data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "*How many odd numbers between 200 and 999 have distinct digits, and no digit greater than 7?*"
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 400,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "-15 + 9 \\times (6 \\div 3) ="
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 401,
"solution": "To solve the expression $-15+9\\times (6\\div 3)$, we follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).\n\n1. **Evaluate the expression inside ... |
DeepScaleR | [
{
"role": "user",
"content": ".4 + .02 + .006 ="
}
] | math | {
"style": "rule",
"ground_truth": ".426"
} | {
"index": 402,
"solution": "To solve the problem, we add the decimal numbers together. It's helpful to align the numbers by their decimal places to ensure accurate addition:\n\n1. Write each number with the same number of decimal places:\n \\[\n 0.4 = 0.400 \\\\\n 0.02 = 0.020 \\\\\n 0.006 = 0.006\n \\]\... |
DeepScaleR | [
{
"role": "user",
"content": "0.8 + 0.02"
}
] | math | {
"style": "rule",
"ground_truth": "0.82"
} | {
"index": 403,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "1-2-3+4+5-6-7+8+9-10-11+\\cdots + 1992+1993-1994-1995+1996="
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 404,
"solution": "1. **Group the terms**: We group the terms in sets of four as follows:\n \\[\n (1-2-3+4) + (5-6-7+8) + (9-10-11+12) + \\cdots + (1993-1994-1995+1996)\n \\]\n\n2. **Calculate the sum of each group**: \n - For the first group: \n \\[\n 1 - 2 - 3 + 4 = (1 + 4) - (2 + 3) = 5... |
DeepScaleR | [
{
"role": "user",
"content": "1. Calculate $\\log _{2.5}6.25+ \\lg 0.01+ \\ln \\sqrt {e}-2\\,^{1+\\log _{2}3}$\n2. Given $\\tan \\alpha=-3$, and $\\alpha$ is an angle in the second quadrant, find $\\sin \\alpha$ and $\\cos \\alpha$."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac { \\sqrt {10}}{10}"
} | {
"index": 405,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "1. Calculate the value of the following expression:\n$(1)(2\\frac{7}{9})^{\\frac{1}{2}} - (2\\sqrt{3} - \\pi)^{0} - (2\\frac{10}{27})^{-\\frac{2}{3}} + 0.25^{-\\frac{3}{2}}$\n2. Given that $x + x^{-1} = 4 (0 < x < 1)$, find the value of ${x^{\\frac{1}{2}}} + {x^{-\\frac{1}{2}}}... | math | {
"style": "rule",
"ground_truth": "\\frac{389}{48}"
} | {
"index": 406,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "1. Convert the parametric equations of the conic curve $C$: \n$$\n\\begin{cases} \nx=t^{2}+ \\frac {1}{t^{2}}-2 \\\\ \ny=t- \\frac {1}{t}\n\\end{cases}\n$$\n($t$ is the parameter) into a Cartesian coordinate equation.\n\n2. If the polar equations of two curves are $\\rho=1$ and... | math | {
"style": "rule",
"ground_truth": "\\sqrt {3}"
} | {
"index": 407,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "1. Given $$\\cos\\left(\\alpha+ \\frac {\\pi}{6}\\right)-\\sin\\alpha= \\frac {3 \\sqrt {3}}{5}$$, find the value of $$\\sin\\left(\\alpha+ \\frac {5\\pi}{6}\\right)$$; \n2. Given $$\\sin\\alpha+\\sin\\beta= \\frac {1}{2}$$ and $$\\cos\\alpha+\\cos\\beta= \\frac {\\sqrt {2}}{2... | math | {
"style": "rule",
"ground_truth": "-\\frac {5}{8}"
} | {
"index": 408,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "1. Given $\\sin\\alpha + \\cos\\alpha = \\frac{7}{13}$, with $\\alpha \\in (0, \\pi)$, find the value of $\\tan\\alpha$.\n2. Find the minimum value for $y=\\sin 2x + 2\\sqrt{2}\\cos\\left(\\frac{\\pi}{4}+x\\right)+3$."
}
] | math | {
"style": "rule",
"ground_truth": "2 - 2\\sqrt{2}"
} | {
"index": 409,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "1. Given $\\tan \\frac{\\alpha}{2} = \\frac{1}{2}$, find the value of $\\sin\\left(\\alpha + \\frac{\\pi}{6}\\right)$.\n2. Given $\\alpha \\in \\left(\\pi, \\frac{3\\pi}{2}\\right)$ and $\\cos\\alpha = -\\frac{5}{13}$, $\\tan \\frac{\\beta}{2} = \\frac{1}{3}$, find the value of... | math | {
"style": "rule",
"ground_truth": "-\\frac{17\\sqrt{13}}{65}"
} | {
"index": 410,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "1. Given non-negative real numbers \\( x, y, z \\) satisfying \\( x^{2} + y^{2} + z^{2} + x + 2y + 3z = \\frac{13}{4} \\), determine the maximum value of \\( x + y + z \\).\n\n2. Given \\( f(x) \\) is an odd function defined on \\( \\mathbb{R} \\) with a period of 3, and when \... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 411,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "1. Given that ${(3x-2)^{6}}={a_{0}}+{a_{1}}(2x-1)+{a_{2}}{(2x-1)^{2}}+ \\cdots +{a_{6}}{(2x-1)^{6}}$, find the value of $\\dfrac{{a_{1}}+{a_{3}}+{a_{5}}}{{a_{0}}+{a_{2}}+{a_{4}}+{a_{6}}}$.\n2. A group of 6 volunteers is to be divided into 4 teams, with 2 teams of 2 people and t... | math | {
"style": "rule",
"ground_truth": "\\dfrac{ \\sqrt{2}}{2}"
} | {
"index": 412,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "1. Given that $α$ and $β$ are acute angles, and $\\cos α= \\frac{4}{5}$, $\\cos (α+β)=- \\frac{16}{65}$, find the value of $\\cos β$.\n2. Given that $0 < β < \\frac{π}{4} < α < \\frac{3}{4}π$, $\\cos ( \\frac{π}{4}-α)= \\frac{3}{5}$, $\\sin ( \\frac{3π}{4}+β)= \\frac{5}{13}$, f... | math | {
"style": "rule",
"ground_truth": "\\frac{56}{65}"
} | {
"index": 413,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "1. Given that the line $l$ passing through the point $M(-3,0)$ is intercepted by the circle $x^{2}+(y+2)^{2}=25$ to form a chord of length $8$, what is the equation of line $l$?\n2. Circles $C_{1}$: $x^{2}+y^{2}+2x+8y-8=0$ and $C_{2}$: $x^{2}+y^{2}-4x-4y-2=0$ intersect. What is... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 414,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "1. Given that the terminal side of angle $\\alpha$ passes through point $P(4, -3)$, find the value of $2\\sin\\alpha + \\cos\\alpha$.\n2. Given that the terminal side of angle $\\alpha$ passes through point $P(4a, -3a)$ ($a \\neq 0$), find the value of $2\\sin\\alpha + \\cos\\a... | math | {
"style": "rule",
"ground_truth": "-\\frac{2}{5}"
} | {
"index": 415,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "1. How many four-digit numbers with no repeated digits can be formed using the digits 1, 2, 3, 4, 5, 6, 7, and the four-digit number must be even?\n2. How many five-digit numbers with no repeated digits can be formed using the digits 0, 1, 2, 3, 4, 5, and the five-digit number ... | math | {
"style": "rule",
"ground_truth": "216"
} | {
"index": 416,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "1. If $A_{10}^{m} =10×9×…×5$, then $m=$ ______.\n2. The number of ways for A, B, C, and D to take turns reading the same book, with A reading first, is ______.\n3. If five boys and two girls are to be arranged in a row for a photo, with boy A required to stand in the middle and... | math | {
"style": "rule",
"ground_truth": "192"
} | {
"index": 417,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "1. In the Cartesian coordinate system, given three points A(5, 4), B(k, 10), and C(12, -2), for what value of k are the vectors $$\\overrightarrow {AB}$$ and $$\\overrightarrow {BC}$$ collinear?\n2. In the Cartesian coordinate system, given O as the origin, $$\\overrightarrow {... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 418,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "1. Let $[x]$ denote the greatest integer less than or equal to the real number $x$. Given a sequence of positive integers $\\{a_{n}\\}$ such that $a_{1} = a$, and for any positive integer $n$, the sequence satisfies the recursion\n$$\na_{n+1} = a_{n} + 2 \\left[\\sqrt{a_{n}}\\r... | math | {
"style": "rule",
"ground_truth": "82"
} | {
"index": 419,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "1. Let \\( x_i \\in \\{0,1\\} \\) (for \\( i=1,2,\\cdots,n \\)). If the value of the function \\( f=f(x_1, x_2, \\cdots, x_n) \\) only takes 0 or 1, then \\( f \\) is called an n-ary Boolean function. We denote\n \\[\n D_{n}(f)=\\left\\{(x_1, x_2, \\cdots, x_n) \\mid f(x_1,... | math | {
"style": "rule",
"ground_truth": "565"
} | {
"index": 420,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "1. On a semicircle with AB as the diameter, besides points A and B, there are 6 other points. Since there are also 4 other points on AB, making a total of 12 points, how many quadrilaterals can be formed with these 12 points as vertices?\n2. On one side of angle A, there are fi... | math | {
"style": "rule",
"ground_truth": "200"
} | {
"index": 421,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "1. Simplify and evaluate the expression: $\\log_{\\frac{1}{3}} \\sqrt{27} + \\lg 25 + \\lg 4 + 7^{-\\log_{7} 2} + (-0.98)^0$\n2. Given a point $P(\\sqrt{2}, -\\sqrt{6})$ on the terminal side of angle $\\alpha$, evaluate: $\\frac{\\cos \\left( \\frac{\\pi}{2} + \\alpha \\right) ... | math | {
"style": "rule",
"ground_truth": "\\frac{-\\sqrt{3} - 1}{3}"
} | {
"index": 422,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "1. Solve the inequality $\\frac{2x+1}{3-x} \\geq 1$.\n2. Given that $x > 0$ and $y > 0$, and $x + y = 1$, find the minimum value of $\\frac{4}{x} + \\frac{9}{y}$."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 423,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "1. Solve the trigonometric inequality: $\\cos x \\geq \\frac{1}{2}$\n2. In $\\triangle ABC$, if $\\sin A + \\cos A = \\frac{\\sqrt{2}}{2}$, find the value of $\\tan A$."
}
] | math | {
"style": "rule",
"ground_truth": "-2 - \\sqrt{3}"
} | {
"index": 424,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "1. The converse of the proposition \"If $x > 1$, then ${x}^{2} > 1$\" is ________.\n2. Let $P$ be a point on the parabola ${{y}^{2}=4x}$ such that the distance from $P$ to the line $x+2=0$ is $6$. The distance from $P$ to the focus $F$ of the parabola is ________.\n3. In a geom... | math | {
"style": "rule",
"ground_truth": "\\frac{32}{5}"
} | {
"index": 425,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "1. The focal distance of the parabola $4x^{2}=y$ is \\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\n2. The equation of the hyperbola that has the same asymptotes as the hyperbola $\\frac{x^{2}}{2} -y^{2}=1$ and passes through $(2,0)$ is \\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\n3. In the pla... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"index": 426,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "1. There are 5 different books, and we need to choose 3 books to give to 3 students, one book per student. There are a total of different ways to do this.\n2. There are 5 different books, and we want to buy 3 books to give to 3 students, one book per student. There are a to... | math | {
"style": "rule",
"ground_truth": "125"
} | {
"index": 427,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "1. When a die (with faces numbered 1 through 6) is thrown twice in succession, find the probability that the sum of the numbers facing up is at least 10.\n\n2. On a line segment MN of length 16cm, a point P is chosen at random. A rectangle is formed with MP and NP as adjacent s... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 428,
"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"
} | {
"index": 429,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "10 - 1.05 ÷ [5.2 × 14.6 - (9.2 × 5.2 + 5.4 × 3.7 - 4.6 × 1.5)] = ?"
}
] | math | {
"style": "rule",
"ground_truth": "9.93"
} | {
"index": 430,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "10 chatterboxes sat in a circle. Initially, one of them told one joke, the next one clockwise told two jokes, the next one three jokes, and so on in a circle until one of them told 100 jokes at once. Then the chatterboxes got tired, and the next one clockwise told 99 jokes, the... | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"index": 431,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "10 pairs of distinct shoes are mixed in a bag. If 4 shoes are randomly drawn, how many possible outcomes are there for the following results?\n(1) None of the 4 shoes form a pair;\n(2) Among the 4 shoes, there is one pair, and the other two shoes do not form a pair;\n(3) The 4 ... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 432,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "10 people attend a meeting. Everyone at the meeting exchanges business cards with everyone else. How many exchanges of business cards occur?"
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 433,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "10 students (one of whom is the captain, and 9 are team members) formed a team to participate in a math competition and won first prize. The organizing committee decided to award each team member 200 yuan as a prize. The captain received 90 yuan more than the average prize of a... | math | {
"style": "rule",
"ground_truth": "300"
} | {
"index": 434,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "10 times 0.1 equals to ____, 10 times 0.01 equals to ____, 10 times 0.001 equals to ____."
}
] | math | {
"style": "rule",
"ground_truth": "0.01"
} | {
"index": 435,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "10 times 10,000 is ; 10 times is 10 million; times 10 million is 100 million. There are 10,000s in 100 million."
}
] | math | {
"style": "rule",
"ground_truth": "10000"
} | {
"index": 436,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "10.25 people are lined up in a row, each of whom either tells the truth or lies.\nThe person at the front of the line says: \"Everyone behind me is lying.\"\nEveryone else says: \"The person in front of me (the one directly next to the speaker) is lying.\"\nAmong these 25 peopl... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 437,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "100 \\times 19.98 \\times 1.998 \\times 1000="
}
] | math | {
"style": "rule",
"ground_truth": "(1998)^2"
} | {
"index": 438,
"solution": "1. **Calculate the product of $19.98$ and $100$:**\n \\[\n 19.98 \\times 100 = 1998\n \\]\n This step involves multiplying $19.98$ by $100$, which effectively shifts the decimal point two places to the right.\n\n2. **Calculate the product of $1.998$ and $1000$:**\n \\[\n 1.9... |
DeepScaleR | [
{
"role": "user",
"content": "100 participants came to the international StarCraft championship. The game is played in a knockout format, meaning two players participate in each match, the loser is eliminated from the championship, and the winner remains. Find the maximum possible number of participants who... | math | {
"style": "rule",
"ground_truth": "49"
} | {
"index": 439,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "100 people participated in a quick calculation test consisting of 10 questions. The number of people who answered each question correctly is given in the table below:\n\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline \nProblem Number & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10... | math | {
"style": "rule",
"ground_truth": "62"
} | {
"index": 440,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "100 people who were surveyed and asked the question: ``Is fuchsia kinda pink, or purply?'' Of them, 60 believe that fuchsia is ``kinda pink'' and 27 believe it is both ``kinda pink,'' and also ``purply.'' Another 17 think that fuchsia is neither ``kinda pink'' nor ``purply.''\... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 441,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "11 people were standing in line under the rain, each holding an umbrella. They stood so close together that the umbrellas touched each other. Once the rain stopped, people closed their umbrellas and maintained a distance of 50 cm between each other. By how many times did the le... | math | {
"style": "rule",
"ground_truth": "2.2"
} | {
"index": 442,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "12 Smurfs are seated around a round table. Each Smurf dislikes the 2 Smurfs next to them, but does not dislike the other 9 Smurfs. Papa Smurf wants to form a team of 5 Smurfs to rescue Smurfette, who was captured by Gargamel. The team must not include any Smurfs who dislike eac... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 443,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "12 balls numbered 1 through 12 are placed in a bin. In how many ways can 3 balls be drawn, in order, from the bin, if each ball remains outside the bin after it is drawn?"
}
] | math | {
"style": "rule",
"ground_truth": "1320"
} | {
"index": 444,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "12 balls numbered 1 through 12 are placed in a bin. Joe produces a list of three numbers by performing the following sequence three times: he chooses a ball, records the number, and places the ball back in the bin. How many different lists are possible?"
}
] | math | {
"style": "rule",
"ground_truth": "1728"
} | {
"index": 445,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "12 points are placed around the circumference of a circle. How many ways are there to draw 6 non-intersecting chords joining these points in pairs?"
}
] | math | {
"style": "rule",
"ground_truth": "132"
} | {
"index": 446,
"solution": "$C$ (number of chords) $=C(6)=132$."
} |
DeepScaleR | [
{
"role": "user",
"content": "12 real numbers x and y satisfy \\( 1 + \\cos^2(2x + 3y - 1) = \\frac{x^2 + y^2 + 2(x+1)(1-y)}{x-y+1} \\). Find the minimum value of xy."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{25}"
} | {
"index": 447,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "12 soccer teams participate in a round-robin tournament. Each pair of teams plays a match, with the winning team earning 2 points and the losing team earning 0 points. If the match results in a draw, each team earns 1 point. Three referees recorded the total points for all team... | math | {
"style": "rule",
"ground_truth": "47"
} | {
"index": 448,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "12 students are standing in two rows, with 4 in the front row and 8 in the back row. Now, 2 students from the back row are to be selected to stand in the front row. If the relative order of the other students remains unchanged, the number of different rearrangement methods is _... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 449,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "12. If $p$ is the smallest positive prime number such that there exists an integer $n$ for which $p$ divides $n^{2}+5n+23$, then $p=$ ______"
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 450,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "120 granite slabs weighing 7 tons each and 80 slabs weighing 9 tons each have been stockpiled at the quarry. A railroad platform can hold up to 40 tons. What is the minimum number of platforms required to transport all the slabs?"
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 451,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "120 schools each send 20 people to form 20 teams, with each team having exactly 1 person from each school. Find the smallest positive integer \\( k \\) such that when \\( k \\) people are selected from each team, there will be at least 20 people from the same school among all t... | math | {
"style": "rule",
"ground_truth": "115"
} | {
"index": 452,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "13. Given that $a$, $b$, $c$, are the lengths of the sides opposite to angles $A$, $B$, $C$ in $\\triangle ABC$ respectively, with $a=2$, and $(2+b)(\\sin A-\\sin B)=(c-b)\\sin C$, find the maximum area of $\\triangle ABC$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 453,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "132009 students are taking a test which comprises ten true or false questions. Find the minimum number of answer scripts required to guarantee two scripts with at least nine identical answers."
}
] | math | {
"style": "rule",
"ground_truth": "513"
} | {
"index": 454,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "14 students attend the IMO training camp. Every student has at least $k$ favourite numbers. The organisers want to give each student a shirt with one of the student's favourite numbers on the back. Determine the least $k$ , such that this is always possible if: $a)$ The stu... | math | {
"style": "rule",
"ground_truth": "k = 2"
} | {
"index": 455,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "15 balls numbered 1 through 15 are placed in a bin. Joe produces a list of four numbers by performing the following sequence four times: he chooses a ball, records the number, and places the ball back in the bin. Finally, Joe chooses to make a unique list by selecting 3 numbers... | math | {
"style": "rule",
"ground_truth": "202500"
} | {
"index": 456,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "15 boys and 20 girls sat around a round table. It turned out that the number of pairs of boys sitting next to each other is one and a half times less than the number of pairs of girls sitting next to each other. Find the number of boy-girl pairs sitting next to each other."
}... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 457,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "15 plus 16 equals."
}
] | math | {
"style": "rule",
"ground_truth": "31"
} | {
"index": 458,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "15. If \\( a = 1.69 \\), \\( b = 1.73 \\), and \\( c = 0.48 \\), find the value of\n$$\n\\frac{1}{a^{2} - a c - a b + b c} + \\frac{2}{b^{2} - a b - b c + a c} + \\frac{1}{c^{2} - a c - b c + a b}.\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 459,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "15. Let $a_{n}$ denote the number of ternary strings of length $n$ so that there does not exist a $k<n$ such that the first $k$ digits of the string equals the last $k$ digits. What is the largest integer $m$ such that $3^{m} \\mid a_{2023}$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "2022"
} | {
"index": 460,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "150 people were surveyed and asked: \"Do you think teal is more green or blue?\" Of them, 90 believe teal is \"more green,\" and 50 believe it's \"more blue.\" Additionally, 40 believe it's both \"more green\" and \"more blue.\" Another 20 think teal is neither \"more green\" n... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"index": 461,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "16 students took part in a competition. All problems were multiple choice style. Each problem had four choices. It was said that any two students had at most one answer in common, find the maximum number of problems."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 462,
"solution": "\nLet 16 students take part in a competition where each problem is multiple choice with four choices. We are to find the maximum number of problems such that any two students have at most one answer in common.\n\nLet \\( T \\) denote the number of triples \\((S_i, S_j, Q_k)\\) such that... |
DeepScaleR | [
{
"role": "user",
"content": "18. Given the function $f(x)=x^3+ax^2+bx+5$, the equation of the tangent line to the curve $y=f(x)$ at the point $x=1$ is $3x-y+1=0$.\n\nⅠ. Find the values of $a$ and $b$;\n\nⅡ. Find the maximum and minimum values of $y=f(x)$ on the interval $[-3,1]$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{95}{27}"
} | {
"index": 463,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "18.14 People are participating in a round-robin Japanese chess tournament. Each person plays against 13 others, with no draws in the matches. Find the maximum number of \"circular triples\" (where each of the three participants wins against one and loses to another) in the tour... | math | {
"style": "rule",
"ground_truth": "112"
} | {
"index": 464,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "180 grams of 920 purity gold was alloyed with 100 grams of 752 purity gold. What is the purity of the resulting alloy?"
}
] | math | {
"style": "rule",
"ground_truth": "860"
} | {
"index": 465,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "19) A puck is kicked up a ramp, which makes an angle of $30^{\\circ}$ with the horizontal. The graph below depicts the speed of the puck versus time. What is the coefficient of friction between the puck and the ramp?\n\nA) 0.07\nB) 0.15\nC) 0.22\nD) 0.29\nE) 0.37"
}
] | math | {
"style": "rule",
"ground_truth": "0.29"
} | {
"index": 466,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "199 people signed up to participate in a tennis tournament. In the first round, pairs of opponents are selected by drawing lots. The same method is used to pair opponents in the second, third, and all subsequent rounds. After each match, one of the two opponents is eliminated, ... | math | {
"style": "rule",
"ground_truth": "198"
} | {
"index": 467,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "1990-1980+1970-1960+\\cdots -20+10 ="
}
] | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"index": 468,
"solution": "1. **Identify the pattern and the number of terms**: The sequence given is $1990 - 1980 + 1970 - 1960 + \\cdots - 20 + 10$. We observe that the sequence alternates between addition and subtraction, starting with a subtraction. The sequence starts at $1990$ and ends at $10$, decreasing b... |
DeepScaleR | [
{
"role": "user",
"content": "2 diagonals of a regular decagon (a 10-sided polygon) are chosen. What is the probability that their intersection lies inside the decagon?"
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{42}{119}"
} | {
"index": 469,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "2 diagonals of a regular heptagon (a 7-sided polygon) are chosen. What is the probability that they intersect inside the heptagon?"
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{5}{13}"
} | {
"index": 470,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "2 diagonals of a regular nonagon (a 9-sided polygon) are chosen. What is the probability that their intersection lies inside the nonagon?"
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{14}{39}"
} | {
"index": 471,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "2 distinct vertices of an octagon are chosen at random. What is the probability that they are adjacent?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{7}"
} | {
"index": 472,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "20 balls numbered 1 through 20 are placed in a bin. In how many ways can 4 balls be drawn, in order, from the bin, if each ball remains outside the bin after it is drawn and each drawn ball has a consecutive number as the previous one?"
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 473,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "20 different villages are located along the coast of a circular island. Each of these villages has 20 fighters, with all 400 fighters having different strengths.\n\nTwo neighboring villages $A$ and $B$ now have a competition in which each of the 20 fighters from village $A$ com... | math | {
"style": "rule",
"ground_truth": "290"
} | {
"index": 474,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "20 players are playing in a Super Smash Bros. Melee tournament. They are ranked $1-20$, and player $n$ will always beat player $m$ if $n<m$. Out of all possible tournaments where each player plays 18 distinct other players exactly once, one is chosen uniformly at random. Find t... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 475,
"solution": "Consider instead the complement of the tournament: The 10 possible matches that are not played. In order for each player to play 18 games in the tournament, each must appear once in these 10 unplayed matches. Players $n$ and $n+1$ will win the same number of games if, in the matching, t... |
DeepScaleR | [
{
"role": "user",
"content": "20 points were marked inside a square and connected with non-intersecting segments to each other and to the vertices of the square, such that the square was divided into triangles. How many triangles were formed?"
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"index": 476,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "20 shareholders are seated around a round table. What is the minimum total number of their shares if it is known that:\na) any three of them together have more than 1000 shares,\nb) any three consecutive shareholders together have more than 1000 shares?"
}
] | math | {
"style": "rule",
"ground_truth": "6674"
} | {
"index": 477,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "20. Given an ellipse $C$: $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 \\left( a > b > 0 \\right)$ passes through point $M\\left( 1,\\frac{3}{2} \\right)$, $F_1$ and $F_2$ are the two foci of ellipse $C$, and $\\left| MF_1 \\right|+\\left| MF_2 \\right|=4$, $O$ is the center of ell... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}"
} | {
"index": 478,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "200 people stand in a circle. Each of them is either a liar or a conformist. Liars always lie. A conformist standing next to two conformists always tells the truth. A conformist standing next to at least one liar can either tell the truth or lie. 100 of the standing people said... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"index": 479,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "2000 people are sitting around a round table. Each one of them is either a truth-sayer (who always tells the truth) or a liar (who always lies). Each person said: \"At least two of the three people next to me to the right are liars\". How many truth-sayers are there in the circ... | math | {
"style": "rule",
"ground_truth": "666"
} | {
"index": 480,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "2001 coins, each valued at 1, 2, or 3, are arranged in a row. The coins are placed such that:\n- Between any two coins of value 1, there is at least one other coin.\n- Between any two coins of value 2, there are at least two other coins.\n- Between any two coins of value 3, the... | math | {
"style": "rule",
"ground_truth": "501"
} | {
"index": 481,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "2002 is a palindromic year, meaning it reads the same backward and forward. The previous palindromic year was 11 years ago (1991). What is the maximum number of non-palindromic years that can occur consecutively (between the years 1000 and 9999)?"
}
] | math | {
"style": "rule",
"ground_truth": "109"
} | {
"index": 482,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "2005^{2} + 2 \\times 2005 \\times 1995 + 1995^{2} divided by 800."
}
] | math | {
"style": "rule",
"ground_truth": "20000"
} | {
"index": 483,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "2008 persons take part in a programming contest. In one round, the 2008 programmers are divided into two groups. Find the minimum number of groups such that every two programmers ever be in the same group."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 484,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "2011 warehouses are connected by roads in such a way that from any warehouse you can reach any other, possibly passing through several roads. There are $x_{1}, \\ldots, x_{2011}$ kg of cement in the warehouses, respectively. In one trip, an arbitrary amount of cement can be tra... | math | {
"style": "rule",
"ground_truth": "2010"
} | {
"index": 485,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "2019 points are chosen independently and uniformly at random on the interval $[0,1]$. Tairitsu picks 1000 of them randomly and colors them black, leaving the remaining ones white. Hikari then computes the sum of the positions of the leftmost white point and the rightmost black ... | math | {
"style": "rule",
"ground_truth": "\\frac{1019}{2019}"
} | {
"index": 486,
"solution": "Note that each point is chosen uniformly and independently from 0 to 1, so we can apply symmetry. Given any coloring, suppose that we flip all the positions of the black points: then the problem becomes computing the probability that the leftmost white point is to the left of the leftmo... |
DeepScaleR | [
{
"role": "user",
"content": "2019 students are voting on the distribution of \\(N\\) items. For each item, each student submits a vote on who should receive that item, and the person with the most votes receives the item (in case of a tie, no one gets the item). Suppose that no student votes for the same p... | math | {
"style": "rule",
"ground_truth": "1009"
} | {
"index": 487,
"solution": "To get an item, a student must receive at least 2 votes on that item. Since each student receives at most 2019 votes, the number of items one student can receive does not exceed \\(\\frac{2019}{2}=1009.5\\). So, the answer is at most 1009. This occurs when \\(N=2018\\) and item \\(i\\) ... |
DeepScaleR | [
{
"role": "user",
"content": "2022 knights and liars are lined up in a row, with the ones at the far left and right being liars. Everyone except the ones at the extremes made the statement: \"There are 42 times more liars to my right than to my left.\" Provide an example of a sequence where there is exactly... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 488,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "22. In the polar coordinate system, the polar equation of circle $C$ is $\\rho =4\\cos \\theta$. Taking the pole as the origin and the direction of the polar axis as the positive direction of the $x$-axis, a Cartesian coordinate system is established using the same unit length.... | math | {
"style": "rule",
"ground_truth": "\\dfrac{\\sqrt{46}}{2}"
} | {
"index": 489,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "22. Let the function $f : \\mathbb{Z} \\to \\mathbb{Z}$ take only integer inputs and have integer outputs. For any integers $x$ and $y$ , f satisfies $$ f(x) + f(y) = f(x + 1) + f(y - 1) $$ If $f(2016) = 6102$ and $f(6102) = 2016$ , what is $f(1)$ ?\n\n23. Let $d$ ... | math | {
"style": "rule",
"ground_truth": "8117"
} | {
"index": 490,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "23 people attend a party. Each person shakes hands with at least two other people. What is the minimum possible number of handshakes?"
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"index": 491,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "23 people attend a party. Each person shakes hands with at most 22 other people. What is the maximum possible number of handshakes, assuming that any two people can shake hands at most once?"
}
] | math | {
"style": "rule",
"ground_truth": "253"
} | {
"index": 492,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "23. Two friends, Marco and Ian, are talking about their ages. Ian says, \"My age is a zero of a polynomial with integer coefficients.\"\nHaving seen the polynomial \\( p(x) \\) Ian was talking about, Marco exclaims, \"You mean, you are seven years old? Oops, sorry I miscalculat... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 493,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "24 people participate in a training competition consisting of 12 rounds. After each round, every participant receives a certain score \\( a_k \\) based on their ranking \\( k \\) in that round, where \\( a_{k} \\in \\mathbb{N}_+, k = 1, 2, \\ldots, n, a_1 > a_2 > \\cdots > a_n ... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 494,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "24×12, my approach is to first calculate \\_\\_\\_\\_\\_\\_, then calculate \\_\\_\\_\\_\\_\\_, and finally calculate \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "288"
} | {
"index": 495,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "2500 chess kings have to be placed on a $100 \\times 100$ chessboard so that\n\n[b](i)[/b] no king can capture any other one (i.e. no two kings are placed in two squares sharing a common vertex);\n[b](ii)[/b] each row and each column contains exactly 25 kings.\n\nFind the numbe... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 496,
"solution": "\nLet us consider a \\(100 \\times 100\\) chessboard and the placement of 2500 kings such that:\n\n1. No king can capture another king, meaning no two kings can be placed on squares that share a common vertex.\n2. Each row and each column contains exactly 25 kings.\n\nThe primary challe... |
DeepScaleR | [
{
"role": "user",
"content": "27 identical dice were glued together to form a $3 \\times 3 \\times 3$ cube in such a way that any two adjacent small dice have the same number of dots on the touching faces. How many dots are there on the surface of the large cube?"
}
] | math | {
"style": "rule",
"ground_truth": "189"
} | {
"index": 497,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "27 people went to a mall to buy water to drink. There was a promotion in the mall where three empty bottles could be exchanged for one bottle of water. The question is: For 27 people, the minimum number of bottles of water that need to be purchased so that each person can have ... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 498,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "27 points P and a square with side length 1 lie on the same plane. Let the vertices of the square be \\( A, B, C, \\) and \\( D \\) (counterclockwise), and the distances from P to \\( A, B, \\) and \\( C \\) are \\( u, v, \\) and \\( w \\) respectively. If \\( u^2 + v^2 = w^2 \... | math | {
"style": "rule",
"ground_truth": "2 + \\sqrt{2}"
} | {
"index": 499,
"solution": ""
} |
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