data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "The minimum possible sum of the three dimensions of a rectangular box with a volume of 3003 in^3 is what value?"
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 33200,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The minimum sum of the distances from a point in space to the vertices of a regular tetrahedron with side length 1 is:"
}
] | math | {
"style": "rule",
"ground_truth": "$\\sqrt{6}$"
} | {
"index": 33201,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The minimum value of $\\frac{b^{2}+1}{\\sqrt{3}a}$ is what occurs when the eccentricity of the hyperbola $\\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}=1$ is 2."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {4 \\sqrt {3}}{3}"
} | {
"index": 33202,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The minimum value of \\\\(f(x)=\\sin x+\\cos x-\\sin x\\cos x\\\\) is"
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac{1}{2}- \\sqrt{2}"
} | {
"index": 33203,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The minimum value of the function $f(x) = \\cos^2 x + \\sin x$ is given by $\\frac{-1 + \\sqrt{2}}{2}$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 33204,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The minimum value of the function $y = \\sin 2 \\cos 2x$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac{1}{2}"
} | {
"index": 33205,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The minimum value of the function $y=|x-1|$ is 0, the minimum value of the function $y=|x-1|+|x-2|$ is 1, and the minimum value of the function $y=|x-1|+|x-2|+|x-3|$ is 2. What is the minimum value of the function $y=|x-1|+|x-2|+\\ldots+|x-10|$?"
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 33206,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The minimum value of the function \\( f(x)=(x+1)(x+2)(x+3)(x+4)+35 \\) is:"
}
] | math | {
"style": "rule",
"ground_truth": "34"
} | {
"index": 33207,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The minimum value of the function \\( y = \\sin^4{x} + \\cos^4{x} + \\sec^4{x} + \\csc^4{x} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{17}{2}"
} | {
"index": 33208,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The minimum value of the function \\( y = |\\cos x| + |\\cos 2x| \\) (for \\( x \\in \\mathbf{R} \\)) is ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"index": 33209,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The minimum value of the polynomial $x^2 + y^2 - 6x + 8y + 7$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "-18"
} | {
"index": 33210,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The minimum value of the quotient of a (base ten) number of three different non-zero digits divided by the sum of its digits is"
}
] | math | {
"style": "rule",
"ground_truth": "10.5"
} | {
"index": 33211,
"solution": "1. **Expression Setup**: We start by expressing the number with three different non-zero digits $a$, $b$, and $c$ as $100a + 10b + c$. The sum of its digits is $a + b + c$. Thus, the quotient of the number by the sum of its digits is:\n \\[\n \\frac{100a + 10b + c}{a + b + c}\n ... |
DeepScaleR | [
{
"role": "user",
"content": "The minimum value of the sum of the distances from a point in space to the vertices of a regular tetrahedron with edge length 1 is:"
}
] | math | {
"style": "rule",
"ground_truth": "$\\sqrt{6}$"
} | {
"index": 33212,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The minute hand on a clock points at the 12. After rotating $120^{\\circ}$ clockwise, which number will it point at?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 33213,
"solution": "Since there are 12 equally spaced numbers and the total angle in a complete circle is $360^{\\circ}$, then the angle between two consecutive numbers is $360^{\\circ} \\div 12=30^{\\circ}$. To rotate $120^{\\circ}$, the minute hand must move by $120^{\\circ} \\div 30^{\\circ}=4$ number... |
DeepScaleR | [
{
"role": "user",
"content": "The mode and median of the data $9.30$, $9.05$, $9.10$, $9.40$, $9.20$, $9.10$ are ______ and ______, respectively."
}
] | math | {
"style": "rule",
"ground_truth": "9.15"
} | {
"index": 33214,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The moisture content of freshly cut grass is $60\\%$, and the moisture content of hay is $15\\%$. How much hay will be obtained from one ton of freshly cut grass?"
}
] | math | {
"style": "rule",
"ground_truth": "470.588"
} | {
"index": 33215,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The moisture content of freshly cut grass is $70\\%$, while the moisture content of hay is $16\\%. How much grass needs to be cut to obtain 1 ton of hay?"
}
] | math | {
"style": "rule",
"ground_truth": "2800"
} | {
"index": 33216,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The monkey has 100 bananas and its home is 50 meters away. The monkey can carry at most 50 bananas at a time and eats one banana for every meter walked. Calculate the maximum number of bananas the monkey can bring home."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 33217,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The movie \"Thirty Thousand Miles in Chang'an\" allows the audience to experience the unique charm of Tang poetry that has been passed down for thousands of years and the beauty of traditional Chinese culture. In the film, Li Bai was born in the year $701$ AD. If we represent t... | math | {
"style": "rule",
"ground_truth": "-551"
} | {
"index": 33218,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The national security agency's wiretap recorded a conversation between two spies and found that on a 30-minute tape, starting from the 30-second mark, there was a 10-second segment of conversation containing information about the spies' criminal activities. Later, it was discov... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{45}"
} | {
"index": 33219,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The national student loan is a credit loan subsidized by the finance department, aimed at helping college students from families with financial difficulties to pay for tuition, accommodation, and living expenses during their study period in college. The total amount applied for... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"index": 33220,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The natural domain of the function \\( y = f\\left(\\frac{2x}{3x^2 + 1}\\right) \\) is \\(\\left[\\frac{1}{4}, a\\right]\\). Find the value of \\( a \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"index": 33221,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The natural number \\( a \\) is divisible by 21 and has 105 distinct divisors, including 1 and \\( a \\). Find the smallest such \\( a \\)."
}
] | math | {
"style": "rule",
"ground_truth": "254016"
} | {
"index": 33222,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The natural number \\( a \\) is divisible by 55 and has 117 distinct divisors, including 1 and \\( a \\). Find the smallest such \\( a \\)."
}
] | math | {
"style": "rule",
"ground_truth": "12390400"
} | {
"index": 33223,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The natural number \\(a\\) is divisible by 35 and has 75 different divisors, including 1 and \\(a\\). Find the smallest such \\(a\\)."
}
] | math | {
"style": "rule",
"ground_truth": "490000"
} | {
"index": 33224,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The natural numbers from 1951 to 1982 are arranged in a certain order one after another. A computer reads two consecutive numbers from left to right (i.e., the 1st and 2nd, the 2nd and 3rd, etc.) until the last two numbers. If the larger number is on the left, the computer swap... | math | {
"style": "rule",
"ground_truth": "1982"
} | {
"index": 33225,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The necessary and sufficient condition for the graph of the function $y=f(x)$ to be symmetric with respect to the origin is that the function $y=f(x)$ is an odd function. This can be extended as follows: The necessary and sufficient condition for the graph of the function $y=f(... | math | {
"style": "rule",
"ground_truth": "-8086"
} | {
"index": 33226,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The necessary and sufficient condition for the lines $ax+2y+1=0$ and $3x+(a-1)y+1=0$ to be parallel is \"$a=$ ______\"."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 33227,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The negation of the proposition \"For all pairs of real numbers $a,b$, if $a=0$, then $ab=0$\" is: There are real numbers $a,b$ such that"
}
] | math | {
"style": "rule",
"ground_truth": "$a=0$ and $ab \\ne 0$"
} | {
"index": 33228,
"solution": "To find the negation of the given proposition, we start by analyzing the original statement and then apply the rules of logical negation.\n\n**Original Proposition:**\n\"For all pairs of real numbers $a,b$, if $a=0$, then $ab=0$.\"\n\nThis statement can be written in logical form as:\... |
DeepScaleR | [
{
"role": "user",
"content": "The net change in the population over these four years is a 20% increase, then a 30% decrease, then a 20% increase, and finally a 30% decrease. Calculate the net change in the population over these four years."
}
] | math | {
"style": "rule",
"ground_truth": "-29"
} | {
"index": 33229,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The new PUMaC tournament hosts $2020$ students, numbered by the following set of labels $1, 2, . . . , 2020$ . The students are initially divided up into $20$ groups of $101$ , with each division into groups equally likely. In each of the groups, the contestant with the l... | math | {
"style": "rule",
"ground_truth": "2123"
} | {
"index": 33230,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The new individual income tax law has been implemented since January 1, 2019. According to the \"Individual Income Tax Law of the People's Republic of China,\" it is known that the part of the actual wages and salaries (after deducting special, additional special, and other leg... | math | {
"style": "rule",
"ground_truth": "9720"
} | {
"index": 33231,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The nine delegates to the Economic Cooperation Conference include $2$ officials from Mexico, $3$ officials from Canada, and $4$ officials from the United States. During the opening session, three of the delegates fall asleep. Assuming that the three sleepers were determined ran... | math | {
"style": "rule",
"ground_truth": "139"
} | {
"index": 33232,
"solution": "Like in the solution above, there are $84$ ways to pick $3$ delegates. We can use casework to find the probability that there aren't exactly $2$ sleepers from a county, then subtract from $1$.\nIf no country has at least $2$ delegates sleeping, then every country must have $1$ delegat... |
DeepScaleR | [
{
"role": "user",
"content": "The nine horizontal and nine vertical lines on an $8\\times8$ checkerboard form $r$ rectangles, of which $s$ are squares. The number $s/r$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$"
}
] | math | {
"style": "rule",
"ground_truth": "125"
} | {
"index": 33233,
"solution": "To determine the two horizontal sides of a rectangle, we have to pick two of the horizontal lines of the checkerboard, or ${9\\choose 2} = 36$. Similarly, there are ${9\\choose 2}$ ways to pick the vertical sides, giving us $r = 1296$ rectangles.\n For $s$, there are $8^2$ unit square... |
DeepScaleR | [
{
"role": "user",
"content": "The nine points of this grid are equally spaced horizontally and vertically. The distance between two neighboring points is 1 unit. What is the area, in square units, of the region where the two triangles overlap?\n\n[asy]\nsize(80);\ndot((0,0)); dot((0,1));dot((0,2));dot((1,... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 33234,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The nine squares in the table shown are to be filled so that every row and every column contains each of the numbers $1,2,3$. Then $A+B=$ \\begin{tabular}{|c|c|c|}\\hline 1 & &\\\\ \\hline & 2 & A\\\\ \\hline & & B\\\\ \\hline\\end{tabular}"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 33235,
"solution": "1. **Fill in the first row**: We start with the given number in the top left corner, which is $1$. Since each row and column must contain each of the numbers $1, 2, 3$, the middle cell in the top row cannot be $1$. It also cannot be $2$ because there is already a $2$ in the middle col... |
DeepScaleR | [
{
"role": "user",
"content": "The non-negative difference between two numbers \\(a\\) and \\(b\\) is \\(a-b\\) or \\(b-a\\), whichever is greater than or equal to 0. For example, the non-negative difference between 24 and 64 is 40. In the sequence \\(88, 24, 64, 40, 24, \\ldots\\), each number after the sec... | math | {
"style": "rule",
"ground_truth": "760"
} | {
"index": 33236,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The non-negative numbers \\(a, b, c\\) sum up to 1. Find the maximum possible value of the expression\n\n$$\n(a + 3b + 5c) \\cdot \\left(a + \\frac{b}{3} + \\frac{c}{5}\\right)\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{5}"
} | {
"index": 33237,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The nonzero roots of the equation $x^2 + 6x + k = 0$ are in the ratio $2:1$. What is the value of $k$?"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 33238,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The noon temperatures for seven consecutive days were $80^{\\circ}$, $79^{\\circ}$, $81^{\\circ}$, $85^{\\circ}$, $87^{\\circ}$, $89^{\\circ}$, and $87^{\\circ}$ Fahrenheit. What is the mean noon temperature, in degrees Fahrenheit, for the week?"
}
] | math | {
"style": "rule",
"ground_truth": "84"
} | {
"index": 33239,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The noon temperatures for ten consecutive days were $78^{\\circ}$, $80^{\\circ}$, $82^{\\circ}$, $85^{\\circ}$, $88^{\\circ}$, $90^{\\circ}$, $92^{\\circ}$, $95^{\\circ}$, $97^{\\circ}$, and $95^{\\circ}$ Fahrenheit. The increase in temperature over the weekend days (days 6 to ... | math | {
"style": "rule",
"ground_truth": "88.2"
} | {
"index": 33240,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The notation $[x]$ stands for the greatest integer that is less than or equal to $x$. Calculate $[-1.2]$."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 33241,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The novel takes 630 minutes to read aloud. The disc can hold 80 minutes of reading with at most 4 minutes of unused space. Calculate the number of minutes of reading each disc will contain."
}
] | math | {
"style": "rule",
"ground_truth": "70"
} | {
"index": 33242,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number\n$\\frac 2{\\log_4{2000^6}} + \\frac 3{\\log_5{2000^6}}$\n can be written as $\\frac mn$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 33243,
"solution": "$\\frac 2{\\log_4{2000^6}} + \\frac 3{\\log_5{2000^6}}$\n$=\\frac{\\log_4{16}}{\\log_4{2000^6}}+\\frac{\\log_5{125}}{\\log_5{2000^6}}$\n$=\\frac{\\log{16}}{\\log{2000^6}}+\\frac{\\log{125}}{\\log{2000^6}}$\n$=\\frac{\\log{2000}}{\\log{2000^6}}$\n$=\\frac{\\log{2000}}{6\\log{2000}}$\n$... |
DeepScaleR | [
{
"role": "user",
"content": "The number\n\\[\\text{cis } 75^\\circ + \\text{cis } 83^\\circ + \\text{cis } 91^\\circ + \\dots + \\text{cis } 147^\\circ\\]is expressed in the form $r \\, \\text{cis } \\theta$, where $r > 0$ and $0^\\circ \\le \\theta < 360^\\circ$. Find $\\theta$ in degrees."
}
] | math | {
"style": "rule",
"ground_truth": "111^\\circ"
} | {
"index": 33244,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number\n\\[e^{7\\pi i/60} + e^{17\\pi i/60} + e^{27 \\pi i/60} + e^{37\\pi i /60} + e^{47 \\pi i /60}\\]is expressed in the form $r e^{i \\theta}$, where $0 \\le \\theta < 2\\pi$. Find $\\theta$."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{9\\pi}{20}"
} | {
"index": 33245,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number $15$ is written on a blackboard. A move consists of erasing the number $x$ and replacing it with $x+y$ where $y$ is a randomly chosen number between $1$ and $5$ (inclusive). The game ends when the number on the blackboard exceeds $51$ . Which number is m... | math | {
"style": "rule",
"ground_truth": "54"
} | {
"index": 33246,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number $2021$ leaves a remainder of $11$ when divided by a positive integer. Find the smallest such integer."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 33247,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number $2^{1997}$ has $m$ decimal digits, while the number $5^{1997}$ has $n$ digits. Evaluate $m+n$ ."
}
] | math | {
"style": "rule",
"ground_truth": "1998"
} | {
"index": 33248,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number $$316990099009901=\\frac{32016000000000001}{101}$$ is the product of two distinct prime numbers. Compute the smaller of these two primes."
}
] | math | {
"style": "rule",
"ground_truth": "4002001"
} | {
"index": 33249,
"solution": "Let $x=2000$, so the numerator is $$x^{5}+x^{4}+1=\\left(x^{2}+x+1\\right)\\left(x^{3}-x+1\\right)$$ (This latter factorization can be noted by the fact that plugging in $\\omega$ or $\\omega^{2}$ into $x^{5}+x^{4}+1$ gives 0 .) Then $x^{2}+x+1=4002001$ divides the numerator. However,... |
DeepScaleR | [
{
"role": "user",
"content": "The number $(2^{48}-1)$ is exactly divisible by two numbers between $60$ and $70$. These numbers are"
}
] | math | {
"style": "rule",
"ground_truth": "63,65"
} | {
"index": 33250,
"solution": "To solve the problem, we need to find the factors of $2^{48} - 1$ that lie between $60$ and $70$. We start by factoring $2^{48} - 1$ using the difference of squares repeatedly.\n\n1. **Factorization using difference of squares:**\n \\[\n 2^{48} - 1 = (2^{24} - 1)(2^{24} + 1)\n \... |
DeepScaleR | [
{
"role": "user",
"content": "The number $0.324375$ can be written as a fraction $\\frac{a}{b}$ for positive integers $a$ and $b$. When this fraction is in simplest terms, what is $a+b$?"
}
] | math | {
"style": "rule",
"ground_truth": "2119"
} | {
"index": 33251,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number $0.428125$ can be written as a fraction $\\frac{a}{b}$ for positive integers $a$ and $b$. When this fraction is in simplest terms, what is $a+b$?"
}
] | math | {
"style": "rule",
"ground_truth": "457"
} | {
"index": 33252,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number $10!$ ($10$ is written in base $10$), when written in the base $12$ system, ends with exactly $k$ zeros. The value of $k$ is"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 33253,
"solution": "1. **Understanding the problem**: We need to find how many zeros are at the end of $10!$ when it is expressed in base $12$. This means we need to determine the highest power of $12$ that divides $10!$.\n\n2. **Factorizing $12$**: Since $12 = 2^2 \\cdot 3$, we need to find the number o... |
DeepScaleR | [
{
"role": "user",
"content": "The number $1000!$ has a long tail of zeroes. How many zeroes are there? (Reminder: The number $n!$ is the product of the integers from 1 to $n$. For example, $5!=5\\cdot 4\\cdot3\\cdot2\\cdot 1= 120$.)"
}
] | math | {
"style": "rule",
"ground_truth": "249"
} | {
"index": 33254,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number $101$ is the smallest three-digit palindromic prime. What is the second-smallest one?"
}
] | math | {
"style": "rule",
"ground_truth": "131"
} | {
"index": 33255,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number $1025$ can be written as $23q + r$ where $q$ and $r$ are positive integers. What is the greatest possible value of $q - r$?"
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 33256,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number $121_b$, written in the integral base $b$, is the square of an integer, for"
}
] | math | {
"style": "rule",
"ground_truth": "$b > 2$"
} | {
"index": 33257,
"solution": "1. **Convert the base-b number to base-10**: The number $121_b$ in base $b$ can be expressed in base 10 as:\n \\[\n 1 \\cdot b^2 + 2 \\cdot b^1 + 1 \\cdot b^0 = b^2 + 2b + 1\n \\]\n\n2. **Factorize the expression**: The expression $b^2 + 2b + 1$ can be rewritten by recognizing i... |
DeepScaleR | [
{
"role": "user",
"content": "The number $17!$ has a certain number of positive integer divisors. What is the probability that one of them is odd?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{16}"
} | {
"index": 33258,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number $18!=6,402,373,705,728,000$ has many positive integer divisors. One of them is chosen at random. What is the probability that it is odd?\nA) $\\frac{1}{16}$\nB) $\\frac{1}{18}$\nC) $\\frac{1}{15}$\nD) $\\frac{1}{20}$\nE) $\\frac{1}{21}$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{16}"
} | {
"index": 33259,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number $2.29^{\\star \\star} N$ is an integer. Its representation in base $b$ is 777. Find the smallest positive integer $b$ such that $N$ is a perfect fourth power."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 33260,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number $2.5252525\\ldots$ can be written as a fraction.\nWhen reduced to lowest terms the sum of the numerator and denominator of this fraction is:"
}
] | math | {
"style": "rule",
"ground_truth": "349"
} | {
"index": 33261,
"solution": "\n#### Step 1: Represent the repeating decimal as a fraction\nLet $x = 2.5252525\\ldots$. We can express $x$ in terms of itself shifted by two decimal places:\n\\[ 100x = 252.5252525\\ldots \\]\n\n#### Step 2: Set up the equation\nSubtract the original $x$ from $100x$:\n\\[ 100x - x =... |
DeepScaleR | [
{
"role": "user",
"content": "The number $2013$ has the property that its units digit is the sum of its other digits, that is $2+0+1=3$. How many integers less than $2013$ but greater than $1000$ have this property?"
}
] | math | {
"style": "rule",
"ground_truth": "46"
} | {
"index": 33262,
"solution": "We are tasked with finding how many integers between 1000 and 2013 have the property that their units digit is the sum of the other digits. We will consider two cases based on the thousands digit, which can be either 1 or 2.\n\n#### Case 1: Thousands digit is 1\nThe number is of the f... |
DeepScaleR | [
{
"role": "user",
"content": "The number $2013$ is expressed in the form $2013 = \\frac {a_1!a_2!...a_m!}{b_1!b_2!...b_n!}$,where $a_1 \\ge a_2 \\ge \\cdots \\ge a_m$ and $b_1 \\ge b_2 \\ge \\cdots \\ge b_n$ are positive integers and $a_1 + b_1$ is as small as possible. What is $|a_1 - b_1|$?"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 33263,
"solution": "1. **Prime Factorization of 2013**: \n The prime factorization of $2013$ is $2013 = 3 \\times 11 \\times 61$.\n\n2. **Form of the Expression**:\n We need to express $2013$ in the form $\\frac{a_1!a_2!\\cdots a_m!}{b_1!b_2!\\cdots b_n!}$, where $a_1 \\ge a_2 \\ge \\cdots \\ge a_m$ ... |
DeepScaleR | [
{
"role": "user",
"content": "The number $2017$ is prime. Let $S = \\sum \\limits_{k=0}^{62} \\dbinom{2014}{k}$. What is the remainder when $S$ is divided by $2017?$\n$\\textbf{(A) }32\\qquad \\textbf{(B) }684\\qquad \\textbf{(C) }1024\\qquad \\textbf{(D) }1576\\qquad \\textbf{(E) }2016\\qquad$"
}
] | math | {
"style": "rule",
"ground_truth": "1024"
} | {
"index": 33264,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number $2022$ has the following property: it is a multiple of $6$ and the sum of its digits is $6$. Such positive integers are called \"auspicious numbers.\" Among all three-digit positive integers, the number of \"auspicious numbers\" is ____."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 33265,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number $2027$ is prime. Let $T = \\sum \\limits_{k=0}^{72} \\binom{2024}{k}$. What is the remainder when $T$ is divided by $2027$?"
}
] | math | {
"style": "rule",
"ground_truth": "1369"
} | {
"index": 33266,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number $21! = 51,090,942,171,709,440,000$ has over $60,000$ positive integer divisors. One of them is chosen at random. What is the probability that it is odd?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{19}"
} | {
"index": 33267,
"solution": "1. **Prime Factorization of $21!$:** \n To find the prime factorization of $21!$, we need to consider the contributions of each prime number up to 21. We count the number of times each prime divides any of the numbers from 1 to 21. This can be done using the formula for the highest... |
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?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{23}"
} | {
"index": 33268,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number $25^{64} \\cdot 64^{25}$ is the square of a positive integer $N$. In decimal representation, the sum of the digits of $N$ is"
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 33269,
"solution": "1. **Identify the expression for $N$:** \n Given that $N^2 = 25^{64} \\cdot 64^{25}$, we find $N$ by taking the square root:\n \\[\n N = \\sqrt{25^{64} \\cdot 64^{25}}\n \\]\n\n2. **Simplify the expression using properties of exponents:** \n Recognize that $25 = 5^2$ and $... |
DeepScaleR | [
{
"role": "user",
"content": "The number $27,000,001$ has exactly four prime factors. Find their sum."
}
] | math | {
"style": "rule",
"ground_truth": "652"
} | {
"index": 33270,
"solution": "First, we factor $$\\begin{aligned} 27 x^{6}+1 & =\\left(3 x^{2}\\right)^{3}+1 \\\\ & =\\left(3 x^{2}+1\\right)\\left(9 x^{4}-3 x^{2}+1\\right) \\\\ & =\\left(3 x^{2}+1\\right)\\left(\\left(9 x^{4}+6 x^{2}+1\\right)-9 x^{2}\\right) \\\\ & =\\left(3 x^{2}+1\\right)\\left(\\left(3 x^{2}... |
DeepScaleR | [
{
"role": "user",
"content": "The number $2^{1993}+3^{1993}$ is a multiple of $5.$ What is the units digit of the quotient $\\frac{2^{1993}+3^{1993}}{5}?$"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 33271,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number $5\\,41G\\,507\\,2H6$ is divisible by $40.$ Determine the sum of all distinct possible values of the product $GH.$"
}
] | math | {
"style": "rule",
"ground_truth": "225"
} | {
"index": 33272,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number $5\\,41G\\,507\\,2H6$ is divisible by $72.$ If $G$ and $H$ each represent a single digit, what is the sum of all distinct possible values of the product $GH?$ (Count each possible value of $GH$ only once, even if it results from multiple $G,$ $H$ pairs.)"
}
] | math | {
"style": "rule",
"ground_truth": "59"
} | {
"index": 33273,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$. How many pairs of integers $(m,n)$ are there such that $1\\leq m\\leq 2012$ and $5^n<2^m<2^{m+2}<5^{n+1}$?"
}
] | math | {
"style": "rule",
"ground_truth": "279"
} | {
"index": 33274,
"solution": "To solve this problem, we need to understand the relationship between the powers of $5$ and $2$ and how they are distributed between $5^n$ and $5^{n+1}$.\n\n1. **Understanding the relationship between $5^n$ and $2^m$:**\n We know that $5^{867}$ is between $2^{2013}$ and $2^{2014}$. ... |
DeepScaleR | [
{
"role": "user",
"content": "The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$. How many pairs of integers $(m,n)$ are there such that $1\\leq m\\leq 2012$ and $5^n<2^m<2^{m+2}<5^{n+1}$?"
}
] | math | {
"style": "rule",
"ground_truth": "279"
} | {
"index": 33275,
"solution": "To solve this problem, we need to determine how many pairs of integers $(m, n)$ satisfy the inequalities $5^n < 2^m < 2^{m+2} < 5^{n+1}$ for $1 \\leq m \\leq 2012$.\n\n1. **Understanding the inequalities:**\n - $5^n < 2^m$ implies that $n \\log 5 < m \\log 2$.\n - $2^{m+2} < 5^{n+... |
DeepScaleR | [
{
"role": "user",
"content": "The number $6545$ can be written as a product of a pair of positive two-digit numbers. What is the sum of this pair of numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "162"
} | {
"index": 33276,
"solution": "To solve the problem, we first need to find the prime factorization of the number $6545$.\n\n1. **Prime Factorization of $6545$:**\n \\[\n 6545 = 5 \\times 1309\n \\]\n We need to check if $1309$ is prime or can be further factored. Testing divisibility by small primes:\n \\... |
DeepScaleR | [
{
"role": "user",
"content": "The number $695$ is to be written with a factorial base of numeration, that is, $695=a_1+a_2\\times2!+a_3\\times3!+ \\ldots a_n \\times n!$ where $a_1, a_2, a_3 ... a_n$ are integers such that $0 \\le a_k \\le k,$ and $n!$ means $n(n-1)(n-2)...2 \\times 1$. Find $a_4$"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 33277,
"solution": "To solve this problem, we need to express the number 695 in a factorial base. This means we need to find coefficients $a_1, a_2, a_3, \\ldots, a_n$ such that:\n\\[ 695 = a_1 + a_2 \\times 2! + a_3 \\times 3! + \\ldots + a_n \\times n! \\]\nwhere $0 \\leq a_k \\leq k$ for each $k$.\n\n... |
DeepScaleR | [
{
"role": "user",
"content": "The number $7.21\\times 10^{11}$ has how many digits in the original number."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 33278,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number $839$ can be written as $19q+r$ where $q$ and $r$ are positive integers. What is the greatest possible value of $q-r$?"
}
] | math | {
"style": "rule",
"ground_truth": "41"
} | {
"index": 33279,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number $947$ can be written as $23q + r$ where $q$ and $r$ are positive integers. What is the greatest possible value of $q - r$?"
}
] | math | {
"style": "rule",
"ground_truth": "37"
} | {
"index": 33280,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number $989 \\cdot 1001 \\cdot 1007+320$ can be written as the product of three distinct primes $p, q, r$ with $p<q<r$. Find $(p, q, r)$."
}
] | math | {
"style": "rule",
"ground_truth": "(991,997,1009)"
} | {
"index": 33281,
"solution": "Let $f(x)=x(x-12)(x+6)+320=x^{3}-6 x^{2}-72 x+320$, so that $f(1001)=989 \\cdot 1001 \\cdot 1007+320$. But $f(4)=4(-8)(10)+320=0$, so $f(x)=(x-4)\\left(x^{2}-2 x-80\\right)=(x-4)(x-10)(x+8)$ Thus $f(1001)=991 \\cdot 997 \\cdot 1009$, as desired."
} |
DeepScaleR | [
{
"role": "user",
"content": "The number $\\overline{x y z t}$ is a perfect square such that the number $\\overline{t z y x}$ is also a perfect square, and the quotient of the numbers $\\overline{x y z t}$ and $\\overline{t z y x}$ is also a perfect square. Determine the number $\\overline{x y z t}$. (The o... | math | {
"style": "rule",
"ground_truth": "9801"
} | {
"index": 33282,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number $\\sqrt{104\\sqrt{6}+468\\sqrt{10}+144\\sqrt{15}+2006}$ can be written as $a\\sqrt{2}+b\\sqrt{3}+c\\sqrt{5},$ where $a, b,$ and $c$ are positive integers. Find $abc$."
}
] | math | {
"style": "rule",
"ground_truth": "936"
} | {
"index": 33283,
"solution": "We begin by equating the two expressions:\n\\[a\\sqrt{2}+b\\sqrt{3}+c\\sqrt{5} = \\sqrt{104\\sqrt{6}+468\\sqrt{10}+144\\sqrt{15}+2006}\\]\n Squaring both sides yields:\n\\[2ab\\sqrt{6} + 2ac\\sqrt{10} + 2bc\\sqrt{15} + 2a^2 + 3b^2 + 5c^2 = 104\\sqrt{6}+468\\sqrt{10}+144\\sqrt{15}+2006... |
DeepScaleR | [
{
"role": "user",
"content": "The number $a+\\sqrt{b}$ and its radical conjugate have a sum of $-4$ and a product of $1$. Find $a+b$."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 33284,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number $a=\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, has the property that the sum of all real numbers $x$ satisfying\n\\[\\lfloor x \\rfloor \\cdot \\{x\\} = a \\cdot x^2\\]is $420$, where $\\lfloor x \\rfloor$ denotes the greatest integer les... | math | {
"style": "rule",
"ground_truth": "929"
} | {
"index": 33285,
"solution": "1. **Define Variables**:\n Let $w = \\lfloor x \\rfloor$ and $f = \\{x\\}$ denote the whole part and the fractional part of $x$, respectively. Thus, $x = w + f$ where $0 \\leq f < 1$.\n\n2. **Rewrite the Equation**:\n The given equation is:\n \\[\n \\lfloor x \\rfloor \\cdot \... |
DeepScaleR | [
{
"role": "user",
"content": "The number $m$ is a prime number between 30 and 50. If you divide $m$ by 12, the remainder is 7. What is the value of $m$?"
}
] | math | {
"style": "rule",
"ground_truth": "43"
} | {
"index": 33286,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number $m$ is a three-digit positive integer and is the product of the three distinct prime factors $x$, $y$ and $10x+y$, where $x$ and $y$ are each less than 10. What is the largest possible value of $m$?"
}
] | math | {
"style": "rule",
"ground_truth": "795"
} | {
"index": 33287,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number $n$ can be written in base $14$ as $\\underline{a}\\text{ }\\underline{b}\\text{ }\\underline{c}$, can be written in base $15$ as $\\underline{a}\\text{ }\\underline{c}\\text{ }\\underline{b}$, and can be written in base $6$ as $\\underline{a}\\text{ }\\underline{c}\... | math | {
"style": "rule",
"ground_truth": "925"
} | {
"index": 33288,
"solution": "We have these equations: $196a+14b+c=225a+15c+b=222a+37c$. Taking the last two we get $3a+b=22c$. Because $c \\neq 0$ otherwise $a \\ngtr 0$, and $a \\leq 5$, $c=1$.\n Then we know $3a+b=22$. Taking the first two equations we see that $29a+14c=13b$. Combining the two gives $a=4, b=10,... |
DeepScaleR | [
{
"role": "user",
"content": "The number $n$ is a four-digit positive integer and is the product of three distinct prime factors $x$, $y$ and $10y+x$, where $x$ and $y$ are each less than 10. What is the largest possible value of $n$?"
}
] | math | {
"style": "rule",
"ground_truth": "1533"
} | {
"index": 33289,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number $n$ is a prime number between 20 and 30. If you divide $n$ by 8, the remainder is 5. What is the value of $n$?"
}
] | math | {
"style": "rule",
"ground_truth": "29"
} | {
"index": 33290,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number $n$ is a three-digit integer and is the product of two distinct prime factors $x$ and $10x+y$, where $x$ and $y$ are each less than 10, with no restrictions on $y$ being prime. What is the largest possible value of $n$?"
}
] | math | {
"style": "rule",
"ground_truth": "553"
} | {
"index": 33291,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number $r$ can be expressed as a four-place decimal $0.abcd,$ where $a, b, c,$ and $d$ represent digits, any of which could be zero. It is desired to approximate $r$ by a fraction whose numerator is 1 or 2 and whose denominator is an integer. The closest such fraction to $r... | math | {
"style": "rule",
"ground_truth": "417"
} | {
"index": 33292,
"solution": "The nearest fractions to $\\frac 27$ with numerator $1$ are $\\frac 13, \\frac 14$; and with numerator $2$ are $\\frac 26, \\frac 28 = \\frac 13, \\frac 14$ anyway. For $\\frac 27$ to be the best approximation for $r$, the decimal must be closer to $\\frac 27 \\approx .28571$ than to ... |
DeepScaleR | [
{
"role": "user",
"content": "The number $x$ satisfies $5x^2 + 4 = 3x + 9$. Find the value of $(10x - 3)^2$."
}
] | math | {
"style": "rule",
"ground_truth": "109"
} | {
"index": 33293,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number 119 has the following properties:\n\n(a) Division by 2 leaves a remainder of 1;\n\n(b) Division by 3 leaves a remainder of 2;\n\n(c) Division by 4 leaves a remainder of 3;\n\n(d) Division by 5 leaves a remainder of 4;\n\n(e) Division by 6 leaves a remainder of 5.\n\n... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"index": 33294,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number 119 has the following property:\n\n- Division by 2 leaves a remainder of 1;\n- Division by 3 leaves a remainder of 2;\n- Division by 4 leaves a remainder of 3;\n- Division by 5 leaves a remainder of 4;\n- Division by 6 leaves a remainder of 5.\n\nHow many positive in... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 33295,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number 121 is a palindrome, because it reads the same backwards as forward. How many integer palindromes are between 100 and 500?"
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 33296,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number 123456789 is written on the board. Two adjacent digits are selected from the number, if neither of them is 0, 1 is subtracted from each digit, and the selected digits are swapped (for example, from 123456789, one operation can result in 123436789). What is the smalle... | math | {
"style": "rule",
"ground_truth": "101010101"
} | {
"index": 33297,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number 2015 can be represented as a sum of consecutive integers in several ways, for example, $2015 = 1007 + 1008$ or $2015 = 401 + 402 + 403 + 404 + 405$. How many ways can this be done?"
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 33298,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The number 2015 is split into 12 terms, and then all the numbers that can be obtained by adding some of these terms (from one to nine) are listed. What is the minimum number of numbers that could have been listed?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 33299,
"solution": ""
} |
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