data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "If \\( S = \\sum_{k=1}^{99} \\frac{(-1)^{k+1}}{\\sqrt{k(k+1)}(\\sqrt{k+1}-\\sqrt{k})} \\), find the value of \\( 1000 S \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1100"
} | {
"split": "train",
"index": 19049,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $b= \\sqrt {2}a$, $\\sqrt {3}\\cos B= \\sqrt {2}\\cos A$, $c= \\sqrt {3}+1$. Find the area of $\\triangle ABC$. Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "\\frac { \\sqrt {3}+1}{2}"
} | {
"split": "train",
"index": 19050,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\Delta ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, where $b=6$, and the area of $\\Delta ABC$ is $15$. The radius of the circumcircle is $5$.\n\n(Ⅰ) Find the value of $\\sin 2B$;\n\n(Ⅱ) Find the perimeter of $\\De... | math | {
"style": "rule",
"ground_truth": "6+6 \\sqrt {6}"
} | {
"split": "train",
"index": 19051,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The length of the longer side of rectangle $R$ is $10$ percent more than the length of a side of square $S.$ The length of the shorter side of rectangle $R$ is $10$ percent less than the length of a side of square $S.$ What is the ratio of the area of rectangle $R$ to the area ... | math | {
"style": "rule",
"ground_truth": "\\frac{99}{100}"
} | {
"split": "train",
"index": 19052,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two numbers are independently selected from the set of positive integers less than or equal to 7. What is the probability that the sum of the two numbers is less than their product? 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{36}{49}"
} | {
"split": "train",
"index": 19053,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of complex solutions to\n\\[\\frac{z^3 - 1}{z^2 + z - 2} = 0.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 19054,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "During the National Day military parade, the marching sequences of three formations, namely A, B, and C, pass by the viewing stand in a certain order. If the order is randomly arranged, the probability that B passes before both A and C is ( ). Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 19055,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given non-zero vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ satisfy $|\\overrightarrow{a}|=2|\\overrightarrow{b}|$, and $(\\overrightarrow{a}-\\overrightarrow{b})\\bot \\overrightarrow{b}$, calculate the angle between $\\overrightarrow{a}$ and $\\overrightarrow{b}$. ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"split": "train",
"index": 19056,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jeffrey jogs $3$ miles at $4$ miles per hour. Then, he walks $4$ miles at $3$ miles an hour. What was Jeffrey's average speed in miles per hour for the entire trip? Express your answer as a decimal to the nearest hundredth. Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "3.36"
} | {
"split": "train",
"index": 19057,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many different points could be the fin... | math | {
"style": "rule",
"ground_truth": "221"
} | {
"split": "train",
"index": 19058,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a peculiar computer with a button. If the current number on the screen is a multiple of 3, pressing the button will divide it by 3. If the current number is not a multiple of 3, pressing the button will multiply it by 6. Xiaoming pressed the button 6 times without look... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 19059,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of real solutions to the equation\n\\[\n\\frac{1}{x - 1} + \\frac{2}{x - 2} + \\frac{3}{x - 3} + \\dots + \\frac{50}{x - 50} = x + 5.\n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "51"
} | {
"split": "train",
"index": 19060,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For breakfast, Milan is eating a piece of toast shaped like an equilateral triangle. On the piece of toast rests a single sesame seed that is one inch away from one side, two inches away from another side, and four inches away from the third side. He places a circular piece of ... | math | {
"style": "rule",
"ground_truth": "\\frac{49 \\pi}{9}"
} | {
"split": "train",
"index": 19061,
"solution": "Suppose the toast has side length $s$. If we draw the three line segments from the sesame seed to the three vertices of the triangle, we partition the triangle into three smaller triangles, with areas $\\frac{s}{2}, s$, and $2 s$, so the entire piece of toast has a... |
deepscaler | [
{
"role": "user",
"content": "On each side of a square, a point is taken. It turns out that these points are the vertices of a rectangle whose sides are parallel to the diagonals of the square. Find the perimeter of the rectangle if the diagonal of the square is 6. Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 19062,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bag contains four pieces of paper, each labeled with one of the digits $1$, $2$, $3$ or $4$, with no repeats. Three of these pieces are drawn, one at a time without replacement, to construct a three-digit number. What is the probability that the three-digit number is a multip... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 19063,
"solution": "1. **Total number of ways to draw three digits from four**: \n The bag contains four distinct digits: $1, 2, 3, 4$. We are to draw three digits without replacement to form a three-digit number. The number of ways to choose the first digit is $4$, the second digit... |
deepscaler | [
{
"role": "user",
"content": "Given a circle $O: x^2 + y^2 = 6$, and $P$ is a moving point on circle $O$. A perpendicular line $PM$ is drawn from $P$ to the x-axis at $M$, and $N$ is a point on $PM$ such that $\\overrightarrow{PM} = \\sqrt{2} \\overrightarrow{NM}$.\n\n(Ⅰ) Find the equation of the trajectory... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 19064,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Assume $n$ is a positive integer. Considers sequences $a_0, a_1, \\ldots, a_n$ for which $a_i \\in \\{1, 2, \\ldots , n\\}$ for all $i$ and $a_n = a_0$. \r\n\r\n(a) Suppose $n$ is odd. Find the number of such sequences if $a_i - a_{i-1} \\not \\equiv i \\pmod{n}$ for all $i = 1... | math | {
"style": "rule",
"ground_truth": "(n-1)(n-2)^{n-1} - \\frac{2^{n-1} - 1}{n} - 1"
} | {
"split": "train",
"index": 19065,
"solution": "\nLet \\( n \\) be a positive integer. Consider sequences \\( a_0, a_1, \\ldots, a_n \\) for which \\( a_i \\in \\{1, 2, \\ldots , n\\} \\) for all \\( i \\) and \\( a_n = a_0 \\).\n\n### Part (a)\nSuppose \\( n \\) is odd. We need to find the number of such sequen... |
deepscaler | [
{
"role": "user",
"content": "A regular polygon of $n$ sides is inscribed in a circle of radius $R$. The area of the polygon is $3R^2$. Then $n$ equals: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 19066,
"solution": "1. **Understanding the Problem**: We are given a regular polygon with $n$ sides inscribed in a circle of radius $R$. The area of the polygon is given as $3R^2$. We need to find the value of $n$.\n\n2. **Area of a Regular Polygon**: The area $A$ of a regular polygon... |
deepscaler | [
{
"role": "user",
"content": "My five friends and I play doubles badminton every weekend. Each weekend, two of us play as a team against another two, while the remaining two rest. How many different ways are there for us to choose the two teams and the resting pair? Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 19067,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In trapezoid $PQRS$ with $PQ$ parallel to $RS$, the diagonals $PR$ and $QS$ intersect at $T$. If the area of triangle $PQT$ is 75 square units, and the area of triangle $PST$ is 30 square units, calculate the area of trapezoid $PQRS$. Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "147"
} | {
"split": "train",
"index": 19068,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $$\\sum_{n=0}^{\\infty} \\frac{n}{n^{4}+n^{2}+1}$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 19069,
"solution": "Note that $$n^{4}+n^{2}+1=\\left(n^{4}+2 n^{2}+1\\right)-n^{2}=\\left(n^{2}+1\\right)^{2}-n^{2}=\\left(n^{2}+n+1\\right)\\left(n^{2}-n+1\\right)$$ Decomposing into partial fractions, we find that $$\\frac{n}{n^{4}+n^{2}+1}=\\frac{1}{2}\\left(\\frac{1}{n^{2}-n+1}-\\... |
deepscaler | [
{
"role": "user",
"content": "Let the set \\( T \\) consist of integers between 1 and \\( 2^{30} \\) whose binary representations contain exactly two 1s. If one number is randomly selected from the set \\( T \\), what is the probability that it is divisible by 9? Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "5/29"
} | {
"split": "train",
"index": 19070,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that point $M$ lies on the circle $C:x^{2}+y^{2}-4x-14y+45=0$, and point $Q(-2,3)$.\n\n(1) If $P(a,a+1)$ is on circle $C$, find the length of segment $PQ$ and the slope of line $PQ$;\n\n(2) Find the maximum and minimum values of $|MQ|$;\n\n(3) If $M(m,n)$, find the maximu... | math | {
"style": "rule",
"ground_truth": "2- \\sqrt {3}"
} | {
"split": "train",
"index": 19071,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the set $\\{0.34,0.304,0.034,0.43\\}$. Find the sum of the smallest and largest numbers in the set. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.464"
} | {
"split": "train",
"index": 19072,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The Songjiang tram project is in full swing. After the tram starts operating, it will bring convenience to the public's travel. It is known that after the opening of a certain route, the tram interval $t$ (unit: minutes) satisfies $2 \\leq t \\leq 20$. According to market surve... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 19073,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "One bag contains 2 red marbles and 2 blue marbles. A second bag contains 2 red marbles, 2 blue marbles, and $g$ green marbles, with $g>0$. For each bag, Maria calculates the probability of randomly drawing two marbles of the same colour in two draws from that bag, without repla... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 19074,
"solution": "First, we consider the first bag, which contains a total of $2+2=4$ marbles. There are 4 possible marbles that can be drawn first, leaving 3 possible marbles that can be drawn second. This gives a total of $4 \\times 3=12$ ways of drawing two marbles. For both marb... |
deepscaler | [
{
"role": "user",
"content": "\nCalculate the definite integral:\n\n$$\n\\int_{\\frac{\\pi}{2}}^{2 \\operatorname{arctg} 2} \\frac{d x}{\\sin x(1+\\sin x)}\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\ln 2 - \\frac{1}{3}"
} | {
"split": "train",
"index": 19075,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive integer with six positive odd integer divisors and twelve positive even integer divisors? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "180"
} | {
"split": "train",
"index": 19076,
"solution": "Somewhat similar to the first solution, we see that the number $n$ has two even factors for every odd factor. Thus, if $x$ is an odd factor of $n$, then $2x$ and $4x$ must be the two corresponding even factors. So, the prime factorization of $n$ is $2^2 3^a 5^b 7^c... |
deepscaler | [
{
"role": "user",
"content": "Given triangle $\\triangle ABC$ with sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$ respectively, satisfying $\\frac{a}{{2\\cos A}}=\\frac{b}{{3\\cos B}}=\\frac{c}{{6\\cos C}}$, then $\\sin 2A=$____. Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{11}}{10}"
} | {
"split": "train",
"index": 19077,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A batch of fragile goods totaling $10$ items is transported to a certain place by two trucks, A and B. Truck A carries $2$ first-class items and $2$ second-class items, while truck B carries $4$ first-class items and $2$ second-class items. Upon arrival at the destination, it w... | math | {
"style": "rule",
"ground_truth": "\\frac{29}{48}"
} | {
"split": "train",
"index": 19078,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are four identical balls numbered $1$, $2$, $3$, $4$, and four boxes also numbered $1$, $2$, $3$, $4$.\n\n$(1)$ If each box contains one ball, find the number of ways such that exactly one box has the same number as the ball inside it.\n\n$(2)$ Find the number of ways suc... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 19079,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The square quilt block shown is made from 16 unit squares, four of which have been divided in half to form triangles. Additionally, two squares are completely filled while others are empty. What fraction of the square quilt is shaded? Express your answer as a common fraction. L... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 19080,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A two-digit number, when three times the sum of its units and tens digits is subtracted by -2, still results in the original number. Express this two-digit number algebraically. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 19081,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the coordinates of $A$, $B$, and $C$ are $A(4,0)$, $B(0,4)$, and $C(3\\cos \\alpha,3\\sin \\alpha)$ respectively: \n$(1)$ If $\\alpha \\in (-\\pi,0)$ and $|\\overrightarrow{AC}|=|\\overrightarrow{BC}|$, find the value of $\\alpha$; \n$(2)$ If $\\overrightarrow{AC} \\cdo... | math | {
"style": "rule",
"ground_truth": "-\\frac{7}{16}"
} | {
"split": "train",
"index": 19082,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a right triangle JKL, the hypotenuse KL measures 13 units, and side JK measures 5 units. Determine $\\tan L$ and $\\sin L$. 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": 19083,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Club Truncator is in a soccer league with six other teams, each of which it plays once. In any of its 6 matches, the probabilities that Club Truncator will win, lose, or tie are each $\\frac {1}{3}$. The probability that Club Truncator will finish the season with more wins than... | math | {
"style": "rule",
"ground_truth": "341"
} | {
"split": "train",
"index": 19084,
"solution": "At first, it wins $6$ games, only one way\n Secondly, it wins $5$ games, the other game can be either win or loss, there are $\\binom{6}{5}\\cdot 2=12$ ways\n Thirdly, it wins $4$ games, still the other two games can be either win or loss, there are $\\binom{6}{4}\... |
deepscaler | [
{
"role": "user",
"content": "Given a tetrahedron \\(ABCD\\). Points \\(M\\), \\(N\\), and \\(K\\) lie on edges \\(AD\\), \\(BC\\), and \\(DC\\) respectively, such that \\(AM:MD = 1:3\\), \\(BN:NC = 1:1\\), and \\(CK:KD = 1:2\\). Construct the section of the tetrahedron with the plane \\(MNK\\). In what rat... | math | {
"style": "rule",
"ground_truth": "2/3"
} | {
"split": "train",
"index": 19085,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 85 beads in total on a string, arranged in the pattern \"three green, four red, one yellow, three green, four red, one yellow, ...\". How many red beads are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 19086,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sum of the first $n$ terms of the positive arithmetic geometric sequence $\\{a_n\\}$ is $S_n$, and $\\frac{a_{n+1}}{a_n} < 1$, if $a_3 + a_5 = 20$ and $a_2 \\cdot a_6 = 64$, calculate $S_6$. Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "126"
} | {
"split": "train",
"index": 19087,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Of the following expressions the one equal to $\\frac{a^{-1}b^{-1}}{a^{-3} - b^{-3}}$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{a^2b^2}{b^3 - a^3}"
} | {
"split": "train",
"index": 19088,
"solution": "1. Start by rewriting the given expression using the properties of exponents:\n \\[\n \\frac{a^{-1}b^{-1}}{a^{-3} - b^{-3}} = \\frac{\\frac{1}{ab}}{\\frac{1}{a^3} - \\frac{1}{b^3}}\n \\]\n\n2. To simplify the expression, multiply the numerator and the denomin... |
deepscaler | [
{
"role": "user",
"content": "Given a circle $x^2 + (y-1)^2 = 1$ with its tangent line $l$, which intersects the positive x-axis at point A and the positive y-axis at point B. Determine the y-intercept of the tangent line $l$ when the distance AB is minimized. Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "\\frac{3+\\sqrt{5}}{2}"
} | {
"split": "train",
"index": 19089,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $a$ and $b$ are single-digit positive integers chosen independently and at random. What is the probability that the point $(a,b)$ lies above the parabola $y=ax^2-bx$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{19}{81}"
} | {
"split": "train",
"index": 19090,
"solution": "1. **Understanding the Problem**: We need to find the probability that the point $(a,b)$ lies above the parabola $y = ax^2 - bx$. This translates to finding the probability that $b > ax^2 - bx$ for all $x$.\n\n2. **Setting up the Inequality**: To ensure that $(a,b)... |
deepscaler | [
{
"role": "user",
"content": "Given that the sequence $\\{a_n\\}$ is a geometric sequence, with $a_1=2$, common ratio $q>0$, and $a_2$, $6$, $a_3$ forming an arithmetic sequence.\n(I) Find the general term formula for the sequence $\\{a_n\\}$;\n(II) Let $b_n=\\log_2{a_n}$, and $$T_{n}= \\frac {1}{b_{1}b_{2}... | math | {
"style": "rule",
"ground_truth": "98"
} | {
"split": "train",
"index": 19091,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC,\\,$ angle $C$ is a right angle and the altitude from $C\\,$ meets $\\overline{AB}\\,$ at $D.\\,$ The lengths of the sides of $\\triangle ABC\\,$ are integers, $BD=29^3,\\,$ and $\\cos B=m/n\\,$, where $m\\,$ and $n\\,$ are relatively prime positive integers. F... | math | {
"style": "rule",
"ground_truth": "450"
} | {
"split": "train",
"index": 19092,
"solution": "We will solve for $\\cos B$ using $\\triangle CBD$, which gives us $\\cos B = \\frac{29^3}{BC}$. By the Pythagorean Theorem on $\\triangle CBD$, we have $BC^2 - DC^2 = (BC + DC)(BC - DC) = 29^6$. Trying out factors of $29^6$, we can either guess and check or just g... |
deepscaler | [
{
"role": "user",
"content": "An ATM password at Fred's Bank consists of four digits from $0$ to $9$. No password may begin with the sequence \"123\", and if a password begins with \"123\", the fourth digit cannot be $4$ or $5$. Calculate the number of valid passwords that are possible. Let's think step by ... | math | {
"style": "rule",
"ground_truth": "9992"
} | {
"split": "train",
"index": 19093,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Last year, a bicycle cost $200, a cycling helmet $50, and a water bottle $15. This year the cost of each has increased by 6% for the bicycle, 12% for the helmet, and 8% for the water bottle respectively. Find the percentage increase in the combined cost of the bicycle, helmet, ... | math | {
"style": "rule",
"ground_truth": "7.25\\%"
} | {
"split": "train",
"index": 19094,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\overline{AB}$ have a length of 8 units and $\\overline{A'B'}$ have a length of 6 units. $D$ is located 3 units away from $A$ on $\\overline{AB}$ and $D'$ is located 1 unit away from $A'$ on $\\overline{A'B'}$. If $P$ is a point on $\\overline{AB}$ such that $x$ (the dist... | math | {
"style": "rule",
"ground_truth": "\\frac{10}{3}"
} | {
"split": "train",
"index": 19095,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A town's population increased by $1,200$ people, and then this new population decreased by $11\\%$. The town now had $32$ less people than it did before the $1,200$ increase. What is the original population? Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "10000"
} | {
"split": "train",
"index": 19096,
"solution": "1. **Define the variables**: Let $n$ be the original population of the town.\n\n2. **Calculate the population after the increase**: When $1,200$ people are added to the town, the new population becomes $n + 1,200$.\n\n3. **Calculate the population after the decreas... |
deepscaler | [
{
"role": "user",
"content": " $x$ is a base- $10$ number such that when the digits of $x$ are interpreted as a base- $20$ number, the resulting number is twice the value as when they are interpreted as a base- $13$ number. Find the sum of all possible values of $x$ . Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "198"
} | {
"split": "train",
"index": 19097,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $2a^{2}\\sin B\\sin C=\\sqrt{3}(a^{2}+b^{2}-c^{2})\\sin A$. Find:\n\n$(1)$ Angle $C$;\n\n$(2)$ If $a=1$, $b=2$, and the midpoint of sid... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{7}}{2}"
} | {
"split": "train",
"index": 19098,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the sequence $\\{a_n\\}$, $a_n+a_{n+1}+a_{n+2}=(\\sqrt{2})^{n}$. Find the sum of the first $9$ terms of the sequence $\\{a_n\\}$ (express the answer as a numerical value). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4+9\\sqrt{2}"
} | {
"split": "train",
"index": 19099,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mr. Reader has six different Spiderman comic books, five different Archie comic books and four different Garfield comic books. When stacked, all of the Spiderman comic books are grouped together, all of the Archie comic books are grouped together and all of the Garfield comic b... | math | {
"style": "rule",
"ground_truth": "12,\\!441,\\!600"
} | {
"split": "train",
"index": 19100,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cube \\(ABCDA_1B_1C_1D_1\\) has edge length 1. Point \\(M\\) is taken on the side diagonal \\(A_1D\\), and point \\(N\\) is taken on \\(CD_1\\), such that the line segment \\(MN\\) is parallel to the diagonal plane \\(A_1ACC_1\\). Find the minimum length of \\(MN\\). Let's th... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 19101,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute: $8 + 6(3-8)^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "158"
} | {
"split": "train",
"index": 19102,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A farmer's rectangular field is partitioned into a $2$ by $2$ grid of $4$ rectangular sections. In each section the farmer will plant one crop: corn, wheat, soybeans, or potatoes. The farmer does not want to grow corn and wheat in any two sections that share a border, and the f... | math | {
"style": "rule",
"ground_truth": "84"
} | {
"split": "train",
"index": 19103,
"solution": "We will analyze the problem by considering different cases based on the distribution of crops in the sections. The key restrictions are that corn cannot be adjacent to wheat, and soybeans cannot be adjacent to potatoes.\n\n#### Case 1: All sections have the same cr... |
deepscaler | [
{
"role": "user",
"content": "Princeton has an endowment of $5$ million dollars and wants to invest it into improving campus life. The university has three options: it can either invest in improving the dorms, campus parties or dining hall food quality. If they invest $a$ million dollars in the dorms, t... | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 19104,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "It is known that none of the digits of a three-digit number is zero, and the sum of all possible two-digit numbers composed of the digits of this number is equal to the number itself. Find the largest such three-digit number. Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "396"
} | {
"split": "train",
"index": 19105,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $F\\_1$ and $F\\_2$ are the left and right foci of the ellipse $(E)$: $\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1 (a > b > 0)$, $M$ and $N$ are the endpoints of its minor axis, and the perimeter of the quadrilateral $MF\\_1NF\\_2$ is $4$, let line $(l)$ pass... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 19106,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the digits $0$, $1$, $2$, $3$, $4$, $5$, find the number of six-digit numbers that can be formed without repetition and with odd and even digits alternating. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 19107,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that point \\( P \\) lies on the hyperbola \\( \\Gamma: \\frac{x^{2}}{463^{2}} - \\frac{y^{2}}{389^{2}} = 1 \\). A line \\( l \\) passes through point \\( P \\) and intersects the asymptotes of hyperbola \\( \\Gamma \\) at points \\( A \\) and \\( B \\), respectively. If ... | math | {
"style": "rule",
"ground_truth": "180107"
} | {
"split": "train",
"index": 19108,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest base-4 number that has four digits? Express your answer in base 10. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "255"
} | {
"split": "train",
"index": 19109,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the units digit of $13 \\cdot 41$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 19110,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A 10 by 10 checkerboard has alternating black and white squares. How many distinct squares, with sides on the grid lines of the checkerboard (horizontal and vertical) and containing at least 6 black squares, can be drawn on the checkerboard? Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "140"
} | {
"split": "train",
"index": 19111,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A natural number is called a square if it can be written as the product of two identical numbers. For example, 9 is a square because \\(9 = 3 \\times 3\\). The first squares are 1, 4, 9, 16, 25, ... A natural number is called a cube if it can be written as the product of three ... | math | {
"style": "rule",
"ground_truth": "2067"
} | {
"split": "train",
"index": 19112,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A right circular cone is sliced into five pieces of equal height by planes parallel to its base. Determine the ratio of the volume of the second-largest piece to the volume of the largest piece. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{37}{61}"
} | {
"split": "train",
"index": 19113,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$A, B, C, D, E, F, G$ are seven people sitting around a circular table. If $d$ is the total number of ways that $B$ and $G$ must sit next to $C$, find the value of $d$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 19114,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of two numbers is 25 and their product is 126. What is the absolute value of the difference of the two numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 19115,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Positive numbers \\(a\\), \\(b\\), and \\(c\\) satisfy the equations\n\n\\[ \na^{2} + ab + b^{2} = 1, \\quad b^{2} + bc + c^{2} = 3, \\quad c^{2} + ca + a^{2} = 4 \n\\]\n\nFind the value of \\(a + b + c\\). Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "\\sqrt{7}"
} | {
"split": "train",
"index": 19116,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the equation $\\frac {x(x - 1) - (m + 1)}{(x - 1)(m - 1)} = \\frac {x}{m}$ the roots are equal when 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": 19117,
"solution": "1. Start by multiplying both sides of the given equation by $(x-1)(m-1)m$ to eliminate the denominators:\n \\[\n \\frac{x(x - 1) - (m + 1)}{(x - 1)(m - 1)} = \\frac{x}{m} \\implies xm(x-1) - m(m+1) = x(x-1)(m-1)\n \\]\n\n2. Expand and simplify both sides:\n ... |
deepscaler | [
{
"role": "user",
"content": "An ellipse with equation\n\\[\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1\\]contains the circles $(x - 1)^2 + y^2 = 1$ and $(x + 1)^2 +y^2 = 1.$ Then the smallest possible area of the ellipse can be expressed in the form $k \\pi.$ Find $k.$ Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "\\frac{3 \\sqrt{3}}{2}"
} | {
"split": "train",
"index": 19118,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$\\frac{10^7}{5\\times 10^4}=$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 19119,
"solution": "1. **Simplify the expression**: We start by simplifying the fraction given in the problem:\n \\[\n \\frac{10^7}{5 \\times 10^4}\n \\]\n We can separate the powers of ten and the constant:\n \\[\n \\frac{10^7}{5 \\times 10^4} = \\frac{10^7}{10^4} \\times... |
deepscaler | [
{
"role": "user",
"content": "Let the sides opposite to the internal angles $A$, $B$, and $C$ of triangle $\\triangle ABC$ be $a$, $b$, and $c$ respectively. It is known that $\\left(\\sin C+\\sin B\\right)\\left(c-b\\right)=a\\left(\\sin A-\\sin B\\right)$. \n$(1)$ Find the measure of angle $C$. \n$(2)$ ... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 19120,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of t... | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 19121,
"solution": "1. **Define the sequences**: Let the quarterly scores for the Raiders be $a, ar, ar^2, ar^3$ and for the Wildcats be $b, b+d, b+2d, b+3d$. Given that the sequences are increasing and the Raiders' sequence is geometric while the Wildcats' sequence is arithmetic.\n\n... |
deepscaler | [
{
"role": "user",
"content": "Find the number of positive integers \\(n \\le 500\\) that can be expressed in the form\n\\[\n\\lfloor x \\rfloor + \\lfloor 3x \\rfloor + \\lfloor 4x \\rfloor = n\n\\]\nfor some real number \\(x\\). Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "248"
} | {
"split": "train",
"index": 19122,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of permutations \\(a_1, a_2, \\ldots, a_{10}\\) of the numbers \\(1, 2, \\ldots, 10\\) such that \\(a_{i+1}\\) is not less than \\(a_i - 1\\) for \\(i = 1, 2, \\ldots, 9\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "512"
} | {
"split": "train",
"index": 19123,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each of $6$ balls is randomly and independently painted either black or white with equal probability. What is the probability that every ball is different in color from more than half of the other $5$ balls? Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{16}"
} | {
"split": "train",
"index": 19124,
"solution": "1. **Understanding the Problem**: We need to find the probability that each of the 6 balls is painted a color such that it is different from the color of more than half of the other 5 balls. This means each ball must be different from at least 3 of the other balls.... |
deepscaler | [
{
"role": "user",
"content": "A regular polygon has interior angles of 162 degrees. How many sides does the polygon have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 19125,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For any two positive integers, define the operation (represented by the operator ⊕): when both $m$ and $n$ are positive even numbers or both are positive odd numbers, $m⊕n=m+n$; when one of $m$ and $n$ is a positive even number and the other is a positive odd number, $m⊕n=m×n$.... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 19126,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Eight consecutive three-digit positive integers have the following property: each of them is divisible by its last digit. What is the sum of the digits of the smallest of these eight integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 19127,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ket $f(x) = x^{2} +ax + b$ . If for all nonzero real $x$ $$ f\\left(x + \\dfrac{1}{x}\\right) = f\\left(x\\right) + f\\left(\\dfrac{1}{x}\\right) $$ and the roots of $f(x) = 0$ are integers, what is the value of $a^{2}+b^{2}$ ? Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 19128,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For each nonnegative integer $n$ we define $A_n = 2^{3n}+3^{6n+2}+5^{6n+2}$. Find the greatest common divisor of the numbers $A_0,A_1,\\ldots, A_{1999}$.\n\n[i]Romania[/i] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 19129,
"solution": "\nTo find the greatest common divisor (GCD) of the sequence \\( A_n = 2^{3n} + 3^{6n+2} + 5^{6n+2} \\) for each nonnegative integer \\( n \\), we begin by examining the expression for \\( A_n \\):\n\n\\[\nA_n = 2^{3n} + 3^{6n+2} + 5^{6n+2}.\n\\]\n\nOur goal is to d... |
deepscaler | [
{
"role": "user",
"content": "How can we connect 50 cities with the minimum number of flight routes so that it's possible to travel from any city to any other city with no more than two layovers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 19130,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a,$ $b,$ $c,$ $x,$ $y,$ and $z$ are positive and $a^2 + b^2 + c^2 = 25,$ $x^2 + y^2 + z^2 = 36,$ and $ax + by + cz = 30,$ compute\n\\[\\frac{a + b + c}{x + y + z}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{6}"
} | {
"split": "train",
"index": 19131,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A new dump truck delivered sand to a construction site, forming a conical pile with a diameter of $12$ feet. The height of the cone was $50\\%$ of its diameter. However, the pile was too large, causing some sand to spill, forming a cylindrical layer directly around the base of ... | math | {
"style": "rule",
"ground_truth": "98\\pi"
} | {
"split": "train",
"index": 19132,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given equations of the form $x^2 + bx + c = 0$, determine the number of such equations that have real roots and have coefficients $b$ and $c$ selected from the set of integers $\\{1, 2, 3, 4, 5, 7\\}$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 19133,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $XYZ$, we have $\\angle Z = 90^\\circ$, $XY = 10$, and $YZ = \\sqrt{51}$. What is $\\tan X$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{51}}{7}"
} | {
"split": "train",
"index": 19134,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let set $\\mathcal{C}$ be a 70-element subset of $\\{1,2,3,\\ldots,120\\},$ and let $P$ be the sum of the elements of $\\mathcal{C}.$ Find the number of possible values of $P.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3501"
} | {
"split": "train",
"index": 19135,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a cone with vertex $S$, and generatrices $SA$, $SB$ perpendicular to each other, and the angle between $SA$ and the base of the cone is $30^{\\circ}$. If the area of $\\triangle SAB$ is $8$, then the volume of this cone is ______. Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "8\\pi"
} | {
"split": "train",
"index": 19136,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\ln (1+x)- \\frac {x(1+λx)}{1+x}$, if $f(x)\\leqslant 0$ when $x\\geqslant 0$, calculate the minimum value of $λ$. 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": 19137,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define $E(n)$ as the sum of the even digits of $n$ and $O(n)$ as the sum of the odd digits of $n$. Find the value of $E(1) + O(1) + E(2) + O(2) + \\dots + E(150) + O(150)$.\nA) 1200\nB) 1300\nC) 1350\nD) 1400\nE) 1450 Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "1350"
} | {
"split": "train",
"index": 19138,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The values of $f$, $g$, $h$ and $j$ are 5, 6, 7 and 8, but not necessarily in that order. What is the largest possible value of the sum of the four products $fg$, $gh$, $hj$ and $fj$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "169"
} | {
"split": "train",
"index": 19139,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\binom{23}{5}=33649$, $\\binom{23}{6}=42504$, and $\\binom{23}{7}=33649$, find $\\binom{25}{7}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "152306"
} | {
"split": "train",
"index": 19140,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the least common multiple of the numbers 1332 and 888? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2664"
} | {
"split": "train",
"index": 19141,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways are there to arrange the letters of the word $\\text{CA}_1\\text{N}_1\\text{A}_2\\text{N}_2\\text{A}_3\\text{T}_1\\text{T}_2$, where there are three A's, two N's, and two T's, with each A, N, and T considered distinct? Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "5040"
} | {
"split": "train",
"index": 19142,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $y$ is a multiple of $45678$, what is the greatest common divisor of $g(y)=(3y+4)(8y+3)(14y+9)(y+14)$ and $y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1512"
} | {
"split": "train",
"index": 19143,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number \\( x \\) is such that \\( \\log _{2}\\left(\\log _{4} x\\right) + \\log _{4}\\left(\\log _{8} x\\right) + \\log _{8}\\left(\\log _{2} x\\right) = 1 \\). Find the value of the expression \\( \\log _{4}\\left(\\log _{2} x\\right) + \\log _{8}\\left(\\log _{4} x\\right... | math | {
"style": "rule",
"ground_truth": "0.87"
} | {
"split": "train",
"index": 19144,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $3^{\\frac{1}{3}} \\cdot 9^{\\frac{1}{9}} \\cdot 27^{\\frac{1}{27}} \\cdot 81^{\\frac{1}{81}} \\dotsm.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt[4]{27}"
} | {
"split": "train",
"index": 19145,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Divide 6 volunteers into 4 groups, with two groups having 2 people each and the other two groups having 1 person each, to serve at four different pavilions of the World Expo. How many different allocation schemes are there? (Answer with a number). Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "1080"
} | {
"split": "train",
"index": 19146,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Samantha has 10 different colored marbles in her bag. In how many ways can she choose five different marbles such that at least one of them is red? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "126"
} | {
"split": "train",
"index": 19147,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABC$ be a triangle with circumcenter $O$ such that $AC=7$. Suppose that the circumcircle of $AOC$ is tangent to $BC$ at $C$ and intersects the line $AB$ at $A$ and $F$. Let $FO$ intersect $BC$ at $E$. Compute $BE$. Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{2}"
} | {
"split": "train",
"index": 19148,
"solution": "$E B=\\frac{7}{2} \\quad O$ is the circumcenter of $\\triangle ABC \\Longrightarrow AO=CO \\Longrightarrow \\angle OCA=\\angle OAC$. Because $AC$ is an inscribed arc of circumcircle $\\triangle AOC, \\angle OCA=\\angle OFA$. Furthermore $BC$ is tangent to circumcir... |
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