data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "The polynomial $P(x)$ is a monic, quartic polynomial with real coefficients, and two of its roots are $\\cos \\theta + i \\sin \\theta$ and $\\sin \\theta + i \\cos \\theta,$ where $0 < \\theta < \\frac{\\pi}{4}.$ When the four roots of $P(x)$ are plotted in the complex plane,... | math | {
"style": "rule",
"ground_truth": "1 + \\sqrt{3}"
} | {
"split": "train",
"index": 33849,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A chord which is the perpendicular bisector of a radius of length 12 in a circle, has length Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12\\sqrt{3}"
} | {
"split": "train",
"index": 33850,
"solution": "1. **Draw the Diagram**: Consider a circle with center \\( O \\) and a radius \\( OA \\) of length 12. Let \\( CD \\) be the chord that is the perpendicular bisector of \\( OA \\). Since \\( CD \\) bisects \\( OA \\) perpendicularly, \\( OC \\) and \\( OD \\) are b... |
deepscaler | [
{
"role": "user",
"content": "In triangle \\( \\triangle ABC \\), \\( BD \\) is a median, \\( CF \\) intersects \\( BD \\) at \\( E \\), and \\( BE = ED \\). Point \\( F \\) is on \\( AB \\), and if \\( BF = 5 \\), then the length of \\( BA \\) is: Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 33851,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two real numbers $a$ and $b$ satisfy $a+b=8$ and $a^3+b^3=172$. Compute the value of $ab$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{85}{6}"
} | {
"split": "train",
"index": 33852,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\log 2 = .3010$ and $\\log 3 = .4771$, the value of $x$ when $3^{x+3} = 135$ is approximately Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1.47"
} | {
"split": "train",
"index": 33853,
"solution": "1. **Rewrite the equation using logarithms**: Given the equation $3^{x+3} = 135$, we take the logarithm (base 10) of both sides:\n \\[\n \\log(3^{x+3}) = \\log(135)\n \\]\n\n2. **Apply the power rule of logarithms**: The power rule states $\\log(a^b) = b \\lo... |
deepscaler | [
{
"role": "user",
"content": "Consider an equilateral triangle $T$ of side length 12. Matthew cuts $T$ into $N$ smaller equilateral triangles, each of which has side length 1,3, or 8. Compute the minimum possible value of $N$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 33854,
"solution": "Matthew can cut $T$ into 16 equilateral triangles with side length 3. If he instead included a triangle of side 8, then let him include $a$ triangles of side length 3. He must include $12^{2}-8^{2}-3^{2} a=80-9 a$ triangles of side length 1. Thus $a \\leq 8$, givin... |
deepscaler | [
{
"role": "user",
"content": "Given that \\( f(x) \\) and \\( g(x) \\) are two quadratic functions both with a leading coefficient of 1, where \\( g(6) = 35 \\) and \\( \\frac{f(-1)}{g(-1)} = \\frac{f(1)}{g(1)} = \\frac{21}{20} \\), what is \\( f(6) \\)? Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 33855,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that there are a total of 25 students in an art class, including 10 boys and 15 girls. In the final art exhibition, the average number of artworks per boy is 25 with a variance of 1, and the average number of artworks per girl is 30 with a variance of 2. Find the variance... | math | {
"style": "rule",
"ground_truth": "\\frac{38}{5}"
} | {
"split": "train",
"index": 33856,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute\n\\[\\frac{1}{\\cos 80^\\circ} - \\frac{\\sqrt{3}}{\\sin 80^\\circ}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 33857,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the minimum number of digits to the right of the decimal point needed to express the fraction $\\frac{123456789}{2^{26}\\cdot 5^4}$ as a decimal? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 33858,
"solution": "1. **Simplify the given fraction**: We start by simplifying the fraction $\\frac{123456789}{2^{26}\\cdot 5^4}$. Notice that $10^4 = 2^4 \\cdot 5^4$. Therefore, we can rewrite the denominator as:\n \\[\n 2^{26} \\cdot 5^4 = 2^{22} \\cdot (2^4 \\cdot 5^4) = 2^{22... |
deepscaler | [
{
"role": "user",
"content": "Find the minimum value, for \\(a, b > 0\\), of the expression\n\n\\[\n\\frac{|a + 3b - b(a + 9b)| + |3b - a + 3b(a - b)|}{\\sqrt{a^{2} + 9b^{2}}}\n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{10}}{5}"
} | {
"split": "train",
"index": 33859,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A point is chosen at random on the number line between 0 and 1, and this point is colored red. Another point is then chosen at random on the number line between 0 and 2, and this point is colored blue. What is the probability that the number of the blue point is greater than th... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 33860,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\tan 20^\\circ + 4 \\sin 20^\\circ.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 33861,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ has vertices $A(-2, 10)$, $B(3, 0)$, $C(10, 0)$. A line through $B$ cuts the area of $\\triangle ABC$ in half; find the sum of the slope and $y$-intercept of this line. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-10"
} | {
"split": "train",
"index": 33862,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(r)=\\sum_{j=2}^{2008} \\frac{1}{j^{r}}=\\frac{1}{2^{r}}+\\frac{1}{3^{r}}+\\cdots+\\frac{1}{2008^{r}}$. Find $\\sum_{k=2}^{\\infty} f(k)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2007}{2008}"
} | {
"split": "train",
"index": 33863,
"solution": "We change the order of summation: $$\\sum_{k=2}^{\\infty} \\sum_{j=2}^{2008} \\frac{1}{j^{k}}=\\sum_{j=2}^{2008} \\sum_{k=2}^{\\infty} \\frac{1}{j^{k}}=\\sum_{j=2}^{2008} \\frac{1}{j^{2}\\left(1-\\frac{1}{j}\\right)}=\\sum_{j=2}^{2008} \\frac{1}{j(j-1)}=\\sum_{j=2}... |
deepscaler | [
{
"role": "user",
"content": "A supermarket purchases two types of goods, $A$ and $B$. Buying 4 items of type $A$ costs $10$ yuan less than buying 5 items of type $B$. Buying 20 items of type $A$ and 10 items of type $B$ costs a total of $160$ yuan. \n$(1)$ Find the cost price per item of goods $A$ and $B$... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 33864,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Kate has four red socks and four blue socks. If she randomly divides these eight socks into four pairs, what is the probability that none of the pairs will be mismatched? That is, what is the probability that each pair will consist either of two red socks or of two blue socks? ... | math | {
"style": "rule",
"ground_truth": "3 / 35"
} | {
"split": "train",
"index": 33865,
"solution": "The number of ways Kate can divide the four red socks into two pairs is $\\binom{4}{2} / 2=3$. The number of ways she can divide the four blue socks into two pairs is also 3 . Therefore, the number of ways she can form two pairs of red socks and two pairs of blue s... |
deepscaler | [
{
"role": "user",
"content": "In a quadrilateral pyramid \\(S A B C D\\):\n\n- The lateral faces \\(S A B, S B C, S C D, S D A\\) have areas of 9, 9, 27, and 27 respectively;\n- The dihedral angles at the edges \\(A B, B C, C D, D A\\) are equal;\n- The quadrilateral \\(A B C D\\) is inscribed in a circle, ... | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 33866,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A necklace is strung with gems in the order of A, B, C, D, E, F, G, H. Now, we want to select 8 gems from it in two rounds, with the requirement that only 4 gems can be taken each time, and at most two adjacent gems can be taken (such as A, B, E, F). How many ways are there to ... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 33867,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a point P $(x, y)$ moves on the circle $x^2 + (y-1)^2 = 1$, the maximum value of $\\frac{y-1}{x-2}$ is \\_\\_\\_\\_\\_\\_, and the minimum value is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 33868,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the triangle \\( \\triangle ABC \\), it is given that the angles are in the ratio \\(\\angle A : \\angle B : \\angle C = 3 : 5 : 10\\). Also, it is known that \\(\\triangle A'B'C \\cong \\triangle ABC\\). What is the ratio \\(\\angle BCA' : \\angle BCB'\\)? Let's think step ... | math | {
"style": "rule",
"ground_truth": "1:4"
} | {
"split": "train",
"index": 33869,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A student's score on a 150-point test is directly proportional to the hours she studies. If she scores 90 points after studying for 2 hours, what would her score be if she studied for 5 hours? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "225"
} | {
"split": "train",
"index": 33870,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nIn an apartment building, each floor in every entrance has the same number of apartments (more than one). Additionally, each entrance has the same number of floors. The number of floors is more than the number of apartments per floor, but less than the number of entrances. Ho... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 33871,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A right triangle has one angle measuring $30^\\circ$. This triangle shares its hypotenuse with a second triangle that is also right-angled. The two triangles together form a quadrilateral. If the other acute angle in the second triangle is $45^\\circ$, find the area of the quad... | math | {
"style": "rule",
"ground_truth": "\\frac{25\\sqrt{3} + 50}{2}"
} | {
"split": "train",
"index": 33872,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a class of 50 students, it is decided to use systematic sampling to select 10 students. The 50 students are randomly assigned numbers from 1 to 50 and divided into groups, with the first group being 1-5, the second group 6-10, ..., and the tenth group 45-50. If a student wit... | math | {
"style": "rule",
"ground_truth": "37"
} | {
"split": "train",
"index": 33873,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In right triangle $ABC$ with $\\angle B = 90^\\circ$, sides $AB=1$ and $BC=3$. The bisector of $\\angle BAC$ meets $\\overline{BC}$ at $D$. Calculate the length of segment $BD$.\nA) $\\frac{1}{2}$\nB) $\\frac{3}{4}$\nC) $1$\nD) $\\frac{5}{4}$\nE) $2$ Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 33874,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point \\( A \\) lies on the line \\( y = \\frac{15}{8} x - 4 \\), and point \\( B \\) on the parabola \\( y = x^{2} \\). What is the minimum length of segment \\( AB \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "47/32"
} | {
"split": "train",
"index": 33875,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( a_1, a_2, \\dots \\) be a sequence of positive real numbers such that\n\\[ a_n = 7a_{n-1} - n \\]for all \\( n > 1 \\). Find the smallest possible value of \\( a_1 \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{36}"
} | {
"split": "train",
"index": 33876,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given point O is the circumcenter of ∆ABC, and |→BA|=2, |→BC|=6, calculate →BO⋅→AC. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 33877,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f\\left( x \\right)={x}^{2}+{\\left( \\ln 3x \\right)}^{2}-2a(x+3\\ln 3x)+10{{a}^{2}}(a\\in \\mathbf{R})$, determine the value of the real number $a$ for which there exists ${{x}_{0}}$ such that $f\\left( {{x}_{0}} \\right)\\leqslant \\dfrac{1}{10}$. Let's t... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{30}"
} | {
"split": "train",
"index": 33878,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\(a, b, c\\) are positive integers and the quadratic equation \\(a x^{2}+b x+c=0\\) has two real roots whose absolute values are both less than \\(\\frac{1}{3}\\), find the minimum value of \\(a + b + c\\). Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 33879,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of two numbers is $12$ and their difference is $20$. What is the smaller number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 33880,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In response to the call of the commander, 55 soldiers came: archers and swordsmen. All of them were dressed either in golden or black armor. It is known that swordsmen tell the truth when wearing black armor and lie when wearing golden armor, while archers do the opposite.\n\n-... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 33881,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\\times 5$ square array of dots?\n(Two rectangles are different if they do not share all four vertices.) Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 33882,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\\( z_{1}, z_{2}, z_{3} \\) are the three roots of the polynomial\n\\[ P(z) = z^{3} + a z + b \\]\nand satisfy the condition\n\\[ \\left|z_{1}\\right|^{2} + \\left|z_{2}\\right|^{2} + \\left|z_{3}\\right|^{2} = 250 \\]\nMoreover, the three points \\( z_{1}, z_{2}, z_{3} \\) in ... | math | {
"style": "rule",
"ground_truth": "5\\sqrt{15}"
} | {
"split": "train",
"index": 33883,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The train arrives at a station randomly between 1:00 PM and 3:00 PM and waits for 30 minutes before departing. If John also arrives randomly at the station within the same time period, what is the probability that he will find the train at the station? Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{32}"
} | {
"split": "train",
"index": 33884,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( a \\) and \\( b \\) be real numbers, and consider the function \\( f(x) = x^3 + ax^2 + bx \\). Suppose there exist three real numbers \\( x_1, x_2, x_3 \\) such that \\( x_1 + 1 \\leq x_2 \\leq x_3 - 1 \\) and \\( f(x_1) = f(x_2) = f(x_3) \\). Find the minimum value of ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 33885,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse $C:\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1 \\left( a > b > 0 \\right)$ with eccentricity $\\dfrac{1}{2}$, and it passes through the point $\\left( 1,\\dfrac{3}{2} \\right)$. If point $M\\left( x_{0},y_{0} \\right)$ is on the ellipse $C$, then the point $N\\left( ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 33886,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $3015x + 3020y = 3025$ and $3018x + 3024y = 3030$, what is the value of $x - y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11.1167"
} | {
"split": "train",
"index": 33887,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An investor placed \\$12,000 in a one-year fixed deposit that yielded a simple annual interest rate of 8%. After one year, the total amount was reinvested in another one-year deposit. At the end of the second year, the total amount was \\$13,500. If the annual interest rate of ... | math | {
"style": "rule",
"ground_truth": "4.17\\%"
} | {
"split": "train",
"index": 33888,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Emily ordered her playing cards by suit in the order $$A,2,3,4,5,6,7,8,9,10,J,Q,K,A,2,3,\\cdots.$$What is the $42$nd card? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 33889,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number $24!$ has many positive integer divisors. What is the probability that a divisor randomly chosen from these is odd? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{23}"
} | {
"split": "train",
"index": 33890,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a round table on which $2014$ people are seated. Suppose that the person at the head of the table receives a giant plate containing all the food for supper. He then serves himself and passes the plate either right or left with equal probability. Each person, upon rec... | math | {
"style": "rule",
"ground_truth": "1/2013"
} | {
"split": "train",
"index": 33891,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $n=2^3 \\cdot 3^2 \\cdot 5$, how many even positive factors does $n$ have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 33892,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the tens digit and the ones digit of the integer form of $(2+4)^{15}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 33893,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the coefficient of the $x^4$ term in the expansion of the binomial $(4x^{2}-2x+1)(2x+1)^{5}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 33894,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate $\\frac{1586_{7}}{131_{5}}-3451_{6}+2887_{7}$. 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": "334"
} | {
"split": "train",
"index": 33895,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the definite integral:\n\n$$\n\\int_{0}^{\\sqrt{2}} \\frac{x^{4} \\cdot d x}{\\left(4-x^{2}\\right)^{3 / 2}}\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5 - \\frac{3\\pi}{2}"
} | {
"split": "train",
"index": 33896,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a connected simple graph \\( G \\) with \\( e \\) edges and pieces placed on each vertex of \\( G \\) (where each piece can only be placed on a single vertex of \\( G \\)), you are allowed to perform the following operation: if the number of pieces on a vertex \\( v \\) i... | math | {
"style": "rule",
"ground_truth": "e"
} | {
"split": "train",
"index": 33897,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the expression $\\frac{2020^3 - 3 \\cdot 2020^2 \\cdot 2021 + 5 \\cdot 2020 \\cdot 2021^2 - 2021^3 + 4}{2020 \\cdot 2021}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4042 + \\frac{3}{4080420}"
} | {
"split": "train",
"index": 33898,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The distance \\( AB \\) is 100 km. Cyclists depart simultaneously from \\( A \\) and \\( B \\) towards each other with speeds of 20 km/h and 30 km/h respectively. Along with the first cyclist from \\( A \\), a fly departs with a speed of 50 km/h. The fly travels until it meets ... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 33899,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Refer to the diagram, $P$ is any point inside the square $O A B C$ and $b$ is the minimum value of $P O + P A + P B + P C$. Find $b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"split": "train",
"index": 33900,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a}=(\\cos α,\\sin α)$, $\\overrightarrow{b}=(\\cos β,\\sin β)$, and $|\\overrightarrow{a}- \\overrightarrow{b}|= \\frac {4 \\sqrt {13}}{13}$.\n(1) Find the value of $\\cos (α-β)$;\n(2) If $0 < α < \\frac {π}{2}$, $- \\frac {π}{2} < β < 0$, and $\... | math | {
"style": "rule",
"ground_truth": "\\frac {16}{65}"
} | {
"split": "train",
"index": 33901,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\sin (2x+ \\frac {π}{3})$, it is shifted right by $\\frac {2π}{3}$ units, and then the resulting function's graph has each point's horizontal coordinate changed to twice its original value while the vertical coordinate remains unchanged, yielding the f... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 33902,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive two-digit integers are factors of $2^{24}-1$?~ pi_is_3.14 Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 33903,
"solution": "1. **Factorize $2^{24}-1$ using difference of squares repeatedly:**\n \\[\n 2^{24}-1 = (2^{12}+1)(2^{12}-1)\n \\]\n \\[\n 2^{12}-1 = (2^6+1)(2^6-1)\n \\]\n \\[\n 2^6-1 = (2^3+1)(2^3-1)\n \\]\n Thus,\n \\[\n 2^{24}-1 = (2^{12}+1)(2^6+1)(2^3+1... |
deepscaler | [
{
"role": "user",
"content": "The number of students in James' graduating class is greater than 100 but fewer than 200 and is 1 less than a multiple of 4, 2 less than a multiple of 5, and 3 less than a multiple of 6. How many students are in James' graduating class? Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "183"
} | {
"split": "train",
"index": 33904,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the geometric sequence $\\{a_n\\}$, if $a_2a_5 = -\\frac{3}{4}$ and $a_2 + a_3 + a_4 + a_5 = \\frac{5}{4}$, calculate the value of $\\frac{1}{a_2} + \\frac{1}{a_3} + \\frac{1}{a_4} + \\frac{1}{a_5}$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "-\\frac{5}{3}"
} | {
"split": "train",
"index": 33905,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle made of wire and a rectangle are arranged in such a way that the circle passes through two vertices $A$ and $B$ and touches the side $CD$. The length of side $CD$ is 32.1. Find the ratio of the sides of the rectangle, given that its perimeter is 4 times the radius of t... | math | {
"style": "rule",
"ground_truth": "4:1"
} | {
"split": "train",
"index": 33906,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square with a side length of 10 centimeters is rotated about its horizontal line of symmetry. Calculate the volume of the resulting cylinder in cubic centimeters and express your answer in terms of $\\pi$. Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "250\\pi"
} | {
"split": "train",
"index": 33907,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function $f(x)$ satisfies\n\\[f(x) + f \\left( \\frac{x + 1}{1 - 3x} \\right) = x\\]for all $x \\neq \\frac{1}{3}.$ Find $f(2).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{48}{35}"
} | {
"split": "train",
"index": 33908,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Medians $AD$ and $CE$ of $\\triangle ABC$ intersect in $M$. The midpoint of $AE$ is $N$. \nLet the area of $\\triangle MNE$ be $k$ times the area of $\\triangle ABC$. Then $k$ equals: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"split": "train",
"index": 33909,
"solution": "1. **Identify the relationships between the triangles**: \n - Since $AD$ and $CE$ are medians, point $M$ is the centroid of $\\triangle ABC$. The centroid divides each median into two segments, one of which is twice the length of the other, specifically, the segm... |
deepscaler | [
{
"role": "user",
"content": "Let \\( S \\) be a set of size 11. A random 12-tuple \\((s_1, s_2, \\ldots, s_{12})\\) of elements of \\( S \\) is chosen uniformly at random. Moreover, let \\(\\pi: S \\rightarrow S\\) be a permutation of \\( S \\) chosen uniformly at random. The probability that \\( s_{i+1} \... | math | {
"style": "rule",
"ground_truth": "1000000000004"
} | {
"split": "train",
"index": 33910,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that point \\( P \\) lies in the plane of triangle \\( \\triangle ABC \\) and satisfies the condition \\( PA - PB - PC = \\overrightarrow{BC} \\), determine the ratio of the area of \\( \\triangle ABP \\) to the area of \\( \\triangle ABC \\). Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "2:1"
} | {
"split": "train",
"index": 33911,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the multiplication problem below $A$, $B$, $C$, $D$ are different digits. What is $A+B$?\n$\\begin{array}{cccc} & A & B & A\\\\ \\times & & C & D\\\\ \\hline C & D & C & D\\\\ \\end{array}$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 33912,
"solution": "1. **Identify the value of $A$:** \n Given the multiplication problem:\n \\[\n \\begin{array}{cccc}\n & A & B & A\\\\ \n \\times & & C & D\\\\ \n \\hline \n C & D & C & D\\\\ \n \\end{array}\n \\]\n We observe that the product of $A$ and $D$ re... |
deepscaler | [
{
"role": "user",
"content": "For positive integers $x$, let $g(x)$ be the number of blocks of consecutive 1's in the binary expansion of $x$. For example, $g(19)=2$ because $19=10011_{2}$ has a block of one 1 at the beginning and a block of two 1's at the end, and $g(7)=1$ because $7=111_{2}$ only has a si... | math | {
"style": "rule",
"ground_truth": "577"
} | {
"split": "train",
"index": 33913,
"solution": "Solution 1. We prove that $g(1)+g(2)+\\cdots+g\\left(2^{n}\\right)=1+2^{n-2}(n+1)$ for all $n \\geq 1$, giving an answer of $1+2^{6} \\cdot 9=577$. First note that $g\\left(2^{n}\\right)=1$, and that we can view $0,1, \\ldots, 2^{n}-1$ as $n$-digit binary sequences... |
deepscaler | [
{
"role": "user",
"content": "What is the value of $1^{3}+2^{3}+3^{3}+4^{3}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10^{2}"
} | {
"split": "train",
"index": 33914,
"solution": "Expanding and simplifying, $1^{3}+2^{3}+3^{3}+4^{3}=1 \\times 1 \\times 1+2 \\times 2 \\times 2+3 \\times 3 \\times 3+4 \\times 4 \\times 4=1+8+27+64=100$. Since $100=10^{2}$, then $1^{3}+2^{3}+3^{3}+4^{3}=10^{2}$."
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive integer $n$ such that all the roots of $z^6 - z^3 + 1 = 0$ are $n^{\\text{th}}$ roots of unity? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 33915,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x$ be a real number such that\n\\[ x^2 + 8 \\left( \\frac{x}{x-3} \\right)^2 = 53. \\]\nFind all possible values of $y = \\frac{(x - 3)^3 (x + 4)}{2x - 5}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{17000}{21}"
} | {
"split": "train",
"index": 33916,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$, its sum of the first $n$ terms is $S_n$. It is known that $a_2=2$, $S_5=15$, and $b_n=\\frac{1}{a_{n+1}^2-1}$. Find the sum of the first 10 terms of the sequence $\\{b_n\\}$. Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "\\frac {175}{264}"
} | {
"split": "train",
"index": 33917,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A,$ $R,$ $M,$ and $L$ be positive real numbers such that\n\\begin{align*}\n\\log_{10} (AL) + \\log_{10} (AM) &= 2, \\\\\n\\log_{10} (ML) + \\log_{10} (MR) &= 3, \\\\\n\\log_{10} (RA) + \\log_{10} (RL) &= 4.\n\\end{align*}Compute the value of the product $ARML.$ Let's think... | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"split": "train",
"index": 33918,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\sin 2x$, its graph intersects with the line $2kx-2y-kπ=0 (k > 0)$ at exactly three common points. The x-coordinates of these points in ascending order are $x_{1}$, $x_{2}$, $x_{3}$. Calculate the value of $(x_{1}-x_{3})\\tan (x_{2}-2x_{3})$. Let's thi... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 33919,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A set of 6 distinct lattice points is chosen uniformly at random from the set $\\{1,2,3,4,5,6\\}^{2}$. Let $A$ be the expected area of the convex hull of these 6 points. Estimate $N=\\left\\lfloor 10^{4} A\\right\\rfloor$. An estimate of $E$ will receive $\\max \\left(0,\\left\... | math | {
"style": "rule",
"ground_truth": "104552"
} | {
"split": "train",
"index": 33920,
"solution": "The main tools we will use are linearity of expectation and Pick's theorem. Note that the resulting polygon is a lattice polygon, and this the expected area $A$ satisfies $$A=I+\\frac{B}{2}-1$$ where $I$ is the expected number of interior points and $B$ is the expe... |
deepscaler | [
{
"role": "user",
"content": "The sequence $6075, 2025, 675 \\ldots$, is made by repeatedly dividing by 3. How many integers are in this sequence? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 33921,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In space, there are 3 planes and a sphere. How many distinct ways can a second sphere be placed in space so that it touches the three given planes and the first sphere? (In this problem, sphere touching is considered, i.e., it is not assumed that the spheres can only touch exte... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 33922,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points $A$, $B$, $C$, and $T$ are in space such that each of $\\overline{TA}$, $\\overline{TB}$, and $\\overline{TC}$ is perpendicular to the other two. If $TA = TB = 10$ and $TC = 9$, then what is the volume of pyramid $TABC$? Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 33923,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A five-digit number is called a \"hill\" if its first three digits are in ascending order and its last three digits are in descending order. For example, 13760 and 28932 are hills, whereas 78821 and 86521 are not hills. How many hills exist that are greater than the number 7777... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 33924,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When two fair 12-sided dice are tossed, the numbers $a$ and $b$ are obtained. What is the probability that both the two-digit number $ab$ (where $a$ and $b$ are digits) and each of $a$ and $b$ individually are divisible by 4? Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{16}"
} | {
"split": "train",
"index": 33925,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest four-digit negative integer congruent to $1 \\pmod{23}?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1011"
} | {
"split": "train",
"index": 33926,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function \\[f(x) = \\left\\{ \\begin{aligned} 2x + 1 & \\quad \\text{ if } x < 3 \\\\ x^2 & \\quad \\text{ if } x \\ge 3 \\end{aligned} \\right.\\] has an inverse $f^{-1}.$ Compute the value of $f^{-1}(-3) + f^{-1}(0) + \\dots + f^{-1}(4) + f^{-1}(9).$ Let's think step by s... | math | {
"style": "rule",
"ground_truth": "3.5"
} | {
"split": "train",
"index": 33927,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A Saxon silver penny, from the reign of Ethelbert II in the eighth century, was sold in 2014 for £78000. A design on the coin depicts a circle surrounded by four equal arcs, each a quarter of a circle. The width of the design is 2 cm. What is the radius of the small circle, in ... | math | {
"style": "rule",
"ground_truth": "2 - \\sqrt{2}"
} | {
"split": "train",
"index": 33928,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 4 children and 2 coaches in the Chess Interest Group of Hongxing Primary School. The ages of the 4 children differ by 2 years sequentially, and the ages of the 2 coaches differ by 2 years. The sum of the squares of the ages of the 6 people is 2796 years. What is the s... | math | {
"style": "rule",
"ground_truth": "106"
} | {
"split": "train",
"index": 33929,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of the house numbers on one side of a street from corner to corner is 117. What is the house number of the fifth house from the beginning of this section? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 33930,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the least whole number that is divisible by 7, but leaves a remainder of 1 when divided by any integer 2 through 6? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "301"
} | {
"split": "train",
"index": 33931,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A semicircle with diameter $d$ is contained in a square whose sides have length 8. Given the maximum value of $d$ is $m - \\sqrt{n},$ find $m+n.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "544"
} | {
"split": "train",
"index": 33932,
"solution": "It is easy after getting the image, after drawing labeling the lengths of those segments, assume the radius is $x$, we can see $x=\\sqrt{2}(8-x)$ and we get $2x=32-\\sqrt{512}$ and we have the answer $\\boxed{544}$ ~bluesoul"
} |
deepscaler | [
{
"role": "user",
"content": "In the regular tetrahedron \\(ABCD\\), points \\(E\\) and \\(F\\) are on edges \\(AB\\) and \\(AC\\) respectively, such that \\(BE = 3\\) and \\(EF = 4\\), and \\(EF\\) is parallel to face \\(BCD\\). What is the area of \\(\\triangle DEF\\)? Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{33}"
} | {
"split": "train",
"index": 33933,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given regular hexagon $ABCDEF$ , compute the probability that a randomly chosen point inside the hexagon is inside triangle $PQR$ , where $P$ is the midpoint of $AB$ , $Q$ is the midpoint of $CD$ , and $R$ is the midpoint of $EF$ . Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{8}"
} | {
"split": "train",
"index": 33934,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A positive five-digit integer is in the form $AB,CBA$; where $A$, $B$ and $C$ are each distinct digits. What is the greatest possible value of $AB,CBA$ that is divisible by eleven? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "96,\\!569"
} | {
"split": "train",
"index": 33935,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the number of perfect cubic divisors in the product $1! \\cdot 2! \\cdot 3! \\cdot \\ldots \\cdot 6!$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 33936,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $N$ be the number of ordered pairs of integers $(x, y)$ such that\n\\[\n 4x^2 + 9y^2 \\le 1000000000.\n\\]\nLet $a$ be the first digit of $N$ (from the left) and let $b$ be the second digit of $N$ . What is the value of $10a + b$ ? Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "52"
} | {
"split": "train",
"index": 33937,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The specific heat capacity of a body with mass $m=2$ kg depends on the temperature as follows: $c=c_{0}(1+\\alpha t)$, where $c_{0}=150$ J/kg$\\cdot^\\circ$C is the specific heat capacity at $0^\\circ$C, $\\alpha=0.05\\ ^\\circ$C$^{-1}$ is the temperature coefficient, and $t$ i... | math | {
"style": "rule",
"ground_truth": "96000"
} | {
"split": "train",
"index": 33938,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In parallelogram $ABCD$ , the longer side is twice the shorter side. Let $XYZW$ be the quadrilateral formed by the internal bisectors of the angles of $ABCD$ . If the area of $XYZW$ is $10$ , find the area of $ABCD$ Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 33939,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the increasing sequence of positive integers whose binary representation has exactly $8$ ones. Let $N$ be the 1000th number in $S$. Find the remainder when $N$ is divided by $1000$.\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 33940,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( a, b, c \\) be positive integers such that \\( \\frac{1}{a} + \\frac{1}{b} = \\frac{1}{c} \\) and \\( \\operatorname{gcd}(a, b, c) = 1 \\). Suppose \\( a + b \\leq 2011 \\). Determine the largest possible value of \\( a + b \\). Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "1936"
} | {
"split": "train",
"index": 33941,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A point $P$ lies in the same plane as a given square of side $2$. Let the vertices of the square, taken counterclockwise, be $A, B, C$ and $D$. Also, let the distances from $P$ to $B, C$ and $D$, respectively, be $v, w$ and $t$. What is the greatest distance that $P$ can be fro... | math | {
"style": "rule",
"ground_truth": "\\sqrt{10}"
} | {
"split": "train",
"index": 33942,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the product sequence $\\frac{5}{3} \\cdot \\frac{6}{5} \\cdot \\frac{7}{6} \\cdot \\ldots \\cdot \\frac{a}{b} = 12$, determine the sum of $a$ and $b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "71"
} | {
"split": "train",
"index": 33943,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest four-digit number that is divisible by 6? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9996"
} | {
"split": "train",
"index": 33944,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $S$ is a finite set of positive integers. If the greatest integer in $S$ is removed from $S$, then the average value (arithmetic mean) of the integers remaining is $32$. If the least integer in $S$ is also removed, then the average value of the integers remaining i... | math | {
"style": "rule",
"ground_truth": "36.8"
} | {
"split": "train",
"index": 33945,
"solution": "Let $S = \\{a_1, a_2, a_3, \\hdots, a_n\\}$ with $a_1 < a_2 < a_3 < \\hdots < a_n.$ We are given the following conditions:\n1. Removing the greatest integer $a_n$ from $S$ gives an average of $32$ for the remaining integers:\n \\[\n \\frac{\\sum_{i=1}^{n-1} a_i... |
deepscaler | [
{
"role": "user",
"content": "Find the product of the values of $x$ that satisfy the equation $|5x| + 7 = 47$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-64"
} | {
"split": "train",
"index": 33946,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two congruent right circular cones each with base radius $3$ and height $8$ have the axes of symmetry that intersect at right angles at a point in the interior of the cones a distance $3$ from the base of each cone. A sphere with radius $r$ lies withing both cones. The maximum ... | math | {
"style": "rule",
"ground_truth": "298"
} | {
"split": "train",
"index": 33947,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If for any \\( x \\in \\mathbf{R} \\), the function \\( f(x) \\) satisfies the equation \\( f(x+2009) = -f(x+2008) \\), and \\( f(2009) = -2009 \\), determine the value of \\( f(-1) \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2009"
} | {
"split": "train",
"index": 33948,
"solution": ""
} |
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