$ ways to choose two points. We consider lines that pass through more than 2 points. - There are $6+6+3=15$ lines that pass through exactly three points. These are: the six sides of the largest... |
deepscaler | [
{
"role": "user",
"content": "The smallest positive integer $x$ for which $1260x=N^3$, where $N$ is an integer, is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7350"
} | {
"split": "train",
"index": 38729,
"solution": "1. **Factorize 1260**: We start by factorizing 1260 into its prime factors:\n \\[\n 1260 = 2^2 \\cdot 3^2 \\cdot 5 \\cdot 7\n \\]\n\n2. **Condition for $N^3$**: For $N^3$ to be a perfect cube, each prime factor in its factorization must have an exponent that ... |
deepscaler | [
{
"role": "user",
"content": "Quadrilateral \\(ABCD\\) is inscribed in a circle with diameter \\(AD\\) having a length of 4. If the lengths of \\(AB\\) and \\(BC\\) are each 1, calculate the length of \\(CD\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{2}"
} | {
"split": "train",
"index": 38730,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $f(2)$ given that $f$ is a real-valued function that satisfies the equation $$ 4f(x)+\\left(\\frac23\\right)(x^2+2)f\\left(x-\\frac2x\\right)=x^3+1. $$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{19}{12}"
} | {
"split": "train",
"index": 38731,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When $p(x) = Ax^5 + Bx^3 + Cx + 4$ is divided by $x - 3,$ the remainder is 11. Find the remainder when $p(x)$ is divided by $x + 3.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 38732,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the sum shown, each letter represents a different digit with $T \\neq 0$ and $W \\neq 0$. How many different values of $U$ are possible? \n\n\\begin{tabular}{rrrrr} \n& $W$ & $X$ & $Y$ & $Z$ \\\\ \n+ & $W$ & $X$ & $Y$ & $Z$ \\\\ \n\\hline & $W$ & $U$ & $Y$ & $V$ \n\\end{tabu... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 38733,
"solution": "Since $WXYZ$ is a four-digit positive integer, then $WXYZ \\leq 9999$. (In fact $WXYZ$ cannot be this large since all of its digits must be different.) Since $WXYZ \\leq 9999$, then $TWUYV \\leq 2(9999) = 19998$. Since $T \\neq 0$, then $T = 1$. Next, we note that ... |
deepscaler | [
{
"role": "user",
"content": "The area of polygon $ABCDEF$, in square units, is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "46"
} | {
"split": "train",
"index": 38734,
"solution": "1. **Identify the area of polygon $ABCDEF$**: Let $\\langle ABCDEF \\rangle$ represent the area of polygon $ABCDEF$. Introduce point $G$ as the intersection of the extended lines $DC$ and $AF$.\n\n2. **Express the area of $ABCDEF$ in terms of other areas**: \n \\... |
deepscaler | [
{
"role": "user",
"content": "Person A and person B start walking towards each other from locations A and B simultaneously. The speed of person B is $\\frac{3}{2}$ times the speed of person A. After meeting for the first time, they continue to their respective destinations, and then immediately return. Give... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 38735,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define $\\#N$ by the formula $\\#N = .5(N) + 1$. Calculate $\\#(\\#(\\#58))$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 38736,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $N$ be the smallest positive integer $N$ such that $2008N$ is a perfect square and $2007N$ is a perfect cube. Find the remainder when $N$ is divided by $25$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 38737,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each of \\( A \\), \\( B \\), \\( C \\), and \\( D \\) is a positive two-digit integer. These integers satisfy each of the equations\n\n\\[\n\\begin{aligned}\nB &= 3C \\\\\nD &= 2B - C \\\\\nA &= B + D\n\\end{aligned}\n\\]\n\nWhat is the largest possible value of \\( A + B + C ... | math | {
"style": "rule",
"ground_truth": "204"
} | {
"split": "train",
"index": 38738,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The price of an article was increased $p\\%$. Later the new price was decreased $p\\%$. If the last price was one dollar, the original price was: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{10000}{10000-p^2}"
} | {
"split": "train",
"index": 38739,
"solution": "1. **Identify the variables and setup the equation:**\n Let $x$ be the original price of the article. The price is first increased by $p\\%$, and then the new price is decreased by $p\\%$. We need to find the value of $x$ such that after these changes, the final ... |
deepscaler | [
{
"role": "user",
"content": "Given that line $MN$ passes through the left focus $F$ of the ellipse $\\frac{x^{2}}{2}+y^{2}=1$ and intersects the ellipse at points $M$ and $N$. Line $PQ$ passes through the origin $O$ and is parallel to $MN$, intersecting the ellipse at points $P$ and $Q$. Find the value of ... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"split": "train",
"index": 38740,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\sin \\alpha + \\cos \\alpha = - \\frac{\\sqrt{5}}{2}$, and $\\frac{5\\pi}{4} < \\alpha < \\frac{3\\pi}{2}$, calculate the value of $\\cos \\alpha - \\sin \\alpha$. 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": 38741,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If\n\\[x + \\sqrt{x^2 - 1} + \\frac{1}{x - \\sqrt{x^2 - 1}} = 20,\\]then find\n\\[x^2 + \\sqrt{x^4 - 1} + \\frac{1}{x^2 + \\sqrt{x^4 - 1}}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{10201}{200}"
} | {
"split": "train",
"index": 38742,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Starting with the display \"1,\" calculate the fewest number of keystrokes needed to reach \"240\" using only the keys [+1] and [x2]. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 38743,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the positive real number $x$ for which\n\\[5 \\sqrt{1 + x} + 5 \\sqrt{1 - x} = 7 \\sqrt{2}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{25}"
} | {
"split": "train",
"index": 38744,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $a_1 = 1$ , and that for all $n \\ge 2$ , $a_n = a_{n-1} + 2a_{n-2} + 3a_{n-3} + \\ldots + (n-1)a_1.$ Suppose furthermore that $b_n = a_1 + a_2 + \\ldots + a_n$ for all $n$ . If $b_1 + b_2 + b_3 + \\ldots + b_{2021} = a_k$ for some $k$ , find $k$ .\n\n*P... | math | {
"style": "rule",
"ground_truth": "2022"
} | {
"split": "train",
"index": 38745,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let point $P$ be a point on the ellipse $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 (a > b > 0)$. Let $F_1$ and $F_2$ respectively be the left and right foci of the ellipse, and let $I$ be the incenter of $\\triangle PF_1F_2$. If $S_{\\triangle IPF_1} + S_{\\triangle IPF_2} = 2S_{... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 38746,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many two-digit numbers can be written in the form $2^n$ where $n$ is an integer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 38747,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x=\\frac{a}{b}$, $a\\neq b$ and $b\\neq 0$, then $\\frac{a+b}{a-b}=$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{x+1}{x-1}"
} | {
"split": "train",
"index": 38748,
"solution": "1. Given that $x = \\frac{a}{b}$, we can express $a$ in terms of $x$ and $b$:\n \\[\n a = bx\n \\]\n\n2. Substitute $a = bx$ into the expression $\\frac{a+b}{a-b}$:\n \\[\n \\frac{a+b}{a-b} = \\frac{bx + b}{bx - b}\n \\]\n\n3. Factor out $b$ from both t... |
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