data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "The symbol $[x]$ represents the greatest integer less than or equal to the real number $x$. Find the solution to the equation $\\left[3 x - 4 \\frac{5}{6}\\right] - 2 x - 1 = 0$."
}
] | math | {
"style": "rule",
"ground_truth": "6.5"
} | {
"index": 34600,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The symbol $\\diamond$ is defined so that $a \\diamond b=\\frac{a+b}{a \\times b}$. What is the value of $3 \\diamond 6$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 34601,
"solution": "Using the definition of the symbol, $3 \\diamond 6=\\frac{3+6}{3 \\times 6}=\\frac{9}{18}=\\frac{1}{2}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "The symbol $\\lfloor x \\rfloor$ denotes the largest integer not exceeding $x$. For example, $\\lfloor 3 \\rfloor = 3,$ and $\\lfloor 9/2 \\rfloor = 4.$ Compute \\[\\lfloor \\sqrt{1} \\rfloor + \\lfloor \\sqrt{2} \\rfloor + \\lfloor \\sqrt{3} \\rfloor + \\cdots + \\lfloor \\sqr... | math | {
"style": "rule",
"ground_truth": "38"
} | {
"index": 34602,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The symbol $|a|$ means $+a$ if $a$ is greater than or equal to zero, and $-a$ if a is less than or equal to zero; the symbol $<$ means \"less than\"; \nthe symbol $>$ means \"greater than.\" \nThe set of values $x$ satisfying the inequality $|3-x|<4$ consists of all $x$ such th... | math | {
"style": "rule",
"ground_truth": "$-1<x<7$"
} | {
"index": 34603,
"solution": "The problem involves solving the inequality $|3-x| < 4$. The absolute value inequality $|a| < b$ where $a$ is an expression and $b$ is a positive number, can be rewritten as $-b < a < b$. Applying this to our inequality:\n\n1. Rewrite the inequality:\n \\[\n |3-x| < 4 \\implies -4... |
DeepScaleR | [
{
"role": "user",
"content": "The symbol $|a|$ means $a$ is a positive number or zero, and $-a$ if $a$ is a negative number. \nFor all real values of $t$ the expression $\\sqrt{t^4+t^2}$ is equal to?"
}
] | math | {
"style": "rule",
"ground_truth": "|t|\\sqrt{1+t^2}"
} | {
"index": 34604,
"solution": "1. **Identify the expression to simplify**: We start with the expression given in the problem:\n \\[\n \\sqrt{t^4 + t^2}\n \\]\n\n2. **Factor out $t^2$ from the expression**: Recognizing that both terms inside the square root are powers of $t$, we can factor out the lowest power... |
DeepScaleR | [
{
"role": "user",
"content": "The symbol \\( R_{k} \\) represents an integer whose decimal representation consists of \\( k \\) consecutive 1s. For example, \\( R_{3} = 111 \\), \\( R_{5} = 11111 \\), and so on. If \\( R_{4} \\) divides \\( R_{24} \\), the quotient \\( Q = \\frac{R_{24}}{R_{4}} \\) is an in... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 34605,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The symphony orchestra has more than 200 members but fewer than 300 members. When they line up in rows of 6, there are two extra members; when they line up in rows of 8, there are three extra members; and when they line up in rows of 9, there are four extra members. How many me... | math | {
"style": "rule",
"ground_truth": "260"
} | {
"index": 34606,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The system of equations \\begin{eqnarray*}\\log_{10}(2000xy) - (\\log_{10}x)(\\log_{10}y) & = & 4 \\\\ \\log_{10}(2yz) - (\\log_{10}y)(\\log_{10}z) & = & 1 \\\\ \\log_{10}(zx) - (\\log_{10}z)(\\log_{10}x) & = & 0 \\\\ \\end{eqnarray*}\n has two solutions $(x_{1},y_{1},z_{1})$ a... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 34607,
"solution": "Let $a = \\log x$, $b = \\log y$ and $c = \\log z$. Then the given equations become:\n\\begin{align*} \\log 2 + a + b - ab = 1 \\\\ \\log 2 + b + c - bc = 1 \\\\ a+c = ac \\\\ \\end{align*}\n Equating the first and second equations, solving, and factoring, we get $a(1-b) = c(1-b) \\im... |
DeepScaleR | [
{
"role": "user",
"content": "The table below displays some of the results of last summer's Frostbite Falls Fishing Festival, showing how many contestants caught $n\\,$ fish for various values of $n\\,$.\n$\\begin{array}{|c|c|c|c|c|c|c|c|c|} \\hline n & 0 & 1 & 2 & 3 & \\dots & 13 & 14 & 15 \\\\ \\hline \\t... | math | {
"style": "rule",
"ground_truth": "943"
} | {
"index": 34608,
"solution": "Let $f$ be the total number of fish caught by the contestants who didn't catch $0, 1, 2, 3, 13, 14$, or $15$ fish and let $a$ be the number of contestants who didn't catch $0, 1, 2, 3, 13, 14$, or $15$ fish. From $\\text{(b)}$, we know that $\\frac{69+f+65+28+15}{a+31}=6\\implies f=6a... |
DeepScaleR | [
{
"role": "user",
"content": "The table below displays the grade distribution of the $30$ students in a mathematics class on the last two tests. For example, exactly one student received a 'D' on Test 1 and a 'C' on Test 2. What percent of the students received the same grade on both tests?"
}
] | math | {
"style": "rule",
"ground_truth": "40\\%"
} | {
"index": 34609,
"solution": "To solve this problem, we need to determine the number of students who received the same grade on both Test 1 and Test 2, and then calculate this as a percentage of the total number of students.\n\n1. **Identify students with the same grades on both tests**: According to the problem, ... |
DeepScaleR | [
{
"role": "user",
"content": "The table below gives the percent of students in each grade at Annville and Cleona elementary schools:\n\\[\\begin{tabular}{rccccccc}&\\textbf{\\underline{K}}&\\textbf{\\underline{1}}&\\textbf{\\underline{2}}&\\textbf{\\underline{3}}&\\textbf{\\underline{4}}&\\textbf{\\underlin... | math | {
"style": "rule",
"ground_truth": "15\\%"
} | {
"index": 34610,
"solution": "1. **Calculate the number of 6th graders in Annville:**\n - Annville has 100 students in total.\n - The percentage of 6th graders in Annville is 11%.\n - Therefore, the number of 6th graders in Annville is:\n \\[\n \\frac{11\\% \\times 100}{100} = 11 \\text{ students}\n ... |
DeepScaleR | [
{
"role": "user",
"content": "The table below lists the air distances in kilometers between several cities. If two different cities from the table are chosen at random, what is the probability that the distance between them is less than 9000 kilometers? Express your answer as a common fraction.\n\n\\begin{t... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 34611,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The table below shows the Gross Domestic Product (GDP) of China from 2012 to 2018 in trillion US dollars:\n\n| Year | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 |\n|------|------|------|------|------|------|------|------|\n| Year Code $x$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 |\n| GD... | math | {
"style": "rule",
"ground_truth": "2028"
} | {
"index": 34612,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The table shown is partially filled in with the results of a survey done by the radio station KMAT. How many of the females surveyed listen to this station?\n\n\\begin{tabular}{| l | c | r |}\n\\hline\n& listen & don't listen \\\\\n\\hline\nmales & 62 & \\\\ \\hline\nfemales & ... | math | {
"style": "rule",
"ground_truth": "68"
} | {
"index": 34613,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The table shows the vertical drops of six roller coasters in Fibonacci Fun Park: \n\\begin{tabular}{|l|c|}\n\\hline\nSpeed Demon & 150 feet \\\\ \n\\hline\nLooper & 230 feet \\\\ \n\\hline\nDare Devil & 160 feet \\\\ \n\\hline\nGiant Drop & 190 feet \\\\ \n\\hline\nSky Scream &... | math | {
"style": "rule",
"ground_truth": "1.67"
} | {
"index": 34614,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The tail of a 1-mile long train exits a tunnel exactly 3 minutes after the front of the train entered the tunnel. If the train is moving 60 miles per hour, how many miles long is the tunnel?"
}
] | math | {
"style": "rule",
"ground_truth": "2\\text{ miles}"
} | {
"index": 34615,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The tangent value of the angle between the slant height and the base is when the lateral area of the cone with volume $\\frac{\\pi}{6}$ is minimum."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 34616,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The tangents of the angles of a triangle are in the ratio $1: 2: 3$. How do the sides of the triangle compare to each other in length?"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{5} : 2\\sqrt{2} : 3"
} | {
"index": 34617,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The target below is made up of concentric circles with diameters $4$ , $8$ , $12$ , $16$ , and $20$ . The area of the dark region is $n\\pi$ . Find $n$ .\n\n[asy]\nsize(150);\ndefaultpen(linewidth(0.8));\nint i;\nfor(i=5;i>=1;i=i-1)\n{\nif (floor(i/2)==i/2)\n{\nfilldraw(... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 34618,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The task is given a finite increasing sequence \\( a_{1}, a_{2}, \\ldots, a_{n} \\) (\\(n \\geq 3\\)) of natural numbers, and for all \\( k \\leq n-2 \\), the equality \\( a_{k+2}=3 a_{k+1}-2 a_{k}-1 \\) holds. The sequence must necessarily contain the term \\( a_{k}=2021 \\). ... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 34619,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The taxi fare in Gotham City is $2.40 for the first $\\frac{1}{2}$ mile and additional mileage charged at the rate $0.20 for each additional 0.1 mile. You plan to give the driver a $2 tip. How many miles can you ride for $10?"
}
] | math | {
"style": "rule",
"ground_truth": "3.3"
} | {
"index": 34620,
"solution": "1. **Understanding the Fare Structure**: The initial fare is $2.40 for the first $\\frac{1}{2}$ mile. After that, each additional $0.1$ mile costs $0.20.\n\n2. **Total Fare and Tip**: The total amount you are willing to spend is $10, which includes a $2 tip. Therefore, the amount avai... |
DeepScaleR | [
{
"role": "user",
"content": "The taxi fare in Metropolis City is $3.00 for the first $\\frac{3}{4}$ mile and additional mileage charged at the rate $0.30 for each additional 0.1 mile. You plan to give the driver a $3 tip. Calculate the number of miles you can ride for $15."
}
] | math | {
"style": "rule",
"ground_truth": "3.75"
} | {
"index": 34621,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The taxicab distance between points $\\left(x_{1}, y_{1}\\right)$ and $\\left(x_{2}, y_{2}\\right)$ is $\\left|x_{2}-x_{1}\\right|+\\left|y_{2}-y_{1}\\right|$. A regular octagon is positioned in the $x y$ plane so that one of its sides has endpoints $(0,0)$ and $(1,0)$. Let $S$... | math | {
"style": "rule",
"ground_truth": "2309"
} | {
"index": 34622,
"solution": "In the taxicab metric, the set of points that lie at most $d$ units away from some fixed point $P$ form a square centered at $P$ with vertices at a distance of $d$ from $P$ in directions parallel to the axes. The diagram above depicts the intersection of an octagon with eight such squ... |
DeepScaleR | [
{
"role": "user",
"content": "The teacher and two boys and two girls stand in a row for a photo, with the requirement that the two girls must stand together and the teacher cannot stand at either end. Calculate the number of different arrangements."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 34623,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The teacher asked the students to calculate \\(\\overline{AB} . C + D . E\\). Xiao Hu accidentally missed the decimal point in \\(D . E\\), getting an incorrect result of 39.6; while Da Hu mistakenly saw the addition sign as a multiplication sign, getting an incorrect result of... | math | {
"style": "rule",
"ground_truth": "26.1"
} | {
"index": 34624,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The teacher asks Bill to calculate $a-b-c$, but Bill mistakenly calculates $a-(b-c)$ and gets an answer of 11. If the correct answer was 3, what is the value of $a-b$?"
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 34625,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The teacher fills some numbers into the circles in the diagram (each circle can and must contain only one number). The sum of the three numbers in each of the left and right closed loops is 30, and the sum of the four numbers in each of the top and bottom closed loops is 40. If... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 34626,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The teacher fills some numbers into the circles in the diagram below (each circle can and must only contain one number). The sum of the three numbers in each of the left and right closed loops is 30, and the sum of the four numbers in each of the top and bottom closed loops is ... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 34627,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The teacher gave each of her $37$ students $36$ pencils in different colors. It turned out that each pair of students received exactly one pencil of the same color. Determine the smallest possible number of different colors of pencils distributed."
}
] | math | {
"style": "rule",
"ground_truth": "666"
} | {
"index": 34628,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The teacher of the summer math camp brought with him several shirts, several pairs of pants, several pairs of shoes, and two jackets for the entire summer. On each lesson, he wore pants, a shirt, and shoes, and wore a jacket for some lessons. On any two lessons, at least one el... | math | {
"style": "rule",
"ground_truth": "216"
} | {
"index": 34629,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The teacher plans to give children a problem of the following type. He will tell them that he has thought of a polynomial \\( P(x) \\) of degree 2017 with integer coefficients, whose leading coefficient is 1. Then he will tell them \\( k \\) integers \\( n_{1}, n_{2}, \\ldots, ... | math | {
"style": "rule",
"ground_truth": "2017"
} | {
"index": 34630,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The teacher wrote a positive number $x$ on the board and asked Kolya, Petya, and Vasya to raise this number to the 3rd, 4th, and 12th powers, respectively. It turned out that Kolya's number had at least 9 digits before the decimal point, and Petya's number had no more than 11 d... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"index": 34631,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The teacher wrote a positive number \\( x \\) on the board and asked Kolya, Petya, and Vasya to raise this number to the 4th, 5th, and 20th power, respectively. It turned out that Kolya's number has at least 8 digits before the decimal point, and Petya's number has no more than... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 34632,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The teacher wrote a two-digit number on the board. Each of the three boys made two statements.\n\n- Andrey: \"This number ends in the digit 6\" and \"This number is divisible by 7.\"\n- Borya: \"This number is greater than 26\" and \"This number ends in the digit 8.\"\n- Sasha:... | math | {
"style": "rule",
"ground_truth": "91"
} | {
"index": 34633,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The teams $T_1$, $T_2$, $T_3$, and $T_4$ are in the playoffs. In the semifinal matches, $T_1$ plays $T_4$, and $T_2$ plays $T_3$. The winners of those two matches will play each other in the final match to determine the champion. When $T_i$ plays $T_j$, the probability that $T_... | math | {
"style": "rule",
"ground_truth": "781"
} | {
"index": 34634,
"solution": "There are two scenarios in which $T_4$ wins. The first scenario is where $T_4$ beats $T_1$, $T_3$ beats $T_2$, and $T_4$ beats $T_3$, and the second scenario is where $T_4$ beats $T_1$, $T_2$ beats $T_3$, and $T_4$ beats $T_2$. Consider the first scenario. The probability $T_4$ beats ... |
DeepScaleR | [
{
"role": "user",
"content": "The ten smallest positive odd numbers \\( 1, 3, \\cdots, 19 \\) are arranged in a circle. Let \\( m \\) be the maximum value of the sum of any one of the numbers and its two adjacent numbers. Find the minimum value of \\( m \\)."
}
] | math | {
"style": "rule",
"ground_truth": "33"
} | {
"index": 34635,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The ten-letter code $\\text{BEST OF LUCK}$ represents the ten digits $0-9$, in order. What 4-digit number is represented by the code word $\\text{CLUE}$?"
}
] | math | {
"style": "rule",
"ground_truth": "8671"
} | {
"index": 34636,
"solution": "The problem states that the ten-letter code $\\text{BEST OF LUCK}$ represents the ten digits $0-9$ in order. We can assign each letter a corresponding digit based on its position in the sequence:\n\n- B = 0\n- E = 1\n- S = 2\n- T = 3\n- O = 4\n- F = 5\n- L = 6\n- U = 7\n- C = 8\n- K =... |
DeepScaleR | [
{
"role": "user",
"content": "The term containing \\(x^7\\) in the expansion of \\((1 + 2x - x^2)^4\\) arises when \\(x\\) is raised to the power of 3 in three factors and \\(-x^2\\) is raised to the power of 1 in one factor."
}
] | math | {
"style": "rule",
"ground_truth": "-8"
} | {
"index": 34637,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The terms $140, a, \\frac{45}{28}$ are the first, second and third terms, respectively, of a geometric sequence. If $a$ is positive, what is the value of $a$?"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 34638,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The terms $x, x + 2, x + 4, \\dots, x + 2n$ form an arithmetic sequence, with $x$ an integer. If each term of the sequence is cubed, the sum of the cubes is $-1197$. What is the value of $n$ if $n > 3$?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 34639,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The terms of an arithmetic sequence add to $715$. The first term of the sequence is increased by $1$, the second term is increased by $3$, the third term is increased by $5$, and in general, the $k$th term is increased by the $k$th odd positive integer. The terms of the new seq... | math | {
"style": "rule",
"ground_truth": "195"
} | {
"index": 34640,
"solution": "After the adding of the odd numbers, the total of the sequence increases by $836 - 715 = 121 = 11^2$. Since the sum of the first $n$ positive odd numbers is $n^2$, there must be $11$ terms in the sequence, so the mean of the sequence is $\\dfrac{715}{11} = 65$. Since the first, last, ... |
DeepScaleR | [
{
"role": "user",
"content": "The terms of the sequence $(a_i)$ defined by $a_{n + 2} = \\frac {a_n + 2009} {1 + a_{n + 1}}$ for $n \\ge 1$ are positive integers. Find the minimum possible value of $a_1 + a_2$."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 34641,
"solution": "If $a_{n} \\ne \\frac {2009}{a_{n+1}}$, then either \\[a_{n} = \\frac {a_{n}}{1} < \\frac {a_{n} + 2009}{1 + a_{n+1}} < \\frac {2009}{a_{n+1}}\\]\n or\n\\[\\frac {2009}{a_{n+1}} < \\frac {2009 + a_{n}}{a_{n+1} + 1} < \\frac {a_{n}}{1} = a_{n}\\]\n All the integers between $a_{n}$ and ... |
DeepScaleR | [
{
"role": "user",
"content": "The terms of the sequence $(b_i)$ defined by $b_{n + 2} = \\frac {b_n + 2021} {1 + b_{n + 1}}$ for $n \\ge 1$ are positive integers. Find the minimum possible value of $b_1 + b_2$."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 34642,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The tetrahedron $ S.ABC$ has the faces $ SBC$ and $ ABC$ perpendicular. The three angles at $ S$ are all $ 60^{\\circ}$ and $ SB \\equal{} SC \\equal{} 1$ . Find the volume of the tetrahedron."
}
] | math | {
"style": "rule",
"ground_truth": "1/8"
} | {
"index": 34643,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The third and fourth terms of a geometric sequence are 12 and 16, respectively. What is the first term of the sequence?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{27}{4}"
} | {
"index": 34644,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The third and twentieth terms of an arithmetic sequence are 10 and 65, respectively. What is the thirty-second term?"
}
] | math | {
"style": "rule",
"ground_truth": "103.8235294118"
} | {
"index": 34645,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The third chick received as much porridge as the first two chicks combined. The fourth chick received as much porridge as the second and third chicks combined. The fifth chick received as much porridge as the third and fourth chicks combined. The sixth chick received as much po... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 34646,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The third exit on a highway is located at milepost 40 and the tenth exit is at milepost 160. There is a service center on the highway located three-fourths of the way from the third exit to the tenth exit. At what milepost would you expect to find this service center?"
}
] | math | {
"style": "rule",
"ground_truth": "130"
} | {
"index": 34647,
"solution": "1. **Identify the distance between the third and tenth exits**: \n The third exit is at milepost 40 and the tenth exit is at milepost 160. Therefore, the distance between these two exits is calculated as:\n \\[\n 160 - 40 = 120 \\text{ miles}\n \\]\n\n2. **Calculate the locati... |
DeepScaleR | [
{
"role": "user",
"content": "The third quartile of the data $13$, $11$, $12$, $15$, $16$, $18$, $21$, $17$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "17.5"
} | {
"index": 34648,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The three different points \\(A(x_1, y_1)\\), \\(B\\left(4, \\frac{9}{5}\\right)\\), and \\(C(x_2, y_2)\\) on the ellipse \\(\\frac{x^{2}}{25} + \\frac{y^{2}}{9} = 1\\), along with the focus \\(F(4,0)\\) have distances that form an arithmetic sequence. If the perpendicular bise... | math | {
"style": "rule",
"ground_truth": "5/4"
} | {
"index": 34649,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The three medians of a triangle has lengths $3, 4, 5$ . What is the length of the shortest side of this triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{10}{3}"
} | {
"index": 34650,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The three numbers $5, a, b$ have an average (mean) of 33. What is the average of $a$ and $b$?"
}
] | math | {
"style": "rule",
"ground_truth": "47"
} | {
"index": 34651,
"solution": "Since $5, a, b$ have an average of 33, then $\\frac{5+a+b}{3}=33$. Multiplying by 3, we obtain $5+a+b=3 \\times 33=99$, which means that $a+b=94$. The average of $a$ and $b$ is thus equal to $\\frac{a+b}{2}=\\frac{94}{2}=47$."
} |
DeepScaleR | [
{
"role": "user",
"content": "The three points $(3,-5)$, $(-a + 2, 3)$, and $(2a+3,2)$ lie on the same line. What is $a$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{-7}{23}"
} | {
"index": 34652,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The three points A, B, C form a triangle. AB=4, BC=5, AC=6. Let the angle bisector of \\angle A intersect side BC at D. Let the foot of the perpendicular from B to the angle bisector of \\angle A be E. Let the line through E parallel to AC meet BC at F. Compute DF."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 34653,
"solution": "Since AD bisects \\angle A, by the angle bisector theorem \\frac{AB}{BD}=\\frac{AC}{CD}, so BD=2 and CD=3. Extend BE to hit AC at X. Since AE is the perpendicular bisector of BX, AX=4. Since B, E, X are collinear, applying Menelaus' Theorem to the triangle ADC, we have \\frac{AE}{ED} ... |
DeepScaleR | [
{
"role": "user",
"content": "The three roots of the cubic $ 30 x^3 \\minus{} 50x^2 \\plus{} 22x \\minus{} 1$ are distinct real numbers between $ 0$ and $ 1$ . For every nonnegative integer $ n$ , let $ s_n$ be the sum of the $ n$ th powers of these three roots. What is the value of the infinite ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 34654,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The three roots of the cubic $ 30 x^3 - 50x^2 + 22x - 1$ are distinct real numbers strictly between $ 0$ and $ 1$. If the roots are $p$, $q$, and $r$, what is the sum\n\\[ \\frac{1}{1-p} + \\frac{1}{1-q} +\\frac{1}{1-r} ?\\]"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 34655,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The three roots of the equation \\[ax^3+bx^2+cx+d=0\\]are $1,$ $2,$ and $3.$ Compute $\\frac{c}{d}.$"
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{11}{6}"
} | {
"index": 34656,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The three row sums and the three column sums of the array \\[\n\\left[\\begin{matrix}4 & 9 & 2\\\\ 8 & 1 & 6\\\\ 3 & 5 & 7\\end{matrix}\\right]\n\\]\nare the same. What is the least number of entries that must be altered to make all six sums different from one another?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 34657,
"solution": "1. **Identify the initial sums**: First, we calculate the row sums and column sums of the given matrix:\n \\[\n \\begin{matrix}\n 4 & 9 & 2\\\\ \n 8 & 1 & 6\\\\ \n 3 & 5 & 7\n \\end{matrix}\n \\]\n - Row sums: $4+9+2=15$, $8+1+6=15$, $3+5+7=15$\n - Column sums: $4+8+... |
DeepScaleR | [
{
"role": "user",
"content": "The three sides of a right triangle form a geometric sequence. Determine the ratio of the length of the hypotenuse to the length of the shorter leg."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1+\\sqrt{5}}{2}"
} | {
"index": 34658,
"solution": "Let the shorter leg have length $\\ell$, and the common ratio of the geometric sequence be $r>1$. Then the length of the other leg is $\\ell r$, and the length of the hypotenuse is $\\ell r^{2}$. Hence, $$\\ell^{2}+(\\ell r)^{2}=\\left(\\ell r^{2}\\right)^{2} \\Longrightarrow \\ell^{2... |
DeepScaleR | [
{
"role": "user",
"content": "The three sides of a triangle are 25, 39, and 40. Find the diameter of its circumscribed circle."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{125}{3}"
} | {
"index": 34659,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The three vertices of an inscribed triangle in a circle divide the circumference into three arcs of lengths $3$, $4$, and $5$. The area of this triangle is:"
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{9}{\\pi^{2}}(\\sqrt{3}+3)$"
} | {
"index": 34660,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The three-digit even numbers \\( A \\, , B \\, , C \\, , D \\, , E \\) satisfy \\( A < B < C < D < E \\). Given that \\( A + B + C + D + E = 4306 \\), find the smallest value of \\( A \\)."
}
] | math | {
"style": "rule",
"ground_truth": "326"
} | {
"index": 34661,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The three-digit integer $63\\underline{\\hphantom{0}}$ is a multiple of 3. What is the greatest possible difference between two of the possibilities for the units digit?"
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 34662,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The three-digit number $2a3$ is added to the number $326$ to give the three-digit number $5b9$. If $5b9$ is divisible by 9, then $a+b$ equals"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 34663,
"solution": "1. **Identify the condition for divisibility by 9**: A number is divisible by 9 if the sum of its digits is a multiple of 9. Given the number $5b9$, the sum of its digits is $5 + b + 9$.\n\n2. **Calculate the sum of the digits**: \n \\[\n 5 + b + 9 = 14 + b\n \\]\n\n3. **Determi... |
DeepScaleR | [
{
"role": "user",
"content": "The three-digit positive integer $N$ has a ones digit of 3. What is the probability that $N$ is divisible by 3? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 34664,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The three-tiered \"pyramid\" shown in the image is built from $1 \\mathrm{~cm}^{3}$ cubes and has a surface area of $42 \\mathrm{~cm}^{2}$. We made a larger \"pyramid\" based on this model, which has a surface area of $2352 \\mathrm{~cm}^{2}$. How many tiers does it have?"
}
... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 34665,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The ticket price for a cinema is: 6 yuan per individual ticket, 40 yuan for a group ticket for every 10 people, and students enjoy a 10% discount. A school with 1258 students plans to watch a movie (teachers get in for free). The school should pay the cinema at least ____ yuan.... | math | {
"style": "rule",
"ground_truth": "4536"
} | {
"index": 34666,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The time it takes for person A to make 90 parts is the same as the time it takes for person B to make 120 parts. It is also known that A and B together make 35 parts per hour. Determine how many parts per hour A and B each make."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 34667,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The time on a cell phone is $3:52$. How many minutes will pass before the phone next shows a time using each of the digits 2, 3, and 5 exactly once?"
}
] | math | {
"style": "rule",
"ground_truth": "91"
} | {
"index": 34668,
"solution": "There are six times that can be made using each of the digits 2, 3, and 5 exactly once: $2:35$, $2:53$, $3:25$, $3:52$, $5:23$, and $5:32$. The first of these that occurs after 3:52 is 5:23. From 3:52 to $4:00$, 8 minutes pass. From 4:00 to 5:00, 60 minutes pass. From 5:00 to 5:23, 23... |
DeepScaleR | [
{
"role": "user",
"content": "The time right now is 9 o'clock. On a 12-hour clock, what time will it be 2004 hours from now?"
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 34669,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The time right now is exactly midnight. What time will it be in 1234 minutes?"
}
] | math | {
"style": "rule",
"ground_truth": "8\\!:\\!34 \\text{ p.m.}"
} | {
"index": 34670,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The times between $7$ and $8$ o'clock, correct to the nearest minute, when the hands of a clock will form an angle of $84^{\\circ}$ are:"
}
] | math | {
"style": "rule",
"ground_truth": "7: 23 and 7: 53"
} | {
"index": 34671,
"solution": "1. **Initial Position at 7:00**: \n - The hour hand at 7:00 is at $\\frac{7}{12} \\times 360^\\circ = 210^\\circ$ from the 12 o'clock position.\n - The minute hand at 7:00 is at $0^\\circ$ (at the 12 o'clock position).\n\n2. **Relative Position of Minute Hand to Hour Hand at 7:00*... |
DeepScaleR | [
{
"role": "user",
"content": "The top of one tree is $16$ feet higher than the top of another tree. The heights of the two trees are in the ratio $3:4$. In feet, how tall is the taller tree?"
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"index": 34672,
"solution": "1. **Define Variables:**\n Let the height of the taller tree be $h$ feet. According to the problem, the top of one tree is $16$ feet higher than the top of another tree. Therefore, the height of the shorter tree is $h - 16$ feet.\n\n2. **Set Up the Ratio:**\n The problem states th... |
DeepScaleR | [
{
"role": "user",
"content": "The top section of an 8 cm by 6 cm rectangular sheet of paper is folded along a straight line so that when the top section lies flat on the bottom section, corner $P$ lies on top of corner $R$. What is the length of the crease?"
}
] | math | {
"style": "rule",
"ground_truth": "7.5"
} | {
"index": 34673,
"solution": "Suppose that the crease intersects $PS$ at $X$, $QR$ at $Y$, and the line $PR$ at $Z$. We want to determine the length of $XY$. Since $P$ folds on top of $R$, then line segment $PZ$ folds on top of line segment $RZ$, since after the fold $Z$ corresponds with itself and $P$ corresponds... |
DeepScaleR | [
{
"role": "user",
"content": "The total GDP of the capital city in 2022 is 41600 billion yuan, express this number in scientific notation."
}
] | math | {
"style": "rule",
"ground_truth": "4.16 \\times 10^{4}"
} | {
"index": 34674,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The total \\( T \\) is obtained as the sum of the integers from 2006 to 2036 inclusive. What is the sum of all the prime factors of \\( T \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "121"
} | {
"index": 34675,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The total area of all the faces of a rectangular solid is $22\\text{cm}^2$, and the total length of all its edges is $24\\text{cm}$. Then the length in cm of any one of its interior diagonals is"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{14}"
} | {
"index": 34676,
"solution": "1. **Define the variables**: Let the edge lengths of the rectangular solid be $a$, $b$, and $c$.\n\n2. **Relate the given total surface area to the variables**: The total surface area of a rectangular solid is given by the formula $2(ab + bc + ac)$. Given that the total surface area i... |
DeepScaleR | [
{
"role": "user",
"content": "The total corn yield in centners, harvested from a certain field area, is expressed as a four-digit number composed of the digits 0, 2, 3, and 5. When the average yield per hectare was calculated, it was found to be the same number of centners as the number of hectares of the f... | math | {
"style": "rule",
"ground_truth": "3025"
} | {
"index": 34677,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The total in-store price for a blender is $\\textdollar 129.95$. A television commercial advertises the same blender for four easy payments of $\\textdollar 29.99$ and a one-time shipping and handling charge of $\\textdollar 14.95$. Calculate the number of cents saved by purcha... | math | {
"style": "rule",
"ground_truth": "496"
} | {
"index": 34678,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The total in-store price for a laptop is $299.99. A radio advertisement offers the same laptop for five easy payments of $55.98 and a one-time shipping and handling charge of $12.99. Calculate the amount of money saved by purchasing the laptop from the radio advertiser."
}
] | math | {
"style": "rule",
"ground_truth": "710"
} | {
"index": 34679,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The total in-store price for an appliance is $99.99$. A television commercial advertises the same product for three easy payments of $29.98$ and a one-time shipping and handling charge of $9.98$. How many cents are saved by buying the appliance from the television advertiser?"
... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 34680,
"solution": "1. **Calculate the total cost of buying the appliance from the television advertiser:**\n The advertisement mentions three payments of $\\textdollar 29.98$ each and an additional one-time shipping and handling charge of $\\textdollar 9.98$. First, calculate the total of the three pa... |
DeepScaleR | [
{
"role": "user",
"content": "The total investment of the Xiangshan Port Sea-Crossing Bridge, about 7.7 billion yuan, should be expressed in scientific notation."
}
] | math | {
"style": "rule",
"ground_truth": "7.7 \\times 10^9"
} | {
"index": 34681,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The total length of the highway from Lishan Town to the provincial capital is 189 kilometers, passing through the county town. The county town is 54 kilometers away from Lishan Town. In the morning at 8:30, a bus departs from Lishan Town to the county town and arrives at 9:15. ... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 34682,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The total number of matches played in the 2006 World Cup competition can be calculated by summing the number of matches determined at each stage of the competition."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"index": 34683,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The total number of toothpicks used to build a rectangular grid 15 toothpicks high and 12 toothpicks wide, with internal diagonal toothpicks, is calculated by finding the sum of the toothpicks."
}
] | math | {
"style": "rule",
"ground_truth": "567"
} | {
"index": 34684,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The total number of workers in Workshop A and Workshop C is $x + y$. If a sample of 45 people is drawn from the factory with 20 people from Workshop A and 10 people from Workshop C, determine the relationship between the number of workers in Workshop A, Workshop B, and Workshop... | math | {
"style": "rule",
"ground_truth": "900"
} | {
"index": 34685,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The total price of 4 floppy disks and 5 CDs is not less than 20 yuan, and the total price of 6 floppy disks and 3 CDs is not more than 24 yuan. What is the minimum amount of money needed to buy 3 floppy disks and 9 CDs?"
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 34686,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The tourists on a hike had several identical packs of cookies. During a daytime break, they opened two packs and divided the cookies equally among all the hikers. One cookie was left over, so they fed it to a squirrel. In the evening break, they opened three more packs and agai... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"index": 34687,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The tower function of twos is defined recursively as follows: $T(1) = 2$ and $T(n + 1) = 2^{T(n)}$ for $n\\ge1$. Let $A = (T(2009))^{T(2009)}$ and $B = (T(2009))^A$. What is the largest integer $k$ for which $\\underbrace{\\log_2\\log_2\\log_2\\ldots\\log_2B}_{k\\text{ times}}$... | math | {
"style": "rule",
"ground_truth": "2010"
} | {
"index": 34688,
"solution": "To solve this problem, we need to understand the recursive definition of the tower function $T(n)$ and how logarithms interact with powers and products. We start by analyzing the expression for $B$ and then determine how many times we can apply the logarithm base 2 before the result i... |
DeepScaleR | [
{
"role": "user",
"content": "The traditional Chinese mathematical masterpiece \"Nine Chapters on the Mathematical Art\" records: \"There are 5 cows and 2 sheep, worth 19 taels of silver; 2 cows and 5 sheep, worth 16 taels of silver. How much is each cow and each sheep worth in silver?\" According to the tr... | math | {
"style": "rule",
"ground_truth": "7.5"
} | {
"index": 34689,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The traffic police brigade of our county is carrying out a comprehensive road traffic safety rectification campaign \"Hundred-Day Battle\" throughout the county, which strictly requires riders of electric bicycles and motorcycles to comply with the rule of \"one helmet, one bel... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 34690,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The train arrives at a station randomly between 1:00 PM and 3:00 PM and waits for 30 minutes before departing. If John also arrives randomly at the station within the same time period, what is the probability that he will find the train at the station?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{32}"
} | {
"index": 34691,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The train schedule in Hummut is hopelessly unreliable. Train A will enter Intersection X from the west at a random time between 9:00 am and 2:30 pm; each moment in that interval is equally likely. Train B will enter the same intersection from the north at a random time between ... | math | {
"style": "rule",
"ground_truth": "\\frac{13}{48}"
} | {
"index": 34692,
"solution": "Suppose we fix the time at which Train B arrives at Intersection X; then call the interval during which Train A could arrive (given its schedule) and collide with Train B the 'disaster window.' We consider two cases: (i) Train B enters Intersection $X$ between 9:30 and 9:45. If Train ... |
DeepScaleR | [
{
"role": "user",
"content": "The transformation $T,$ taking vectors to vectors, has the following properties:\n\n(i) $T(a \\mathbf{v} + b \\mathbf{w}) = a T(\\mathbf{v}) + b T(\\mathbf{w})$ for all vectors $\\mathbf{v}$ and $\\mathbf{w},$ and for all scalars $a$ and $b.$\n(ii) $T(\\mathbf{v} \\times \\math... | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 7 \\\\ 8 \\\\ 11 \\end{pmatrix}"
} | {
"index": 34693,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The triangle $ABC$ is isosceles with $AB=BC$ . The point F on the side $[BC]$ and the point $D$ on the side $AC$ are the feets of the the internals bisectors drawn from $A$ and altitude drawn from $B$ respectively so that $AF=2BD$ . Fine the measure of the angle ... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 34694,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The triangle $ABC$ is isosceles with $AB=AC$, and $\\angle{BAC}<60^{\\circ}$. The points $D$ and $E$ are chosen on the side $AC$ such that, $EB=ED$, and $\\angle{ABD}\\equiv\\angle{CBE}$. Denote by $O$ the intersection point between the internal bisectors of the angles $\\angle... | math | {
"style": "rule",
"ground_truth": "120^\\circ"
} | {
"index": 34695,
"solution": "\nGiven triangle \\( \\triangle ABC \\), where \\( AB = AC \\) and \\( \\angle BAC < 60^\\circ \\), points \\( D \\) and \\( E \\) are chosen on side \\( AC \\) such that \\( EB = ED \\) and \\( \\angle ABD \\equiv \\angle CBE \\). We are tasked with finding \\(\\angle COD\\), where \... |
DeepScaleR | [
{
"role": "user",
"content": "The triangle $\\triangle ABC$ is an isosceles triangle where $AB = 4\\sqrt{2}$ and $\\angle B$ is a right angle. If $I$ is the incenter of $\\triangle ABC,$ then what is $BI$?\n\nExpress your answer in the form $a + b\\sqrt{c},$ where $a,$ $b,$ and $c$ are integers, and $c$ is ... | math | {
"style": "rule",
"ground_truth": "8 - 4\\sqrt{2}"
} | {
"index": 34696,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The triangle $\\triangle ABC$ is an isosceles triangle where $AC = 6$ and $\\angle A$ is a right angle. If $I$ is the incenter of $\\triangle ABC,$ then what is $BI$?"
}
] | math | {
"style": "rule",
"ground_truth": "6\\sqrt{2} - 6"
} | {
"index": 34697,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The triangle $\\triangle ABC$ is equilateral, and the point $P$ is such that $PA = 3 \\, \\text{cm}$, $PB = 4 \\, \\text{cm}$, and $PC = 5 \\, \\text{cm}$. Calculate the length of the sides of the triangle $\\triangle ABC$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{25 + 12 \\sqrt{3}}"
} | {
"index": 34698,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The triangle \\( \\triangle ABC \\) has side \\( AC \\) with length \\( 24 \\text{ cm} \\) and a height from vertex \\( B \\) with length \\( 25 \\text{ cm} \\). Side \\( AB \\) is divided into five equal parts, with division points labeled \\( K, L, M, N \\) from \\( A \\) to ... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 34699,
"solution": ""
} |
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