data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "The stem-and-leaf plot shows the duration of songs (in minutes and seconds) played during a concert by a band. There are 15 songs listed in the plot. In the stem-and-leaf plot, $3 \\ 15$ represents $3$ minutes, $15$ seconds, which is the same as $195$ seconds. Find the median d... | math | {
"style": "rule",
"ground_truth": "170"
} | {
"index": 34400,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The stem-and-leaf plot shows the number of minutes and seconds of one ride on each of the $17$ top-rated roller coasters in the world. In the stem-and-leaf plot, $2 \\ 20$ represents $2$ minutes, $20$ seconds, which is the same as $140$ seconds. What is the median of this data ... | math | {
"style": "rule",
"ground_truth": "163"
} | {
"index": 34401,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The stem-and-leaf plot shows the number of minutes and seconds of one ride on each of the 21 top-rated water slides in the world. In the stem-and-leaf plot, $1 \\ 45$ represents 1 minute, 45 seconds, which is equivalent to 105 seconds. What is the median of this data set? Expre... | math | {
"style": "rule",
"ground_truth": "135"
} | {
"index": 34402,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The storage capacity of two reservoirs, A and B, changes over time. The relationship between the storage capacity of reservoir A (in hundred tons) and time $t$ (in hours) is: $f(t) = 2 + \\sin t$, where $t \\in [0, 12]$. The relationship between the storage capacity of reservoi... | math | {
"style": "rule",
"ground_truth": "6.721"
} | {
"index": 34403,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The store bought a batch of New Year cards at 0.21 yuan each and sold them for a total of 14.57 yuan. If each card was sold at the same price and did not exceed twice the purchase price, how much profit did the store make?"
}
] | math | {
"style": "rule",
"ground_truth": "4.7"
} | {
"index": 34404,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The store has 89 gold coins with numbers ranging from 1 to 89, each priced at 30 yuan. Among them, only one is a \"lucky coin.\" Feifei can ask an honest clerk if the number of the lucky coin is within a chosen subset of numbers. If the answer is \"Yes,\" she needs to pay a con... | math | {
"style": "rule",
"ground_truth": "130"
} | {
"index": 34405,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The store owner bought 2000 pens at $0.15 each and plans to sell them at $0.30 each, calculate the number of pens he needs to sell to make a profit of exactly $150."
}
] | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"index": 34406,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The strengths of the two players are equal, meaning they have equal chances of winning each game. They agreed that the prize would go to the first player to win 6 games. They had to stop the game after the first player won 5 games and the second won 3. In what proportion should... | math | {
"style": "rule",
"ground_truth": "7:1"
} | {
"index": 34407,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The stronger Goldbach conjecture states that any even integer greater than 7 can be written as the sum of two different prime numbers. For such representations of the even number 126, the largest possible difference between the two primes is"
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 34408,
"solution": "To find the largest possible difference between two prime numbers that sum to 126, we need to consider pairs of primes $(p, q)$ such that $p + q = 126$ and $p \\neq q$. We aim to maximize $|p - q|$.\n\n1. **Identify the condition**: We need $p$ and $q$ to be prime numbers, and without... |
DeepScaleR | [
{
"role": "user",
"content": "The student locker numbers at Olympic High are numbered consecutively beginning with locker number $1$. The plastic digits used to number the lockers cost two cents apiece. Thus, it costs two cents to label locker number $9$ and four cents to label locker number $10$. If it cos... | math | {
"style": "rule",
"ground_truth": "2001"
} | {
"index": 34409,
"solution": "To solve this problem, we need to calculate the total cost of labeling all the lockers and match it with the given cost of $137.94. We will calculate the cost for each range of locker numbers based on the number of digits in the locker numbers.\n\n1. **Calculate the cost for lockers w... |
DeepScaleR | [
{
"role": "user",
"content": "The students in Mr. Neatkin's class took a penmanship test. Two-thirds of the boys and $\\frac{3}{4}$ of the girls passed the test, and an equal number of boys and girls passed the test. What is the minimum possible number of students in the class?"
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 34410,
"solution": "1. **Define Variables:**\n Let $b$ represent the number of boys and $g$ represent the number of girls in Mr. Neatkin's class.\n\n2. **Set Up the Equation:**\n According to the problem, two-thirds of the boys and three-fourths of the girls passed the test, and the number of boys an... |
DeepScaleR | [
{
"role": "user",
"content": "The students in Mrs. Reed's English class are reading the same $760$-page novel. Three friends, Alice, Bob and Chandra, are in the class. Alice reads a page in $20$ seconds, Bob reads a page in $45$ seconds and Chandra reads a page in $30$ seconds. Chandra and Bob, who each h... | math | {
"style": "rule",
"ground_truth": "456"
} | {
"index": 34411,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The students' written work has a binary grading system, i.e., a work will either be accepted if it is done well or not accepted if done poorly. Initially, the works are checked by a neural network which makes an error in 10% of the cases. All works identified as not accepted by... | math | {
"style": "rule",
"ground_truth": "66"
} | {
"index": 34412,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sultan gathered 300 court sages and proposed a trial. There are 25 different colors of hats, known in advance to the sages. The sultan informed them that each sage would be given one of these hats, and if they wrote down the number of hats for each color, all these numbers ... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"index": 34413,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum\n$$\n\\frac{1}{1 \\times 2 \\times 3}+\\frac{1}{2 \\times 3 \\times 4}+\\frac{1}{3 \\times 4 \\times 5}+\\cdots+\\frac{1}{100 \\times 101 \\times 102}\n$$\ncan be expressed as $\\frac{a}{b}$, a fraction in its simplest form. Find $a+b$."
}
] | math | {
"style": "rule",
"ground_truth": "12877"
} | {
"index": 34414,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum $10 e^{2 \\pi i/11} + 10 e^{15 \\pi i/22}$ is expressed as $re^{i \\theta}.$ Enter the ordered pair $(r, \\theta).$"
}
] | math | {
"style": "rule",
"ground_truth": "\\left( 10 \\sqrt{2}, \\frac{19 \\pi}{44} \\right)"
} | {
"index": 34415,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum $2\\frac{1}{7} + 3\\frac{1}{2} + 5\\frac{1}{19}$ is between"
}
] | math | {
"style": "rule",
"ground_truth": "$10\\frac{1}{2} \\text{ and } 11$"
} | {
"index": 34416,
"solution": "1. **Convert mixed numbers to improper fractions**:\n - $2\\frac{1}{7} = \\frac{14}{7} + \\frac{1}{7} = \\frac{15}{7}$\n - $3\\frac{1}{2} = \\frac{6}{2} + \\frac{1}{2} = \\frac{7}{2}$\n - $5\\frac{1}{19} = \\frac{95}{19} + \\frac{1}{19} = \\frac{96}{19}$\n\n2. **Estimate the sum... |
DeepScaleR | [
{
"role": "user",
"content": "The sum \\( b_{6} + b_{7} + \\ldots + b_{2018} \\) of the terms of the geometric progression \\( \\left\\{b_{n}\\right\\} \\) with \\( b_{n}>0 \\) is equal to 6. The sum of the same terms taken with alternating signs \\( b_{6} - b_{7} + b_{8} - \\ldots - b_{2017} + b_{2018} \\)... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 34417,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of $25$ consecutive even integers is $10,000$. What is the largest of these $25$ consecutive integers?"
}
] | math | {
"style": "rule",
"ground_truth": "424"
} | {
"index": 34418,
"solution": "1. **Define the sequence**: Let the smallest of the 25 consecutive even integers be $x$. Then the integers are $x, x+2, x+4, \\ldots, x+48$.\n\n2. **Formulate the sum**: The sum of these integers can be expressed as:\n \\[\n x + (x+2) + (x+4) + \\cdots + (x+48)\n \\]\n This is... |
DeepScaleR | [
{
"role": "user",
"content": "The sum of $n$ terms of an arithmetic progression is $153$, and the common difference is $2$. If the first term is an integer, and $n>1$, then the number of possible values for $n$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 34419,
"solution": "1. **Identify the formula for the sum of an arithmetic progression (AP)**:\n The sum $S_n$ of the first $n$ terms of an AP where the first term is $a$ and the common difference is $d$ is given by:\n \\[\n S_n = \\frac{n}{2} \\left(2a + (n-1)d\\right)\n \\]\n Given $d = 2$ an... |
DeepScaleR | [
{
"role": "user",
"content": "The sum of 100 numbers is 1000. The largest of these numbers was doubled, while another number was decreased by 10. After these actions, the sum of all numbers remained unchanged. Find the smallest of the original numbers."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 34420,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of 18 consecutive positive integers is a perfect square. What is the smallest possible value of this sum?"
}
] | math | {
"style": "rule",
"ground_truth": "225"
} | {
"index": 34421,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of 20 consecutive integers is a triangular number. What is the smallest such sum?"
}
] | math | {
"style": "rule",
"ground_truth": "190"
} | {
"index": 34422,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of 36 consecutive integers is $6^4$. What is their median?"
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 34423,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of 49 consecutive integers is $7^5$. What is their median?"
}
] | math | {
"style": "rule",
"ground_truth": "343"
} | {
"index": 34424,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of 81 consecutive integers is $9^5$. What is their median?"
}
] | math | {
"style": "rule",
"ground_truth": "729"
} | {
"index": 34425,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of 9 standard dice rolled has the same probability of occurring as the sum of 20. Find the value of the sum of 9 standard dice that shares this same probability."
}
] | math | {
"style": "rule",
"ground_truth": "43"
} | {
"index": 34426,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of Alice's weight and Clara's weight is 220 pounds. If you subtract Alice's weight from Clara's weight, you get one-third of Clara's weight. How many pounds does Clara weigh?"
}
] | math | {
"style": "rule",
"ground_truth": "88"
} | {
"index": 34427,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of Jim's weight and Bob's weight is 180 pounds. If you subtract Jim's weight from Bob's weight, you get half of Bob's weight. How many pounds does Bob weigh?"
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 34428,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of a negative integer $N$ and its square is 6. What is the value of $N$?"
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 34429,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of a positive number and its square is 156. What is the number?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 34430,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of all digits used in the numbers 1, 2, 3, ..., 999 is ."
}
] | math | {
"style": "rule",
"ground_truth": "13500"
} | {
"index": 34431,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of all integers between 50 and 350 which end in 1 is"
}
] | math | {
"style": "rule",
"ground_truth": "5880"
} | {
"index": 34432,
"solution": "1. **Identify the sequence**: The problem asks for the sum of all integers between 50 and 350 that end in 1. These integers are $51, 61, 71, \\ldots, 341$.\n\n2. **Determine the sequence type**: The sequence is an arithmetic sequence where each term increases by 10.\n\n3. **Find the f... |
DeepScaleR | [
{
"role": "user",
"content": "The sum of all numbers of the form $2k + 1$, where $k$ takes on integral values from $1$ to $n$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "$n(n+2)$"
} | {
"index": 34433,
"solution": "1. **Identify the sequence**: The problem asks for the sum of numbers of the form $2k + 1$ where $k$ ranges from $1$ to $n$. This forms a sequence of odd numbers starting from $3$ (when $k=1$, $2k+1=3$) up to $2n+1$ (when $k=n$, $2k+1=2n+1$).\n\n2. **Write out the sequence explicitly*... |
DeepScaleR | [
{
"role": "user",
"content": "The sum of all of the digits of the integers from 1 to 2008 is to be calculated."
}
] | math | {
"style": "rule",
"ground_truth": "28054"
} | {
"index": 34434,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of all of the digits of the integers from 1 to 2008 is:"
}
] | math | {
"style": "rule",
"ground_truth": "28054"
} | {
"index": 34435,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of all positive integers $m$ such that $\\frac{13!}{m}$ is a perfect square can be written as $2^a3^b5^c7^d11^e13^f,$ where $a,b,c,d,e,$ and $f$ are positive integers. Find $a+b+c+d+e+f.$"
}
] | math | {
"style": "rule",
"ground_truth": "012"
} | {
"index": 34436,
"solution": "We first rewrite $13!$ as a prime factorization, which is $2^{10}\\cdot3^5\\cdot5^2\\cdot7\\cdot11\\cdot13.$\n For the fraction to be a square, it needs each prime to be an even power. This means $m$ must contain $7\\cdot11\\cdot13$. Also, $m$ can contain any even power of $2$ up to $... |
DeepScaleR | [
{
"role": "user",
"content": "The sum of all real roots of the equation $|x^2-3x+2|+|x^2+2x-3|=11$ is ."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5\\sqrt{97}-19}{20}"
} | {
"index": 34437,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of all roots of the equation $2x^2-3x-5=0$ and the equation $x^2-6x+2=0$ is equal to $\\boxed{\\text{blank}}$, and the product of all roots is equal to $\\boxed{\\text{blank}}$."
}
] | math | {
"style": "rule",
"ground_truth": "7\\frac{1}{2}"
} | {
"index": 34438,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of all the positive factors of integer $x$ is 24. If one of the factors is 3, what is the value of $x$?"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 34439,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of all the roots of $4x^3-8x^2-63x-9=0$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 34440,
"solution": "1. **Identify the polynomial and its degree**:\n Given the polynomial equation:\n \\[\n 4x^3 - 8x^2 - 63x - 9 = 0\n \\]\n This is a cubic polynomial of degree 3.\n\n2. **Simplify the polynomial**:\n Factor out the common coefficient of the highest degree term:\n \\[\n ... |
DeepScaleR | [
{
"role": "user",
"content": "The sum of all three-digit numbers that, when divided by 7 give a remainder of 5, when divided by 5 give a remainder of 2, and when divided by 3 give a remainder of 1 is"
}
] | math | {
"style": "rule",
"ground_truth": "4436"
} | {
"index": 34441,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of all three-digit numbers that, when divided by 7 give a remainder of 5, when divided by 5 give a remainder of 2, and when divided by 3 give a remainder of 1, calculate the sum of these numbers."
}
] | math | {
"style": "rule",
"ground_truth": "4436"
} | {
"index": 34442,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of an infinite geometric series is $16$ times the series that results if the first two terms of the original series are removed. What is the value of the series' common ratio?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 34443,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of an infinite geometric series is $27$ times the series that results if the first three terms of the original series are removed. What is the value of the series' common ratio?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 34444,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of an infinite geometric series is $64$ times the series that results if the first four terms of the original series are removed. What is the value of the series' common ratio?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 34445,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of an infinite geometric series is $81$ times the series that results if the first five terms of the original series are removed. What is the value of the series' common ratio?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 34446,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of an infinite geometric series is 64 times the series that results if the first four terms of the original series are removed. What is the value of the series' common ratio?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 34447,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of an infinite geometric series is \\( 16 \\) times the series that results if the first two terms of the original series are removed. What is the value of the series' common ratio?"
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{4}"
} | {
"index": 34448,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of five consecutive odd integers is 125. What is the smallest of these integers?"
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 34449,
"solution": "Suppose that the smallest of the five odd integers is $x$. Since consecutive odd integers differ by 2, the other four odd integers are $x+2, x+4, x+6$, and $x+8$. Therefore, $x + (x+2) + (x+4) + (x+6) + (x+8) = 125$. From this, we obtain $5x + 20 = 125$ and so $5x = 105$, which gives ... |
DeepScaleR | [
{
"role": "user",
"content": "The sum of four different positive integers is 100. The largest of these four integers is $n$. What is the smallest possible value of $n$?"
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 34450,
"solution": "Suppose that the integers $a < b < c < n$ have $a + b + c + n = 100$. Since $a < b < c < n$, then $a + b + c + n < n + n + n + n = 4n$. Thus, $100 < 4n$ and so $n > 25$. Since $n$ is an integer, then $n$ is at least 26. Could $n$ be 26? In this case, we would have $a + b + c = 100 - 2... |
DeepScaleR | [
{
"role": "user",
"content": "The sum of four numbers $p, q, r$, and $s$ is 100. If we increase $p$ by 10, we get the value $M$. If we decrease $q$ by 5, we get the value $M$. If we multiply $r$ by 10, we also get the value $M$. Lastly, if $s$ is divided by 2, it equals $M$. Determine the value of $M$."
}... | math | {
"style": "rule",
"ground_truth": "25.610"
} | {
"index": 34451,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of four numbers is one-half. What is the mean of the four numbers? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"index": 34452,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of four positive integers that form an arithmetic sequence is 46. Of all such possible sequences, what is the greatest possible third term?"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 34453,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of four positive integers that form an arithmetic sequence is 58. Of all such possible sequences, what is the greatest possible third term?"
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 34454,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of four prime numbers $P,$ $Q,$ $P-Q,$ and $P+Q$ must be expressed in terms of a single letter indicating the property of the sum."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 34455,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of n terms of an arithmetic progression is 180, and the common difference is 3. If the first term must be a positive integer, and n > 1, then find the number of possible values for n."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 34456,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of one hundred natural numbers $x, x+1, x+2, \\cdots, x+99$ is denoted as $a$. If the sum of the digits of $a$ is 50, then what is the smallest value of $x$?"
}
] | math | {
"style": "rule",
"ground_truth": "99950"
} | {
"index": 34457,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of one hundred numbers is 1000. The largest of them was doubled, and another number was decreased by 10. It turned out that the sum did not change. Find the smallest of the original numbers."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 34458,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of seven consecutive even numbers is 686. What is the smallest of these seven numbers? Additionally, calculate the median and mean of this sequence."
}
] | math | {
"style": "rule",
"ground_truth": "98"
} | {
"index": 34459,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of seven integers is $-1$. What is the maximum number of the seven integers that can be larger than $13$?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 34460,
"solution": "To solve this problem, we need to determine the maximum number of integers among the seven that can exceed 13 while still achieving a total sum of $-1$.\n\n1. **Assume the maximum number of integers greater than 13**: Let's denote these integers that are greater than 13 as $x_1, x_2, ... |
DeepScaleR | [
{
"role": "user",
"content": "The sum of seven test scores has a mean of 84, a median of 85, and a mode of 88. Calculate the sum of the three highest test scores."
}
] | math | {
"style": "rule",
"ground_truth": "264"
} | {
"index": 34461,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of sides \\( AB \\) and \\( BC \\) of triangle \\( ABC \\) is 11, angle \\( B \\) is \\( 60^\\circ \\), and the radius of the inscribed circle is \\(\\frac{2}{\\sqrt{3}}\\). It is also known that side \\( AB \\) is longer than side \\( BC \\). Find the height of the tri... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{3}"
} | {
"index": 34462,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of six consecutive positive integers is 2013. What is the largest of these six integers?"
}
] | math | {
"style": "rule",
"ground_truth": "338"
} | {
"index": 34463,
"solution": "Let the six consecutive integers be $n, n+1, n+2, n+3, n+4, n+5$. The sum of these integers can be expressed as:\n\\[\nn + (n+1) + (n+2) + (n+3) + (n+4) + (n+5) = 6n + 15\n\\]\nWe are given that this sum equals 2013:\n\\[\n6n + 15 = 2013\n\\]\nTo solve for $n$, we first subtract 15 fr... |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the $x$-coordinates of the vertices of a triangle in the Cartesian plane equals $10$. Find the sum of the $x$-coordinates of the midpoints of the sides of the triangle."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 34464,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the absolute values of the terms of a finite arithmetic progression is 100. If all its terms are increased by 1 or all its terms are increased by 2, the sum of the absolute values of the terms of the resulting progression will also be 100. What values can the quantit... | math | {
"style": "rule",
"ground_truth": "400"
} | {
"index": 34465,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the absolute values of the terms of a finite arithmetic progression is equal to 100. If all its terms are increased by 1 or all its terms are increased by 2, in both cases the sum of the absolute values of the terms of the resulting progression will also be equal to ... | math | {
"style": "rule",
"ground_truth": "400"
} | {
"index": 34466,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the ages of three people A, B, and C, denoted as \\(x, y, z\\), is 120, with \\(x, y, z \\in (20, 60)\\). How many ordered triples \\((x, y, z)\\) satisfy this condition?"
}
] | math | {
"style": "rule",
"ground_truth": "1198"
} | {
"index": 34467,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the areas of all triangles whose vertices are also vertices of a $1$ by $1$ by $1$ cube is $m + \\sqrt{n} + \\sqrt{p},$ where $m, n,$ and $p$ are integers. Find $m + n + p.$"
}
] | math | {
"style": "rule",
"ground_truth": "348"
} | {
"index": 34468,
"solution": "[asy] size(120); import three; unitsize(1cm); draw((0,0,0)--(1,0,0)--(1,1,0)--(0,1,0)--(0,0,0)--(0,0,1)--(0,1,1)--(1,1,1)--(1,0,1)--(0,0,1)); draw((0,1,1)--(0,1,0)); draw((1,1,1)--(1,1,0)); draw((1,0,1)--(1,0,0)); draw((1,1,1)--(1,0,0)--(1,1,0)--(1,1,1), blue); [/asy]\n[asy] size(120)... |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the base-$10$ logarithms of the divisors of $6^n$ is $540$. What is $n$?\nA) 9\nB) 10\nC) 11\nD) 12\nE) 13"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 34469,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the base-10 logarithms of the divisors of $10^n$ is $792$. What is $n$?"
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 34470,
"solution": "\nTo solve the problem, we need to find the value of $n$ such that the sum of the base-$10$ logarithms of the divisors of $10^n$ equals $792$.\n\n#### Step 1: Understanding the divisors of $10^n$\n$10^n = 2^n \\cdot 5^n$. Any divisor of $10^n$ can be expressed in the form $2^a \\cdot ... |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the coefficients of all rational terms in the expansion of $$(2 \\sqrt {x}- \\frac {1}{x})^{6}$$ is \\_\\_\\_\\_\\_\\_ (answer with a number)."
}
] | math | {
"style": "rule",
"ground_truth": "365"
} | {
"index": 34471,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the coefficients of all terms except the constant term in the expansion of $\\left(\\frac{2}{\\sqrt{x}} - x\\right)^9$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "5377"
} | {
"index": 34472,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the coefficients of all terms in the expanded form of $(C_4^1x + C_4^2x^2 + C_4^3x^3 + C_4^4x^4)^2$ is 256."
}
] | math | {
"style": "rule",
"ground_truth": "256"
} | {
"index": 34473,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the coefficients of the expanded form of $(x+ \\frac{a}{x})(2x- \\frac{1}{x})^{5}$ is 2. Find the constant term in the expanded form."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 34474,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the digits in the product of $\\overline{A A A A A A A A A} \\times \\overline{B B B B B B B B B}$."
}
] | math | {
"style": "rule",
"ground_truth": "81"
} | {
"index": 34475,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the digits of a two-digit number is subtracted from the number. The units digit of the result is $6$. How many two-digit numbers have this property?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 34476,
"solution": "1. **Represent the two-digit number**: Let the two-digit number be $10a + b$, where $a$ is the tens digit and $b$ is the units digit. Here, $a$ and $b$ are integers such that $1 \\leq a \\leq 9$ and $0 \\leq b \\leq 9$.\n\n2. **Formulate the given condition**: According to the problem... |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the digits of the integer equal to \\( 777777777777777^2 - 222222222222223^2 \\) can be found by evaluating the expression."
}
] | math | {
"style": "rule",
"ground_truth": "74"
} | {
"index": 34477,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the digits of the integer equal to \\( 777777777777777^2 - 222222222222223^2 \\) is"
}
] | math | {
"style": "rule",
"ground_truth": "74"
} | {
"index": 34478,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the digits of the positive integer $N$ is three times the sum of the digits of $N+1$. What is the smallest possible sum of the digits of $N$?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 34479,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the digits of the positive integer \\(N\\) is three times the sum of the digits of \\(N+1\\). What is the smallest possible sum of the digits of \\(N\\)?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 34480,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the digits of the result of the expression $\\underbrace{99 \\cdots 99}_{2021 \\text{ digits}} \\times \\underbrace{99 \\cdots 99}_{2020 \\text{ digits}}$ is $\\qquad$"
}
] | math | {
"style": "rule",
"ground_truth": "18189"
} | {
"index": 34481,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the digits of the time 19 minutes ago is two less than the sum of the digits of the time right now. Find the sum of the digits of the time in 19 minutes. (Here, we use a standard 12-hour clock of the form hh:mm.)"
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 34482,
"solution": "Let's say the time 19 minutes ago is $h$ hours and $m$ minutes, so the sum of the digits is equivalent to $h+m \\bmod 9$. If $m \\leq 40$, then the time right now is hours and $m+19$ minutes, so the sum of digits is equivalent \\bmod 9 to $h+m+19 \\equiv h+m+1(\\bmod 9)$, which is imp... |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the digits of the year 2004 is 6. What is the first year after 2000 for which the sum of the digits is 12?"
}
] | math | {
"style": "rule",
"ground_truth": "2019"
} | {
"index": 34483,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the dimensions of a rectangular prism is the sum of the number of edges, corners, and faces, where the dimensions are 2 units by 3 units by 4 units. Calculate the resulting sum."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 34484,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the distances from one vertex of a square with sides of length $2$ to the midpoints of each of the sides of the square is"
}
] | math | {
"style": "rule",
"ground_truth": "$2+2\\sqrt{5}$"
} | {
"index": 34485,
"solution": "Let's consider a square with vertices labeled as $A$, $B$, $C$, and $D$, and let the side length of the square be $2$. Assume without loss of generality that $A$ is the vertex from which we are measuring the distances to the midpoints of the sides.\n\n1. **Identify the midpoints:**\n ... |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the first $15$ positive even integers is also the sum of six consecutive even integers. What is the smallest of these six integers?"
}
] | math | {
"style": "rule",
"ground_truth": "35"
} | {
"index": 34486,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the first $20$ positive even integers is also the sum of four consecutive even integers. What is the largest of these four integers?"
}
] | math | {
"style": "rule",
"ground_truth": "108"
} | {
"index": 34487,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the first $2011$ terms of a geometric sequence is $200$. The sum of the first $4022$ terms is $380$. Find the sum of the first $6033$ terms."
}
] | math | {
"style": "rule",
"ground_truth": "542"
} | {
"index": 34488,
"solution": "Since the sum of the first $2011$ terms is $200$, and the sum of the fist $4022$ terms is $380$, the sum of the second $2011$ terms is $180$. This is decreasing from the first 2011, so the common ratio is less than one.\n Because it is a geometric sequence and the sum of the first 201... |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the first $n$ terms in the infinite geometric sequence $\\left\\{1,\\frac{1}{3},\\frac{1}{9},\\frac{1}{27},\\dots \\right\\}$ is $\\frac{121}{81}$. Find $n$."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 34489,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the first $n$ terms in the infinite geometric sequence $\\left\\{\\frac{1}{4},\\frac{1}{8},\\frac{1}{16},\\dots \\right\\}$ is $\\frac{63}{128}$. Find $n$."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 34490,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the first $n$ terms of a geometric sequence $\\{a_n\\}$, where each term is positive, is $S_n$. Given that $S_n = 2$ and $S_{3n} = 14$, calculate $S_{4n}$."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 34491,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the first $n$ terms of an arithmetic sequence $\\{a\\_n\\}$ is $S\\_n$, where the first term $a\\_1 > 0$ and the common difference $d < 0$. For any $n \\in \\mathbb{N}^*$, there exists $k \\in \\mathbb{N}^*$ such that $a\\_k = S\\_n$. Find the minimum value of $k - 2... | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"index": 34492,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the first $n$ terms of an arithmetic sequence $\\{a_n\\}$ is $S_n$, and it is given that $S_4=20$, $S_{n-4}=60$, and $S_n=120$. Find the value of $n$."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 34493,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the first $n$ terms of an arithmetic sequence $\\{a_n\\}$ is $S_n$, and the sequence $\\{b_n\\}$ is a geometric sequence, satisfying $a_1=3$, $b_1=1$, $b_2+S_2=10$, and $a_5-2b_2=a_3$. The sum of the first $n$ terms of the sequence $\\left\\{ \\frac{a_n}{b_n} \\right... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 34494,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the first $n$ terms of an arithmetic sequence $\\{a_n\\}$ is $S_n$. Given that $S_{10}=0$ and $S_{15}=25$, find the minimum value of $nS_n$."
}
] | math | {
"style": "rule",
"ground_truth": "-49"
} | {
"index": 34495,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the first $n$ terms of an arithmetic sequence $\\{a_n\\}$ is denoted as $S_n$. Given that $a_1-a_5-a_10-a_15+a_19=2$, find the value of $S_{19}$."
}
] | math | {
"style": "rule",
"ground_truth": "-38"
} | {
"index": 34496,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the first $n$ terms of the arithmetic sequences ${a_n}$ and ${b_n}$ are $S_n$ and $T_n$ respectively. If $$\\frac {S_{n}}{T_{n}}= \\frac {2n+1}{3n+2}$$, find the value of $$\\frac {a_{3}+a_{11}+a_{19}}{b_{7}+b_{15}}$$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{129}{130}"
} | {
"index": 34497,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the first $n$ terms of the sequence $\\{a_n\\}$ is $S_n=n^2+n+1$, and $b_n=(-1)^n(a_n-2)$ $(n\\in\\mathbb{N}^*)$, then the sum of the first $50$ terms of the sequence $\\{b_n\\}$ is $\\_\\_\\_\\_\\_\\_\\_$."
}
] | math | {
"style": "rule",
"ground_truth": "49"
} | {
"index": 34498,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sum of the first $n$ terms of the sequence $\\{a_n\\}$ is denoted as $S_n$. Given $a_1= \\frac{1}{2}$ and $2a_{n+1}+S_n=0$ for $n=1, 2, 3, \\ldots$. If $S_n \\leqslant k$ always holds, then the minimum value of $k$ is \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 34499,
"solution": ""
} |
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