data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "The keys of a safe with five locks are cloned and distributed among eight people such that any of five of eight people can open the safe. What is the least total number of keys? $"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 32900,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The keystone arch is an ancient architectural feature. It is composed of congruent isosceles trapezoids fitted together along the non-parallel sides, as shown. The bottom sides of the two end trapezoids are horizontal. In an arch made with $9$ trapezoids, let $x$ be the angle m... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 32901,
"solution": "1. **Understanding the Geometry of the Keystone Arch**:\n The keystone arch is composed of $9$ congruent isosceles trapezoids. These trapezoids are arranged such that their non-parallel sides (legs) meet at a common point when extended. The bottom sides of the end trapezoids are hor... |
DeepScaleR | [
{
"role": "user",
"content": "The kindergarten received flashcards for reading: some say \"MA\", and the others say \"NYA\". Each child took three cards to form words. It turned out that 20 children can form the word \"MAMA\" from their cards, 30 children can form the word \"NYANYA\", and 40 children can fo... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 32902,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The kindergarten teacher evenly distributed 270 apples, 180 pears, and 235 oranges to the larger group of children. The remaining quantities of apples, pears, and oranges are in the ratio $3:2:1$. How many children are there in the larger group?"
}
] | math | {
"style": "rule",
"ground_truth": "29"
} | {
"index": 32903,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The king called two wise men. He gave the first one 100 blank cards and ordered him to write a positive number on each (the numbers do not have to be different) without showing them to the second wise man. Then, the first wise man can communicate several different numbers to th... | math | {
"style": "rule",
"ground_truth": "101"
} | {
"index": 32904,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The king distributed ducats to his sons. To the eldest son, he gave a certain number of ducats, to the younger son he gave one less ducat, and to the next younger son again one less ducat, and so he continued until the youngest. Then he returned to the eldest son and gave him o... | math | {
"style": "rule",
"ground_truth": "105"
} | {
"index": 32905,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The kite \\( ABCD \\) is symmetric with respect to diagonal \\( AC \\). The length of \\( AC \\) is 12 cm, the length of \\( BC \\) is 6 cm, and the internal angle at vertex \\( B \\) is a right angle. Points \\( E \\) and \\( F \\) are given on sides \\( AB \\) and \\( AD \\) ... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{3}"
} | {
"index": 32906,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The knights in a certain kingdom come in two colors. $\\frac{2}{7}$ of them are red, and the rest are blue. Furthermore, $\\frac{1}{6}$ of the knights are magical, and the fraction of red knights who are magical is $2$ times the fraction of blue knights who are magical. What fr... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{27}"
} | {
"index": 32907,
"solution": "1. **Define the total number of knights**: Let the total number of knights be $42$. This choice is made because the least common multiple (LCM) of the denominators from the fractions given in the problem ($7$ and $6$) is $42$.\n\n2. **Calculate the number of red and blue knights**:\n ... |
DeepScaleR | [
{
"role": "user",
"content": "The landlord of an apartment building needs to purchase enough digits to label all of the apartments from 100 through 125 on the first floor and 200 through 225 on the second floor. The digits can only be purchased in a package that contains one of each digit 0 through 9. How m... | math | {
"style": "rule",
"ground_truth": "52"
} | {
"index": 32908,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The large cube shown is made up of $27$ identical sized smaller cubes. For each face of the large cube, the opposite face is shaded the same way. The total number of smaller cubes that must have at least one face shaded is"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 32909,
"solution": "To solve this problem, we need to determine the number of smaller cubes in the large cube that have at least one face shaded. We are given that the large cube is composed of $27$ smaller cubes and that the shading pattern on one face is mirrored exactly on the opposite face.\n\n1. **U... |
DeepScaleR | [
{
"role": "user",
"content": "The larger of two consecutive odd integers is three times the smaller. What is their sum?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 32910,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The larger root minus the smaller root of the equation $(7+4\\sqrt{3})x^2+(2+\\sqrt{3})x-2=0$ is"
}
] | math | {
"style": "rule",
"ground_truth": "6-3\\sqrt{3}"
} | {
"index": 32911,
"solution": "1. **Simplify the given quadratic equation**: The given quadratic equation is \\[(7+4\\sqrt{3})x^2+(2+\\sqrt{3})x-2=0.\\] We start by making the quadratic equation monic (i.e., the coefficient of $x^2$ is 1). This is done by dividing the entire equation by $7 + 4\\sqrt{3}$:\n \\[x^2... |
DeepScaleR | [
{
"role": "user",
"content": "The largest and smallest possible three-digit numbers that can be formed using the digits 5, 1, and 9 are found by maximizing and minimizing the order of the digits, respectively. Calculate the difference between these two numbers."
}
] | math | {
"style": "rule",
"ground_truth": "792"
} | {
"index": 32912,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The largest divisor of $2,014,000,000$ is itself. What is its fifth-largest divisor?"
}
] | math | {
"style": "rule",
"ground_truth": "125,875,000"
} | {
"index": 32913,
"solution": "To find the fifth-largest divisor of $2,014,000,000$, we first need to factorize the number.\n\n1. **Prime Factorization of $2,014,000,000$:**\n \\[\n 2,014,000,000 = 2014 \\times 10^6 = (2 \\times 19 \\times 53) \\times (10^6) = 2 \\times 19 \\times 53 \\times (2 \\times 5)^6 = 2... |
DeepScaleR | [
{
"role": "user",
"content": "The largest divisor of a natural number \\( N \\), smaller than \\( N \\), was added to \\( N \\), producing a power of ten. Find all such \\( N \\)."
}
] | math | {
"style": "rule",
"ground_truth": "75"
} | {
"index": 32914,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The largest four-digit number whose digits add to 17 is 9800. The 5th largest four-digit number whose digits have a sum of 17 is:"
}
] | math | {
"style": "rule",
"ground_truth": "9611"
} | {
"index": 32915,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The largest number by which the expression $n^3 - n$ is divisible for all possible integral values of $n$, is:"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 32916,
"solution": "To find the largest number by which the expression $n^3 - n$ is divisible for all possible integral values of $n$, we can start by factoring the expression:\n\n1. **Factorize the expression:**\n \\[\n n^3 - n = n(n^2 - 1) = n(n-1)(n+1)\n \\]\n Here, $n(n-1)(n+1)$ represents th... |
DeepScaleR | [
{
"role": "user",
"content": "The largest ocean in the world is the Pacific Ocean, with an area of 17,996,800 square kilometers. Rewrite this number in terms of \"ten thousand\" as the unit, and round it to the nearest \"ten thousand\" square kilometers."
}
] | math | {
"style": "rule",
"ground_truth": "1800"
} | {
"index": 32917,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The largest prime factor of $999999999999$ is greater than $2006$ . Determine the remainder obtained when this prime factor is divided by $2006$ ."
}
] | math | {
"style": "rule",
"ground_truth": "1877"
} | {
"index": 32918,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The largest prime factor of 101101101101 is a four-digit number $N$. Compute $N$."
}
] | math | {
"style": "rule",
"ground_truth": "9901"
} | {
"index": 32919,
"solution": "Note that $$\\begin{aligned} 101101101101 & =101 \\cdot 1001001001 \\\\ & =101 \\cdot 1001 \\cdot 1000001 \\\\ & =101 \\cdot 1001 \\cdot\\left(100^{3}+1\\right) \\\\ & =101 \\cdot 1001 \\cdot(100+1)\\left(100^{2}-100+1\\right) \\\\ & =101 \\cdot 1001 \\cdot 101 \\cdot 9901 \\\\ & =101... |
DeepScaleR | [
{
"role": "user",
"content": "The largest three-digit number divided by an integer, with the quotient rounded to one decimal place being 2.5, will have the smallest divisor as:"
}
] | math | {
"style": "rule",
"ground_truth": "392"
} | {
"index": 32920,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The largest value of the real number $k$ for which the inequality $\\frac{1+\\sin x}{2+\\cos x} \\geqslant k$ has a solution."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"index": 32921,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The largest whole number such that seven times the number is less than 100 is"
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 32922,
"solution": "1. **Identify the inequality**: We need to find the largest whole number $x$ such that $7x < 100$.\n\n2. **Solve the inequality**:\n - Divide both sides of the inequality $7x < 100$ by 7:\n \\[\n x < \\frac{100}{7}\n \\]\n - Calculate the division:\n \\[\n x < ... |
DeepScaleR | [
{
"role": "user",
"content": "The last 5 digits of $99 \\times 10101 \\times 111 \\times 1001001$ are _____."
}
] | math | {
"style": "rule",
"ground_truth": "88889"
} | {
"index": 32923,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The last three digits of \\( 1978^n \\) and \\( 1978^m \\) are the same. Find the positive integers \\( m \\) and \\( n \\) such that \\( m+n \\) is minimized (here \\( n > m \\geq 1 \\))."
}
] | math | {
"style": "rule",
"ground_truth": "106"
} | {
"index": 32924,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The last two digits of the decimal representation of the square of a natural number are the same and are not zero. What are these digits? Find all solutions."
}
] | math | {
"style": "rule",
"ground_truth": "44"
} | {
"index": 32925,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The lateral edges of a triangular pyramid are mutually perpendicular, and the sides of the base are $\\sqrt{85}$, $\\sqrt{58}$, and $\\sqrt{45}$. The center of the sphere, which touches all the lateral faces, lies on the base of the pyramid. Find the radius of this sphere."
}... | math | {
"style": "rule",
"ground_truth": "14/9"
} | {
"index": 32926,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The lateral face of a regular triangular pyramid \\( SABC \\) is inclined to the base \\( ABC \\) at an angle \\(\\alpha = \\arctan \\frac{3}{4}\\). Points \\( M, N, K \\) are the midpoints of the sides of the base \\( ABC \\). The triangle \\( MNK \\) forms the lower base of a... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 32927,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The lateral sides of a right trapezoid are 10 and 8. The diagonal of the trapezoid, drawn from the vertex of the acute angle, bisects this angle. Find the area of the trapezoid."
}
] | math | {
"style": "rule",
"ground_truth": "104"
} | {
"index": 32928,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The lateral surface area of a circular truncated cone is given by the formula, find the value for the lateral surface area of the cone where the radii of the upper and lower bases are $r=1$ and $R=4$ and the height is $4$."
}
] | math | {
"style": "rule",
"ground_truth": "25\\pi"
} | {
"index": 32929,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The lateral surface area of a regular triangular pyramid is 3 times the area of its base. The area of the circle inscribed in the base is numerically equal to the radius of this circle. Find the volume of the pyramid."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2 \\sqrt{6}}{\\pi^3}"
} | {
"index": 32930,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The lateral surface of a cylinder unfolds into a square. What is the ratio of its lateral surface area to the base area."
}
] | math | {
"style": "rule",
"ground_truth": "4\\pi"
} | {
"index": 32931,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The lattice shown is continued for $9$ rows. What will be the third number in the $9$th row? \\begin{tabular}{rccccc}\nRow 1: & 1 & 2 & 3 & 4 & 5 \\\\\nRow 2: & 6 & 7 & 8 & 9 & 10 \\\\\nRow 3: & 11 & 12 & 13 & 14 & 15 \\\\\nRow 4: & 16 & 17 & 18 & 19 & 20\n\\end{tabular}"
}
] | math | {
"style": "rule",
"ground_truth": "43"
} | {
"index": 32932,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The leadership team of a sports event needs to select 4 volunteers from 5 candidates named A, B, C, D, and E to undertake four different tasks: translation, tour guiding, protocol, and driving. If A and B can only undertake the first three tasks, while the other three candidate... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 32933,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The least common multiple of $a$ and $b$ is $12$, and the least common multiple of $b$ and $c$ is $15$. What is the least possible value of the least common multiple of $a$ and $c$?"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 32934,
"solution": "1. **Identify the constraints given by the problem:**\n - $\\text{lcm}(a, b) = 12$\n - $\\text{lcm}(b, c) = 15$\n\n2. **Determine the possible values for $b$:**\n - Since $b$ must be a common factor of both 12 and 15, we consider the greatest common divisor (gcd) of 12 and 15.\n... |
DeepScaleR | [
{
"role": "user",
"content": "The least common multiple of $a$ and $b$ is $18$, and the least common multiple of $b$ and $c$ is $20$. Find the least possible value of the least common multiple of $a$ and $c$."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 32935,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The least common multiple of $a$ and $b$ is $20$, and the least common multiple of $b$ and $c$ is $21$. Find the least possible value of the least common multiple of $a$ and $c$."
}
] | math | {
"style": "rule",
"ground_truth": "420"
} | {
"index": 32936,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The least common multiple of $x$ and $y$ is $18$, and the least common multiple of $y$ and $z$ is $20$. Determine the least possible value of the least common multiple of $x$ and $z$."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 32937,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The least common multiple of $x$, $10$ and $14$ is $70$. What is the greatest possible value of $x$?"
}
] | math | {
"style": "rule",
"ground_truth": "70"
} | {
"index": 32938,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The least common multiple of two integers is 36 and 6 is their greatest common divisor. What is the product of the two numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "216"
} | {
"index": 32939,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The least common multiple of two numbers is 3780, and the greatest common divisor is 18. Given that one of the numbers is 180, what is the other number?"
}
] | math | {
"style": "rule",
"ground_truth": "378"
} | {
"index": 32940,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The least positive angle $\\alpha$ for which $$ \\left(\\frac34-\\sin^2(\\alpha)\\right)\\left(\\frac34-\\sin^2(3\\alpha)\\right)\\left(\\frac34-\\sin^2(3^2\\alpha)\\right)\\left(\\frac34-\\sin^2(3^3\\alpha)\\right)=\\frac1{256} $$ has a degree measure of $\\tfrac{m}{n}$... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 32941,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The least positive integer with exactly $2021$ distinct positive divisors can be written in the form $m \\cdot 6^k$, where $m$ and $k$ are integers and $6$ is not a divisor of $m$. What is $m+k?$"
}
] | math | {
"style": "rule",
"ground_truth": "58"
} | {
"index": 32942,
"solution": "To find the least positive integer with exactly $2021$ distinct positive divisors, we start by understanding the divisor function. If a number $n$ has a prime factorization of the form $n = p_1^{a_1} p_2^{a_2} \\cdots p_k^{a_k}$, then the number of divisors of $n$ is given by $(a_1+1)... |
DeepScaleR | [
{
"role": "user",
"content": "The least positive integer with exactly \\(2023\\) distinct positive divisors can be written in the form \\(m \\cdot 10^k\\), where \\(m\\) and \\(k\\) are integers and \\(10\\) is not a divisor of \\(m\\). What is \\(m+k?\\)\nA) 999846\nB) 999847\nC) 999848\nD) 999849\nE) 9998... | math | {
"style": "rule",
"ground_truth": "999846"
} | {
"index": 32943,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The left and right foci of a hyperbola are $F_{1}$ and $F_{2}$, respectively. A line passing through $F_{2}$ intersects the right branch of the hyperbola at points $A$ and $B$. If $\\triangle F_{1} A B$ is an equilateral triangle, what is the eccentricity of the hyperbola?"
}... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 32944,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The left and right foci of the ellipse $\\dfrac{x^{2}}{16} + \\dfrac{y^{2}}{9} = 1$ are $F_{1}$ and $F_{2}$, respectively. There is a point $P$ on the ellipse such that $\\angle F_{1}PF_{2} = 30^{\\circ}$. Find the area of triangle $F_{1}PF_{2}$."
}
] | math | {
"style": "rule",
"ground_truth": "18 - 9\\sqrt{3}"
} | {
"index": 32945,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The left and right foci of the hyperbola $C$ are respectively $F_1$ and $F_2$, and $F_2$ coincides with the focus of the parabola $y^2=4x$. Let the point $A$ be an intersection of the hyperbola $C$ with the parabola, and suppose that $\\triangle AF_{1}F_{2}$ is an isosceles tri... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2} + 1"
} | {
"index": 32946,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The left and right foci of the hyperbola $E$: $\\dfrac{x^2}{a^2} - \\dfrac{y^2}{b^2} = 1$ ($a > 0, b > 0$) are $F_1$ and $F_2$, respectively. Point $M$ is a point on the asymptote of hyperbola $E$, and $MF_1 \\perpendicular MF_2$. If $\\sin \\angle MF_1F_2 = \\dfrac{1}{3}$, the... | math | {
"style": "rule",
"ground_truth": "\\dfrac{9}{7}"
} | {
"index": 32947,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The left focus of the hyperbola $C$: $\\dfrac{x^2}{a^2} - \\dfrac{y^2}{b^2} = 1 \\, (a > 0, b > 0)$ is $F$. If the symmetric point $A$ of $F$ with respect to the line $\\sqrt{3}x + y = 0$ is a point on the hyperbola $C$, then the eccentricity of the hyperbola $C$ is \\_\\_\\_\\... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3} + 1"
} | {
"index": 32948,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The leg, the smaller base, and the diagonal of an isosceles trapezoid are 10, 6, and 14 respectively. Find the length of the larger base."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 32949,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The legs \\( AC \\) and \\( CB \\) of the right triangle \\( ABC \\) are 15 and 8, respectively. A circular arc with radius \\( CB \\) is drawn from center \\( C \\), cutting off a part \\( BD \\) from the hypotenuse. Find \\( BD \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{128}{17}"
} | {
"index": 32950,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The legs of a right triangle are 3 and 4. Find the area of the triangle with vertices at the points of tangency of the incircle with the sides of the triangle."
}
] | math | {
"style": "rule",
"ground_truth": "6/5"
} | {
"index": 32951,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The legs of a right triangle have lengths $\\log_4 27$ and $\\log_2 9.$ If the length of the hypotenuse is $h,$ compute $4^h.$"
}
] | math | {
"style": "rule",
"ground_truth": "243"
} | {
"index": 32952,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The length of a chord intercepted on the circle $x^2+y^2-2x+4y-20=0$ by the line $5x-12y+c=0$ is 8. Find the value(s) of $c$."
}
] | math | {
"style": "rule",
"ground_truth": "-68"
} | {
"index": 32953,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The length of a circular race track is $1 \\mathrm{~km}$. Two motorcyclists start simultaneously from a given point $A$ on the track, traveling in opposite directions. One of them travels at a constant speed, while the other accelerates uniformly. They first meet at point $B$ o... | math | {
"style": "rule",
"ground_truth": "\\frac{-1 + \\sqrt{5}}{2}"
} | {
"index": 32954,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The length of a rectangle is increased by $25\\%$, but the width of the rectangle is decreased to keep the area of the rectangle unchanged. By what percent was the rectangle's width decreased?"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 32955,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The length of a rectangle is three times its width. Given that its perimeter and area are both numerically equal to $k>0$, find $k$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{64}{3}"
} | {
"index": 32956,
"solution": "Let $a$ be the width of the rectangle. Then the length of the rectangle is $3 a$, so the perimeter is $2(a+3 a)=8 a$, and the area is $3 a^{2}$. Since the length is numerically equal to the width, we know that $$8 a=3 a^{2}=k$$ Because $k>0$, the rectangle is non-degenerate. It follow... |
DeepScaleR | [
{
"role": "user",
"content": "The length of a rectangle is three times its width. The perimeter is 160 cm. What is the number of square centimeters in the area of the rectangle?"
}
] | math | {
"style": "rule",
"ground_truth": "1200"
} | {
"index": 32957,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The length of a rectangular playground exceeds twice its width by 25 feet, and the perimeter of the playground is 650 feet. What is the area of the playground in square feet?"
}
] | math | {
"style": "rule",
"ground_truth": "22,500"
} | {
"index": 32958,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The length of a rectangular yard exceeds twice its width by 30 feet, and the perimeter of the yard is 700 feet. What is the area of the yard in square feet?"
}
] | math | {
"style": "rule",
"ground_truth": "25955.56"
} | {
"index": 32959,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The length of edge AB is 51 units, and the lengths of the other five edges of the tetrahedron ABCD measure 10, 20, 25, 34, and 45 units. Determine the length of edge CD."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 32960,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The length of edge PQ of a tetrahedron PQRS measures 51 units, and the lengths of the other edges are 12, 19, 24, 33, and 42 units. Determine the length of edge RS."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 32961,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The length of one side of the square \\(ABCD\\) is 4 units. A circle is drawn tangent to \\(\\overline{BC}\\) and passing through the vertices \\(A\\) and \\(D\\). Find the area of the circle."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{25 \\pi}{4}"
} | {
"index": 32962,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The length of rectangle $ABCD$ is 5 inches and its width is 3 inches. Diagonal $AC$ is divided into three equal segments by points $E$ and $F$. The area of triangle $BEF$, expressed in square inches, is:"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}"
} | {
"index": 32963,
"solution": "1. **Draw the rectangle and identify key components**: Let rectangle $ABCD$ have $AB = 5$ inches (length) and $AD = 3$ inches (width). Diagonal $AC$ is drawn, which divides the rectangle into two right triangles, $ABC$ and $ADC$.\n\n2. **Calculate the length of diagonal $AC$ using the... |
DeepScaleR | [
{
"role": "user",
"content": "The length of the chord cut by one of the asymptotes of the hyperbola $\\dfrac{x^2}{a^2} - \\dfrac{y^2}{b^2} = 1 \\ (a > 0, b > 0)$ on the circle $x^2 + y^2 - 6x + 5 = 0$ is $2$. Find the eccentricity of the hyperbola."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{\\sqrt{6}}{2}"
} | {
"index": 32964,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The length of the chord cut by the line $y= \\frac{1}{2}x+1$ on the ellipse $x^2+4y^2=16$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{35}"
} | {
"index": 32965,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The length of the curve given by the parametric equations \\(\\left\\{\\begin{array}{l}x=2 \\cos ^{2} \\theta \\\\ y=3 \\sin ^{2} \\theta\\end{array}\\right.\\) (where \\(\\theta\\) is the parameter) is?"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{13}"
} | {
"index": 32966,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The length of the escalator is 200 steps. When Petya walks down the escalator, he counts 50 steps. How many steps will he count if he runs twice as fast?"
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"index": 32967,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The length of the hypotenuse of a right-angled triangle is 2cm longer than one of its legs, and the other leg is 6cm long. Find the length of its hypotenuse."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 32968,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The length of the interval of solutions of the inequality $a \\le 2x + 3 \\le b$ is $10$. What is $b - a$?"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 32969,
"solution": "1. **Rewrite the inequality in terms of $x$:** \n Given the inequality $a \\le 2x + 3 \\le b$, we first isolate $x$. Subtract $3$ from all parts of the inequality:\n \\[\n a - 3 \\le 2x \\le b - 3\n \\]\n \n2. **Divide by $2$ to solve for $x$:** \n Next, divide the entir... |
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}"
} | {
"index": 32970,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The length of the longer side of rectangle \\(R'\\) is \\(15\\%\\) more than the length of a side of square \\(S'\\). The length of the shorter side of rectangle \\(R'\\) is \\(5\\%\\) less than the length of a side of square \\(S'\\). What is the ratio of the area of rectangle... | math | {
"style": "rule",
"ground_truth": "\\frac{10925}{10000}"
} | {
"index": 32971,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The length of the median to the hypotenuse of an isosceles, right triangle is $10$ units. What is the length of a leg of the triangle, in units? Express your answer in simplest radical form."
}
] | math | {
"style": "rule",
"ground_truth": "10\\sqrt{2}"
} | {
"index": 32972,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The length of the segment between the points $(2a, a-4)$ and $(4, -1)$ is $2\\sqrt{10}$ units. What is the product of all possible values for $a$?"
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 32973,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The length of the shortest trip from $A$ to $B$ along the edges of the cube shown is the length of 3 edges. How many different 3-edge trips are there from $A$ to $B$?\n\n[asy]\n\nsize(4cm,4cm);\n\npair a1, b1, c1, d1;\n\na1=(1,1);\nb1=(0,1);\nc1=(1.6,1.4);\nd1=(1,0);\n\npair e1... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 32974,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The length of the shortest trip from $A$ to $B$ along the edges of the cube shown is the length of 4 edges. How many different 4-edge trips are there from $A$ to $B$?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 32975,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The lengths of a pair of corresponding medians of two similar triangles are 10cm and 4cm, respectively, and the sum of their perimeters is 140cm. The perimeters of these two triangles are , and the ratio of their areas is ."
}
] | math | {
"style": "rule",
"ground_truth": "25:4"
} | {
"index": 32976,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The lengths of the diagonals of a rhombus and the length of its side form a geometric progression. Find the sine of the angle between the side of the rhombus and its longer diagonal, given that it is greater than \\( \\frac{1}{2} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{\\frac{\\sqrt{17}-1}{8}}"
} | {
"index": 32977,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The lengths of the edges of a rectangular parallelepiped extending from one vertex are 8, 8, and 27. Divide the parallelepiped into four parts that can be assembled into a cube."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 32978,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The lengths of the edges of a regular tetrahedron \\(ABCD\\) are 1. \\(G\\) is the center of the base \\(ABC\\). Point \\(M\\) is on line segment \\(DG\\) such that \\(\\angle AMB = 90^\\circ\\). Find the length of \\(DM\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{6}"
} | {
"index": 32979,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The lengths of the sides of a non-degenerate triangle are $x$, 13 and 37 units. How many integer values of $x$ are possible?"
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 32980,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The lengths of the sides of a triangle are $\\sqrt{3}, \\sqrt{4}(=2), \\sqrt{5}$. \n\nIn what ratio does the altitude perpendicular to the middle side divide it?"
}
] | math | {
"style": "rule",
"ground_truth": "1:3"
} | {
"index": 32981,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The lengths of the sides of a triangle are 6 cm, 7 cm and 9 cm. In a similar triangle whose perimeter is 110 cm, what is the length of the longest side, in centimeters?"
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 32982,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The lengths of the sides of a triangle are consecutive integers, and the largest angle is twice the smallest angle. Find the cosine of the smallest angle."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 32983,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The lengths of the sides of a triangle are integers, and its area is also an integer.\nOne side is $21$ and the perimeter is $48$. The shortest side is:"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 32984,
"solution": "1. **Assign Variables to Sides and Perimeter**:\n Let the sides of the triangle be $a = 21$, $b$, and $c$. Given that the perimeter is $48$, we have:\n \\[\n a + b + c = 48\n \\]\n Substituting $a = 21$ into the equation, we get:\n \\[\n 21 + b + c = 48 \\implies b + c =... |
DeepScaleR | [
{
"role": "user",
"content": "The lengths of the sides of a triangle with positive area are $\\log_{10} 12$, $\\log_{10} 75$, and $\\log_{10} n$, where $n$ is a positive integer. Find the number of possible values for $n$."
}
] | math | {
"style": "rule",
"ground_truth": "893"
} | {
"index": 32985,
"solution": "By the Triangle Inequality and applying the well-known logarithmic property $\\log_{c} a + \\log_{c} b = \\log_{c} ab$, we have that\n$\\log_{10} 12 + \\log_{10} n > \\log_{10} 75$\n$\\log_{10} 12n > \\log_{10} 75$\n$12n > 75$\n$n > \\frac{75}{12} = \\frac{25}{4} = 6.25$\n Also,\n$\\l... |
DeepScaleR | [
{
"role": "user",
"content": "The lengths of the sides of a triangle with positive area are $\\log_{10}12$, $\\log_{10}75$, and $\\log_{10}n$, where $n$ is a positive integer. Find the number of possible values for $n$."
}
] | math | {
"style": "rule",
"ground_truth": "893"
} | {
"index": 32986,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The lengths of the sides of a triangle with positive area are $\\log_{2}9$, $\\log_{2}50$, and $\\log_{2}n$, where $n$ is a positive integer. Find the number of possible values for $n$."
}
] | math | {
"style": "rule",
"ground_truth": "445"
} | {
"index": 32987,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The lengths of the six edges of a tetrahedron $ABCD$ are $7, 13, 18, 27, 36, 41$, and $AB = 41$. What is the length of $CD$?"
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 32988,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The lengths of the six edges of a tetrahedron \\(ABCD\\) are 7, 13, 18, 27, 36, and 41, respectively. If \\(AB = 41\\), what is the length of \\(CD\\)?"
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 32989,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The lengths of the three altitudes of a triangle are in the ratio \\(3 : 4 : 6\\). If the radius of the inscribed circle of the triangle is 1, find the radius of the circumcircle of the triangle."
}
] | math | {
"style": "rule",
"ground_truth": "16/5"
} | {
"index": 32990,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The lengths of the three sides of $\\triangle ABC$ are 5, 7, and 8, respectively. The radius of its circumcircle is ______, and the radius of its incircle is ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 32991,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The lengths of the three sides of a triangle are $7$, $x+4$ and $2x+1$. The perimeter of the triangle is 36. What is the length of the longest side of the triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 32992,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The lengths of the three sides of a triangle are 11, 15, and \\( k \\) \\((k \\in Z^{+})\\). The number of values of \\( k \\) that make the triangle an obtuse triangle is:"
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 32993,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The lengths of the three sides of a triangle are \\( 10 \\), \\( y+5 \\), and \\( 3y-2 \\). The perimeter of the triangle is \\( 50 \\). What is the length of the longest side of the triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "25.75"
} | {
"index": 32994,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The lengths of two line segments are $a$ units and $b$ units respectively. Then the correct relation between them is:"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{a+b}{2} \\geq \\sqrt{ab}"
} | {
"index": 32995,
"solution": "To determine the correct relationship between $a$ and $b$, we can use the Arithmetic Mean-Geometric Mean Inequality (AM-GM Inequality). The AM-GM Inequality states that for any non-negative real numbers $x_1, x_2, \\ldots, x_n$, the following inequality holds:\n\\[\n\\frac{x_1 + x_2 +... |
DeepScaleR | [
{
"role": "user",
"content": "The lengths of two sides of a triangle are 33 units and 42 units. The third side also has an integral length. What is the least possible number of units in the perimeter of the triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "85"
} | {
"index": 32996,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The letter T is formed by placing a $2\\:\\text{inch} \\times 6\\:\\text{inch}$ rectangle vertically and a $2\\:\\text{inch} \\times 4\\:\\text{inch}$ rectangle horizontally across the top center of the vertical rectangle. What is the perimeter of the T, in inches?"
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 32997,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The letter T is formed by placing a $2\\:\\text{inch}\\!\\times\\!6\\:\\text{inch}$ rectangle vertically and a $3\\:\\text{inch}\\!\\times\\!2\\:\\text{inch}$ rectangle horizontally on top of the vertical rectangle at its middle, as shown. What is the perimeter of this T, in in... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 32998,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The letter T is formed by placing two $2\\:\\text{inch}\\!\\times\\!4\\:\\text{inch}$ rectangles next to each other, as shown. What is the perimeter of the T, in inches? [asy]\ndraw((1,0)--(3,0)--(3,4)--(4,4)--(4,6)--(0,6)--(0,4)--(1,4)--cycle);\n[/asy]"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 32999,
"solution": ""
} |
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