data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "When $\\sqrt[4]{2^6 \\cdot 3^5 \\cdot 5^2}$ is fully simplified, the result is $x\\sqrt[4]{y}$, where $x$ and $y$ are positive integers. What is $x+y$?"
}
] | math | {
"style": "rule",
"ground_truth": "306"
} | {
"index": 38600,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When $\\sqrt[4]{2^7\\cdot3^3}$ is fully simplified, the result is $a\\sqrt[4]{b}$, where $a$ and $b$ are positive integers. What is $a+b$?"
}
] | math | {
"style": "rule",
"ground_truth": "218"
} | {
"index": 38601,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When $\\sqrt[4]{5^9 \\cdot 7^2}$ is fully simplified, the result is $a\\sqrt[4]{b}$, where $a$ and $b$ are positive integers. What is $a+b$?"
}
] | math | {
"style": "rule",
"ground_truth": "270"
} | {
"index": 38602,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When $\\{a,0,-1\\} = \\{4,b,0\\}$, find the values of $a$ and $b$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 38603,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When $f(x) = ax^3 - 6x^2 + bx - 5$ is divided by $x - 1,$ the remainder is $-5.$ When $f(x)$ is divided by $x + 2,$ the remainder is $-53.$ Find the ordered pair $(a,b).$"
}
] | math | {
"style": "rule",
"ground_truth": "(2,4)"
} | {
"index": 38604,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When $k$ candies were distributed among seven people so that each person received the same number of candies and each person received as many candies as possible, there were 3 candies left over. If instead, $3 k$ candies were distributed among seven people in this way, then how... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 38605,
"solution": "Suppose that each of the 7 people received $q$ candies under the first distribution scheme. Then the people received a total of $7 q$ candies and 3 candies were left over. Since there were $k$ candies, then $k=7 q+3$. Multiplying both sides by 3, we obtain $3 k=21 q+9$. When $21 q+9$ ... |
DeepScaleR | [
{
"role": "user",
"content": "When $n$ is a positive integer, $n! = n \\times (n - 1) \\times \\ldots \\times 2 \\times 1$ is defined as the factorial of $n$ (for example, $10! = 10 \\times 9 \\times \\ldots \\times 2 \\times 1 = 3,628,800$). Then, how many zeros are there at the end of $2010!$?"
}
] | math | {
"style": "rule",
"ground_truth": "501"
} | {
"index": 38606,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When $n$ is an odd number, $\\frac{1+(-1)^{n}}{4}=$______; when $n$ is an even number, $\\frac{1+(-1)^{n}}{4}=\\_\\_\\_\\_\\_\\_$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 38607,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When $n$ is divided by 3, the remainder is 2. What is the remainder when $5n$ is divided by 3?"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 38608,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When $n$ is divided by 6, a remainder of 1 is given. What is the remainder when $n+2010$ is divided by 6?"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 38609,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When $n$ standard 6-sided dice are rolled, find the smallest possible value of $S$ such that the probability of obtaining a sum of 2000 is greater than zero and is the same as the probability of obtaining a sum of $S$."
}
] | math | {
"style": "rule",
"ground_truth": "338"
} | {
"index": 38610,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When $n$ standard 6-sided dice are rolled, the probability of obtaining a sum of 1994 is greater than zero and is the same as the probability of obtaining a sum of $S$. The smallest possible value of $S$ is"
}
] | math | {
"style": "rule",
"ground_truth": "337"
} | {
"index": 38611,
"solution": "1. **Define the transformation**: Let $d_i$ be the number on the $i$-th die. We define a transformation where each die's number is replaced by $d_i' = 7 - d_i$. This transformation is an involution, meaning applying it twice returns the original value: $7 - (7 - d_i) = d_i$.\n\n2. **E... |
DeepScaleR | [
{
"role": "user",
"content": "When $n$ standard 8-sided dice are rolled, the probability of obtaining a sum of 3000 is greater than zero and is the same as the probability of obtaining a sum of S. Find the smallest possible value of S."
}
] | math | {
"style": "rule",
"ground_truth": "375"
} | {
"index": 38612,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When $p(x) = Ax^5 + Bx^3 + Cx + 4$ is divided by $x - 3,$ the remainder is 11. Find the remainder when $p(x)$ is divided by $x + 3.$"
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 38613,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When $x = -1$, the value of the algebraic expression $2ax^{3} - 3bx + 8$ is $18$. Express the value of the algebraic expression $9b - 6a + 2$ in terms of the given information."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 38614,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When $x \\in \\left[-\\frac{\\pi}{3}, \\frac{\\pi}{3}\\right]$, find the minimum value of the function $f(x) = \\sqrt{2}\\sin \\frac{x}{4}\\cos \\frac{x}{4} + \\sqrt{6}\\cos^2 \\frac{x}{4} - \\frac{\\sqrt{6}}{2}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"index": 38615,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When $x$ is divided by each of $4$, $5$, and $6$, remainders of $3$, $4$, and $5$ (respectively) are obtained. What is the smallest possible positive integer value of $x$?"
}
] | math | {
"style": "rule",
"ground_truth": "59"
} | {
"index": 38616,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When $x=$____, the expressions $\\frac{x-1}{2}$ and $\\frac{x-2}{3}$ are opposite in sign."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{5}"
} | {
"index": 38617,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When $x=-2$, what is the value of $(x+1)^{3}$?"
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 38618,
"solution": "When $x=-2$, we have $(x+1)^{3}=(-2+1)^{3}=(-1)^{3}=-1$."
} |
DeepScaleR | [
{
"role": "user",
"content": "When $x=1$, the value of the expression $px^3+qx-10$ is 2006; when $x=-1$, find the value of the expression $px^3+qx-10$."
}
] | math | {
"style": "rule",
"ground_truth": "-2026"
} | {
"index": 38619,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When $x=3$ and $y=4$, what is the value of the expression $xy-x$?"
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 38620,
"solution": "When $x=3$ and $y=4$, we get $xy-x=3 \\times 4-3=12-3=9$. Alternatively, $xy-x=x(y-1)=3 \\times 3=9$."
} |
DeepScaleR | [
{
"role": "user",
"content": "When $x^5$, $x+\\frac{1}{x}$ and $1+\\frac{2}{x} + \\frac{3}{x^2}$ are multiplied, the product is a polynomial of degree."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 38621,
"solution": "1. **Identify the expressions and their multiplication:** We are given three expressions to multiply: $x^5$, $x + \\frac{1}{x}$, and $1 + \\frac{2}{x} + \\frac{3}{x^2}$.\n\n2. **Multiply the first two expressions:** \n \\[\n x^5 \\left(x + \\frac{1}{x}\\right) = x^5 \\cdot x + x^5... |
DeepScaleR | [
{
"role": "user",
"content": "When $x^9-x$ is factored as completely as possible into polynomials and monomials with integral coefficients, the number of factors is:"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 38622,
"solution": "1. **Factor out the greatest common factor**: \n The polynomial $x^9 - x$ can be rewritten by factoring out the common factor of $x$, giving:\n \\[\n x^9 - x = x(x^8 - 1)\n \\]\n\n2. **Factorize $x^8 - 1$ using difference of squares**:\n The expression $x^8 - 1$ can be facto... |
DeepScaleR | [
{
"role": "user",
"content": "When $x^{13}+1$ is divided by $x-1$, the remainder is:"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 38623,
"solution": "\nWe are given the polynomial \\(x^{13} + 1\\) and we need to find the remainder when it is divided by \\(x-1\\).\n\n**Solution 1: Using the Remainder Theorem**\n\nThe Remainder Theorem states that if a polynomial \\(f(x)\\) is divided by \\(x - a\\), the remainder is \\(f(a)\\). Here... |
DeepScaleR | [
{
"role": "user",
"content": "When $x^{2}$ was added to the quadratic polynomial $f(x)$, its minimum value increased by 1. When $x^{2}$ was subtracted from it, its minimum value decreased by 3. How will the minimum value of $f(x)$ change if $2x^{2}$ is added to it?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 38624,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When 100 is divided by a positive integer $x$, the remainder is 10. When 1000 is divided by $x$, what is the remainder?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 38625,
"solution": "When 100 is divided by a positive integer $x$, the remainder is 10. This means that $100-10=90$ is exactly divisible by $x$. It also means that $x$ is larger than 10, otherwise the remainder would be smaller than 10. Since 90 is exactly divisible by $x$, then $11 \\times 90=990$ is al... |
DeepScaleR | [
{
"role": "user",
"content": "When 1524 shi of rice is mixed with an unknown amount of wheat, and in a sample of 254 grains, 28 are wheat grains, calculate the estimated amount of wheat mixed with this batch of rice."
}
] | math | {
"style": "rule",
"ground_truth": "168"
} | {
"index": 38626,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When 2007 bars of soap are packed into \\( N \\) boxes, where \\( N \\) is a positive integer, there is a remainder of 5. How many possible values of \\( N \\) are there?"
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 38627,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When 5 integers are arranged in ascending order, the median is 6. If the only mode of this data set is 8, determine the possible maximum sum of these 5 integers."
}
] | math | {
"style": "rule",
"ground_truth": "31"
} | {
"index": 38628,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When 542 is multiplied by 3, what is the ones (units) digit of the result?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 38629,
"solution": "Since \\( 542 \\times 3 = 1626 \\), the ones digit of the result is 6."
} |
DeepScaleR | [
{
"role": "user",
"content": "When Alia was young, she could cycle 18 miles in 2 hours. Now, as an older adult, she walks 8 kilometers in 3 hours. Given that 1 mile is approximately 1.609 kilometers, determine how many minutes longer it takes for her to walk a kilometer now compared to when she was young."
... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 38630,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When Bendegúz boarded the 78-seat train car with his valid seat reservation, he was shocked to find that all seats were already taken. What had happened was that Dömötör boarded without a seat reservation. The other 77 passengers, including Elek, had purchased a seat reservatio... | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"index": 38631,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When Carlos divides his marbles into groups of six, seven, or eight, he has one marble left over. Assuming Carlos has more than one marble, what is the smallest possible number of marbles in Carlos's collection?"
}
] | math | {
"style": "rule",
"ground_truth": "169"
} | {
"index": 38632,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When Cheenu was a boy, he could run $15$ miles in $3$ hours and $30$ minutes. As an old man, he can now walk $10$ miles in $4$ hours. How many minutes longer does it take for him to walk a mile now compared to when he was a boy?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 38633,
"solution": "1. **Calculate the time Cheenu took per mile as a boy:**\n - Cheenu ran 15 miles in 3 hours and 30 minutes.\n - Convert hours to minutes: \\(3 \\text{ hours} = 3 \\times 60 = 180 \\text{ minutes}\\).\n - Add the extra 30 minutes: \\(180 \\text{ minutes} + 30 \\text{ minutes} = 2... |
DeepScaleR | [
{
"role": "user",
"content": "When Cheenu was a young man, he could run 20 miles in 4 hours. In his middle age, he could jog 15 miles in 3 hours and 45 minutes. Now, as an older man, he walks 12 miles in 5 hours. What is the time difference, in minutes, between his current walking speed and his running spee... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 38634,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When Claire divides her cupcakes into groups of 5, she has 3 remaining, and when she divides her cupcakes into groups of 7, she has 4 remaining. If Claire has fewer than 60 cupcakes, what is the sum of all possible quantities of cupcakes that she could have?"
}
] | math | {
"style": "rule",
"ground_truth": "71"
} | {
"index": 38635,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When Dave walks to school, he averages $90$ steps per minute, and each of his steps is $75$ cm long. It takes him $16$ minutes to get to school. His brother, Jack, going to the same school by the same route, averages $100$ steps per minute, but his steps are only $60$ cm long. ... | math | {
"style": "rule",
"ground_truth": "18 minutes"
} | {
"index": 38636,
"solution": "1. **Calculate Dave's walking speed**: \n Dave walks at a rate of $90$ steps per minute, with each step being $75$ cm long. Therefore, his walking speed is:\n \\[\n 90 \\text{ steps/min} \\times 75 \\text{ cm/step} = 6750 \\text{ cm/min}\n \\]\n\n2. **Calculate the distance t... |
DeepScaleR | [
{
"role": "user",
"content": "When Frederick was born, his grandparents gave him a gift of $\\$2000$, which was invested at an interest rate of $5\\%$ per year, compounded yearly. How much money will Frederick have when he collects the money at the age of $18$? Give your answer to the nearest hundredth of ... | math | {
"style": "rule",
"ground_truth": "\\$4813.24"
} | {
"index": 38637,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When I saw Eleonora, I found her very pretty. After a brief trivial conversation, I told her my age and asked how old she was. She answered:\n\n- When you were as old as I am now, you were three times older than me. And when I will be three times older than I am now, together o... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 38638,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When Joyce counts the pennies in her bank by fives, she has one left over. When she counts them by threes, there are two left over. What is the least possible number of pennies in the bank?"
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 38639,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When Julia divides her apples into groups of nine, ten, or eleven, she has two apples left over. Assuming Julia has more than two apples, what is the smallest possible number of apples in Julia's collection?"
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"index": 38640,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When Lisa squares her favorite $2$ -digit number, she gets the same result as when she cubes the sum of the digits of her favorite $2$ -digit number. What is Lisa's favorite $2$ -digit number?"
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 38641,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When Ma Xiaohu was doing a subtraction problem, he mistakenly wrote the units digit of the minuend as 5 instead of 3, and the tens digit as 0 instead of 6. Additionally, he wrote the hundreds digit of the subtrahend as 2 instead of 7. The resulting difference was 1994. What sho... | math | {
"style": "rule",
"ground_truth": "1552"
} | {
"index": 38642,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When Person A has traveled 100 meters, Person B has traveled 50 meters. When Person A reaches point $B$, Person B is still 100 meters away from $B$. Person A immediately turns around and heads back towards $A$, and they meet 60 meters from point $B$. What is the distance betwee... | math | {
"style": "rule",
"ground_truth": "300"
} | {
"index": 38643,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When Ringo places his marbles into bags with 6 marbles per bag, he has 4 marbles left over. When Paul does the same with his marbles, he has 3 marbles left over. Ringo and Paul pool their marbles and place them into as many bags as possible, with 6 marbles per bag. How many ... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 38644,
"solution": "1. **Define Variables:**\n Let $r$ be the number of marbles Ringo has and $p$ be the number of marbles Paul has.\n\n2. **Set Up Congruences:**\n From the problem, we know:\n \\[ r \\equiv 4 \\pmod{6} \\]\n \\[ p \\equiv 3 \\pmod{6} \\]\n These congruences mean that when Ring... |
DeepScaleR | [
{
"role": "user",
"content": "When Scott completes the square on the quadratic $x^2 + 8x - 1 = 0$, he obtains an equation of the form $(x + a)^2 = b$. What is $b$?"
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 38645,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When Tanya excluded all numbers from 1 to 333 that are divisible by 3 but not by 7, and all numbers that are divisible by 7 but not by 3, she ended up with 215 numbers. Did she solve the problem correctly?"
}
] | math | {
"style": "rule",
"ground_truth": "205"
} | {
"index": 38646,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When Xiaohong was 3 years old, her mother's age was the same as Xiaohong's age this year. When her mother is 78 years old, Xiaohong's age is the same as her mother’s age this year. How old is Xiaohong’s mother this year?"
}
] | math | {
"style": "rule",
"ground_truth": "53"
} | {
"index": 38647,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When \\( N \\) takes all the values from 1, 2, 3, \\ldots, 2015, how many numbers of the form \\( 3^{n} + n^{3} \\) are divisible by 7?"
}
] | math | {
"style": "rule",
"ground_truth": "288"
} | {
"index": 38648,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When \\( N \\) takes all values from 1, 2, 3, ..., to 2015, how many numbers of the form \\( 3^n + n^3 \\) are divisible by 7?"
}
] | math | {
"style": "rule",
"ground_truth": "288"
} | {
"index": 38649,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When \\( \\frac{1}{2222} \\) is expressed as a decimal, what is the sum of the first 50 digits after the decimal point?"
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 38650,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When \\( n \\) is a positive integer, the function \\( f \\) satisfies\n\\[ f(n+3) = \\frac{f(n) - 1}{f(n) + 1}, \\]\nwith \\( f(1) \\neq 0 \\), and \\( f(1) \\neq \\pm 1 \\).\n\nCompute \\( f(1) f(2023) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 38651,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When \\( n \\) is a positive integer, the function \\( f \\) satisfies \\( f(n+3)=\\frac{f(n)-1}{f(n)+1} \\), with \\( f(1) \\neq 0 \\) and \\( f(1) \\neq \\pm 1 \\). Find the value of \\( f(8) \\cdot f(2018) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 38652,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When \\( q(x) = Dx^4 + Ex^2 + Fx + 6 \\) is divided by \\( x - 2 \\), the remainder is 14. Find the remainder when \\( q(x) \\) is divided by \\( x + 2 \\)."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 38653,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When \\( x^{2} \\) is added to the quadratic polynomial \\( f(x) \\), its maximum value increases by \\( \\frac{27}{2} \\), and when \\( 4x^{2} \\) is subtracted from it, its maximum value decreases by 9. How will the maximum value of \\( f(x) \\) change if \\( 2x^{2} \\) is su... | math | {
"style": "rule",
"ground_truth": "\\frac{27}{4}"
} | {
"index": 38654,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When \\(0 < x < \\frac{\\pi}{2}\\), the value of the function \\(y = \\tan 3x \\cdot \\cot^3 x\\) cannot take numbers within the open interval \\((a, b)\\). Find the value of \\(a + b\\)."
}
] | math | {
"style": "rule",
"ground_truth": "34"
} | {
"index": 38655,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When \\(2x^2\\) is added to a quadratic polynomial \\(f(x)\\), its maximum value increases by 10, and when \\(5x^2\\) is subtracted from it, its maximum value decreases by \\(\\frac{15}{2}\\). By how much will the maximum value of \\(f(x)\\) change if \\(3x^2\\) is added to it?... | math | {
"style": "rule",
"ground_truth": "\\frac{45}{2}"
} | {
"index": 38656,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When a bucket is two-thirds full of water, the bucket and water weigh $a$ kilograms. When the bucket is one-half full of water the total weight is $b$ kilograms. In terms of $a$ and $b$, what is the total weight in kilograms when the bucket is full of water?"
}
] | math | {
"style": "rule",
"ground_truth": "3a - 2b"
} | {
"index": 38657,
"solution": "\nLet's denote:\n- $x$ as the weight of the empty bucket.\n- $y$ as the weight of the water when the bucket is full.\n\nFrom the problem, we have two equations based on the given conditions:\n1. When the bucket is two-thirds full, the total weight is $a$ kilograms:\n \\[\n x + \\f... |
DeepScaleR | [
{
"role": "user",
"content": "When a car's brakes are applied, it travels 7 feet less in each second than the previous second until it comes to a complete stop. A car goes 28 feet in the first second after the brakes are applied. How many feet does the car travel from the time the brakes are applied to ... | math | {
"style": "rule",
"ground_truth": "70"
} | {
"index": 38658,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When a certain biased coin is flipped five times, the probability of getting heads exactly once is not equal to $0$ and is the same as that of getting heads exactly twice. Let $\\frac ij$, in lowest terms, be the probability that the coin comes up heads in exactly $3$ out of $5... | math | {
"style": "rule",
"ground_truth": "283"
} | {
"index": 38659,
"solution": "Denote the probability of getting a heads in one flip of the biased coin as $h$. Based upon the problem, note that ${5\\choose1}(h)^1(1-h)^4 = {5\\choose2}(h)^2(1-h)^3$. After canceling out terms, we get $1 - h = 2h$, so $h = \\frac{1}{3}$. The answer we are looking for is ${5\\choose... |
DeepScaleR | [
{
"role": "user",
"content": "When a certain unfair die is rolled, an even number is $3$ times as likely to appear as an odd number. The die is rolled twice. What is the probability that the sum of the numbers rolled is even?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{8}"
} | {
"index": 38660,
"solution": "1. **Determine the probability of rolling an even or odd number:** Given that an even number is 3 times as likely to appear as an odd number, let the probability of rolling an odd number be $p$. Then, the probability of rolling an even number is $3p$. Since the total probability must ... |
DeepScaleR | [
{
"role": "user",
"content": "When a certain unfair die is rolled, an even number is $5$ times as likely to appear as an odd number. The die is rolled twice. Calculate the probability that the sum of the numbers rolled is odd."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{18}"
} | {
"index": 38661,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When a class of math students lined up for a class picture on Picture Day, they found that when they stood in rows of four there was one person left over. When they formed rows of five students, there were two extra students, and when they lined up in rows of seven, there were ... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 38662,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When a die is thrown twice in succession, the numbers obtained are recorded as $a$ and $b$, respectively. The probability that the line $ax+by=0$ and the circle $(x-3)^2+y^2=3$ have no points in common is ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 38663,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When a five-digit number is multiplied by 9, the resulting number consists of the same digits in reverse order. Find the original number."
}
] | math | {
"style": "rule",
"ground_truth": "10989"
} | {
"index": 38664,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When a granary records the arrival of 30 tons of grain as \"+30\", determine the meaning of \"-30\"."
}
] | math | {
"style": "rule",
"ground_truth": "-30"
} | {
"index": 38665,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When a number is divided by 3, the result is 50 more than if the number had been divided by 4. What is the number?"
}
] | math | {
"style": "rule",
"ground_truth": "600"
} | {
"index": 38666,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When a number is divided by 7, the remainder is 2. What is the remainder when three times the number minus 7 is divided by 7?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 38667,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When a number is tripled and then decreased by 5, the result is 16. What is the original number?"
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 38668,
"solution": "To get back to the original number, we undo the given operations. We add 5 to 16 to obtain 21 and then divide by 3 to obtain 7. These are the 'inverse' operations of decreasing by 5 and multiplying by 3."
} |
DeepScaleR | [
{
"role": "user",
"content": "When a polynomial $p(x)$ is divided by $x + 1,$ the remainder is 5. When $p(x)$ is divided by $x + 5,$ the remainder is $-7.$ Find the remainder when $p(x)$ is divided by $(x + 1)(x + 5).$"
}
] | math | {
"style": "rule",
"ground_truth": "3x + 8"
} | {
"index": 38669,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When a polynomial is divided by $-3x^5 + 10x - 11,$ what are the possible degrees of the remainder? Enter all the possible values, separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "0,1,2,3,4"
} | {
"index": 38670,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When a polynomial is divided by $2x^2 - 7x + 18,$ what are the possible degrees of the remainder? Enter all the possible values, separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "0,1"
} | {
"index": 38671,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When a positive integer $N$ is fed into a machine, the output is a number calculated according to the rule shown below.\n\nFor example, starting with an input of $N=7,$ the machine will output $3 \\cdot 7 +1 = 22.$ Then if the output is repeatedly inserted into the machine five... | math | {
"style": "rule",
"ground_truth": "83"
} | {
"index": 38672,
"solution": "We start by understanding the function used by the machine. If $N$ is the input, the output $O$ is given by:\n- If $N$ is odd, $O = 3N + 1$.\n- If $N$ is even, $O = \\frac{N}{2}$.\n\nWe need to find the inverse of this function to trace back from $O = 1$ to the original input $N$ afte... |
DeepScaleR | [
{
"role": "user",
"content": "When a positive integer $x$ is divided by a positive integer $y$, the quotient is $u$ and the remainder is $v$, where $u$ and $v$ are integers.\nWhat is the remainder when $x+2uy$ is divided by $y$?"
}
] | math | {
"style": "rule",
"ground_truth": "v"
} | {
"index": 38673,
"solution": "1. **Understanding the division algorithm**: According to the division algorithm, when a positive integer $x$ is divided by another positive integer $y$, we can express $x$ as:\n \\[\n x = uy + v\n \\]\n where $u$ is the quotient and $v$ is the remainder. Here, $0 \\leq v < y$... |
DeepScaleR | [
{
"role": "user",
"content": "When a positive integer is expressed in base 7, it is $AB_7$, and when it is expressed in base 5, it is $BA_5$. What is the positive integer in decimal?"
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 38674,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When a right triangle is rotated about one leg, the volume of the cone produced is $800\\pi \\;\\textrm{ cm}^3$. When the triangle is rotated about the other leg, the volume of the cone produced is $1920\\pi \\;\\textrm{ cm}^3$. What is the length (in cm) of the hypotenuse of t... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 38675,
"solution": "Let $a$ and $b$ be the two legs of the equation. We can find $\\frac{a}{b}$ by doing $\\frac{1920\\pi}{800\\pi}$. This simplified is $\\frac{12}{5}$. We can represent the two legs as $12x$ and $5x$ for $a$ and $b$ respectively.\n Since the volume of the first cone is $800\\pi$, we use... |
DeepScaleR | [
{
"role": "user",
"content": "When a single number is added to each member of the sequence 20, 50, 100, the sequence becomes expressible as $x, a x, a^{2} x$. Find $a$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{3}"
} | {
"index": 38676,
"solution": "$\\frac{5}{3}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "When a spring is stretched by 5 cm, 29.43 J of work is done. How far will the spring stretch if 9.81 J of work is done?"
}
] | math | {
"style": "rule",
"ground_truth": "0.029"
} | {
"index": 38677,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When a student multiplied the number $66$ by the repeating decimal, \\(1.\\overline{ab}\\), where $a$ and $b$ are digits, he did not notice the notation and just multiplied $66$ times $1.ab$. Later he found that his answer is $0.5$ less than the correct answer. What is the $2$-... | math | {
"style": "rule",
"ground_truth": "75"
} | {
"index": 38678,
"solution": "1. **Understanding the Problem:**\n We are given a repeating decimal $1.\\overline{ab}$, where $a$ and $b$ are digits, and the student mistakenly multiplied $66$ by $1.ab$ instead of $1.\\overline{ab}$. The error in the calculation resulted in an answer that was $0.5$ less than the ... |
DeepScaleR | [
{
"role": "user",
"content": "When a student multiplied the number $66$ by the repeating decimal, \\(1.\\overline{ab}\\), where $a$ and $b$ are digits, he did not notice the notation and just multiplied $66$ times $1.ab.$ Later he found that his answer is $0.5$ less than the correct answer. What is the $2$-... | math | {
"style": "rule",
"ground_truth": "75"
} | {
"index": 38679,
"solution": "1. **Understanding the Problem:**\n We are given a repeating decimal $1.\\overline{ab}$, where $a$ and $b$ are digits, and the student mistakenly multiplied $66$ by $1.ab$ instead of $1.\\overline{ab}$. The error in the calculation resulted in an answer that was $0.5$ less than the ... |
DeepScaleR | [
{
"role": "user",
"content": "When a student used a calculator to find the average of 30 data points, they mistakenly entered one of the data points, 105, as 15. Find the difference between the calculated average and the actual average."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 38680,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When a swing is stationary, with the footboard one foot off the ground, pushing it forward two steps (in ancient times, one step was considered as five feet) equals 10 feet, making the footboard of the swing the same height as a person who is five feet tall, determine the lengt... | math | {
"style": "rule",
"ground_truth": "14.5"
} | {
"index": 38681,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When an integer is divided by 15, the remainder is 7. Find the sum of the remainders when the same integer is divided by 3 and by 5."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 38682,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When any set of $k$ consecutive positive integers necessarily includes at least one positive integer whose digit sum is a multiple of 11, we call each of these sets of $k$ consecutive positive integers a \"dragon\" of length $k.\" Find the shortest dragon length."
}
] | math | {
"style": "rule",
"ground_truth": "39"
} | {
"index": 38683,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When any two numbers are taken from the set {0, 1, 2, 3, 4, 5} to perform division, calculate the number of different sine values that can be obtained."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 38684,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When arranging the performance order of 5 singers, it is required that a certain singer does not perform first, and another singer performs last. The number of different arrangements is ______. (Answer with a number)"
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 38685,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When arranging the performance order of 6 singers, requiring that both singers B and C are either before or after singer A, calculate the total number of different arrangements."
}
] | math | {
"style": "rule",
"ground_truth": "480"
} | {
"index": 38686,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When dividing the numbers 312837 and 310650 by some three-digit natural number, the remainders are the same. Find this remainder."
}
] | math | {
"style": "rule",
"ground_truth": "96"
} | {
"index": 38687,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When drawing 20 numbers from 2005 numbers using systematic sampling, calculate the interval of sampling."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 38688,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When drawing a histogram of the lifespans of 1000 people, if the class interval is uniformly 20, and the height of the vertical axis for the age range 60 to 80 years is 0.03, calculate the number of people aged 60 to 80."
}
] | math | {
"style": "rule",
"ground_truth": "600"
} | {
"index": 38689,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When each edge of a cube is increased by $50\\%$, by what percent is the surface area of the cube increased?"
}
] | math | {
"style": "rule",
"ground_truth": "125\\%"
} | {
"index": 38690,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When each of $702$, $787$, and $855$ is divided by the positive integer $m$, the remainder is always the positive integer $r$. When each of $412$, $722$, and $815$ is divided by the positive integer $n$, the remainder is always the positive integer $s \\neq r$. Find $m+n+r+s$."... | math | {
"style": "rule",
"ground_truth": "62"
} | {
"index": 38691,
"solution": "We know that $702 = am + r, 787 = bm + r,$ and $855 = cm+r$ where $a-c$ are integers.\n Subtracting the first two, the first and third, and the last two, we get $85 = (b-a)m, 153=(c-a)m,$ and $68=(c-b)m.$\n We know that $b-a, c-a$ and $c-b$ must be integers, so all the numbers are div... |
DeepScaleR | [
{
"role": "user",
"content": "When ethane is mixed with chlorine gas under lighting conditions, determine the maximum number of substances that can be generated."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 38692,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When evaluated, the sum of the digits of the integer equal to \\(10^{2021} - 2021\\) is calculated."
}
] | math | {
"style": "rule",
"ground_truth": "18185"
} | {
"index": 38693,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When evaluated, the sum of the digits of the integer equal to \\(10^{2021} - 2021\\) is:"
}
] | math | {
"style": "rule",
"ground_truth": "18185"
} | {
"index": 38694,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When every vector on the line $y = \\frac{5}{2} x + 4$ is projected onto a certain vector $\\mathbf{w},$ the result is always the vector $\\mathbf{p}.$ Find the vector $\\mathbf{p}.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} -40/29 \\\\ 16/29 \\end{pmatrix}"
} | {
"index": 38695,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When f(t)=0.1, solve for t in the function model: $f(t)=\\frac{1}{{1+{e^{-0.22(3t-40)}}}$."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 38696,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When finding the sum $\\frac{1}{2}+\\frac{1}{3}+\\frac{1}{4}+\\frac{1}{5}+\\frac{1}{6}+\\frac{1}{7}$, the least common denominator used is"
}
] | math | {
"style": "rule",
"ground_truth": "420"
} | {
"index": 38697,
"solution": "To find the least common denominator (LCD) for the sum $\\frac{1}{2}+\\frac{1}{3}+\\frac{1}{4}+\\frac{1}{5}+\\frac{1}{6}+\\frac{1}{7}$, we need to determine the least common multiple (LCM) of the denominators: $2, 3, 4, 5, 6, 7$.\n\n1. **Prime Factorization**:\n - $2$ is prime.\n ... |
DeepScaleR | [
{
"role": "user",
"content": "When fitting a set of data with the model $y=ce^{kx}$, in order to find the regression equation, let $z=\\ln y$ and transform it to get the linear equation $z=0.3x+4$. Then, the values of $c$ and $k$ are respectively \\_\\_\\_\\_\\_\\_ and \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "0.3"
} | {
"index": 38698,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "When five students are lining up to take a photo, and two teachers join in, with the order of the five students being fixed, calculate the total number of ways for the two teachers to stand in line with the students for the photo."
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"index": 38699,
"solution": ""
} |
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