data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Two families of three, with a total of $4$ adults and $2$ children, agree to go on a countryside trip together on Sunday using two cars, an \"Audi\" and a \"Jetta\". Each car can accommodate a maximum of $4$ people, and the two children cannot be left alone in one car. Determin... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 36400,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two farmers agree that pigs are worth $\\$300$ and that goats are worth $\\$210$. When one farmer owes the other money, he pays the debt in pigs or goats, with ``change'' received in the form of goats or pigs as necessary. (For example, a $\\$390$ debt could be paid with two pi... | math | {
"style": "rule",
"ground_truth": "\\$30"
} | {
"index": 36401,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two ferries cross a river with constant speeds, turning at the shores without losing time. They start simultaneously from opposite shores and meet for the first time 700 feet from one shore. They continue to the shores, return, and meet for the second time 400 feet from the opp... | math | {
"style": "rule",
"ground_truth": "1400"
} | {
"index": 36402,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two ferries leave simultaneously from opposite shores of a river and cross it perpendicularly to the banks. The speeds of the ferries are constant. The ferries meet each other 720 meters from the nearest shore. Upon reaching the shore, they immediately depart back. On the retur... | math | {
"style": "rule",
"ground_truth": "1280"
} | {
"index": 36403,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two ferries travel between two opposite banks of a river at constant speeds. Upon reaching a bank, each immediately starts moving back in the opposite direction. The ferries departed from opposite banks simultaneously, met for the first time 700 meters from one of the banks, co... | math | {
"style": "rule",
"ground_truth": "1700"
} | {
"index": 36404,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two friends agree to meet at a specific place between 12:00 PM and 12:30 PM. The first one to arrive waits for the other for 20 minutes before leaving. Find the probability that the friends will meet, assuming each chooses their arrival time randomly (between 12:00 PM and 12:30... | math | {
"style": "rule",
"ground_truth": "8/9"
} | {
"index": 36405,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two friends are playing tic-tac-toe. If Amy wins $\\frac{3}{8}$ of the time, Lily wins $\\frac{3}{10}$ of the time, and they tie the rest of the time, then what fraction of the time do they tie?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{40}"
} | {
"index": 36406,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two friends, Alice and Bob, start cycling towards a park 80 miles away. Alice cycles 3 miles per hour slower than Bob. Upon reaching the park, Bob immediately turns around and starts cycling back, meeting Alice 15 miles away from the park. Find the speed of Alice."
}
] | math | {
"style": "rule",
"ground_truth": "6.5"
} | {
"index": 36407,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two geometric sequences $a_1, a_2, a_3, \\ldots$ and $b_1, b_2, b_3, \\ldots$ have the same common ratio, with $a_1 = 27$, $b_1=99$, and $a_{15}=b_{11}$. Find $a_9$."
}
] | math | {
"style": "rule",
"ground_truth": "363"
} | {
"index": 36408,
"solution": "Call the common ratio $r.$ Now since the $n$th term of a geometric sequence with first term $x$ and common ratio $y$ is $xy^{n-1},$ we see that $a_1 \\cdot r^{14} = b_1 \\cdot r^{10} \\implies r^4 = \\frac{99}{27} = \\frac{11}{3}.$ But $a_9$ equals $a_1 \\cdot r^8 = a_1 \\cdot (r^4)^2... |
DeepScaleR | [
{
"role": "user",
"content": "Two groups have an equal number of students. Each student studies at least one language: English or French. It is known that 5 people in the first group and 5 in the second group study both languages. The number of students studying French in the first group is three times less... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 36409,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two hundred people were surveyed. Of these, 150 indicated they liked Beethoven, and 120 indicated they liked Chopin. Additionally, it is known that of those who liked both Beethoven and Chopin, 80 people also indicated they liked Vivaldi. What is the minimum number of people su... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"index": 36410,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two hunters, $A$ and $B$, went duck hunting. Assume that each of them hits a duck as often as they miss it. Hunter $A$ encountered 50 ducks during the hunt, while hunter $B$ encountered 51 ducks. What is the probability that hunter $B$'s catch exceeds hunter $A$'s catch?"
}
] | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"index": 36411,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two identical CDs regularly cost a total of $\\$28.$ What is the cost in dollars of five of these CDs?"
}
] | math | {
"style": "rule",
"ground_truth": "70"
} | {
"index": 36412,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two identical cars are traveling in the same direction. The speed of one is $36 \\kappa \\mu / h$, and the other is catching up with a speed of $54 \\mathrm{kм} / h$. It is known that the reaction time of the driver in the rear car to the stop signals of the preceding car is 2 ... | math | {
"style": "rule",
"ground_truth": "42.5"
} | {
"index": 36413,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two identical cylindrical containers are connected at the bottom by a small tube with a tap. While the tap was closed, water was poured into the first container, and oil was poured into the second one, so that the liquid levels were the same and equal to $h = 40$ cm. At what le... | math | {
"style": "rule",
"ground_truth": "16.47"
} | {
"index": 36414,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two identical cylindrical sheets are cut open along the dotted lines and glued together to form one bigger cylindrical sheet. The smaller sheets each enclose a volume of 100. What volume is enclosed by the larger sheet?"
}
] | math | {
"style": "rule",
"ground_truth": "400"
} | {
"index": 36415,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two identical cylindrical vessels are connected at the bottom with a small-diameter tube with a valve. While the valve was closed, water was poured into the first vessel and oil was poured into the second vessel, so that the liquid levels were identical and equal to $h = 40 \\t... | math | {
"style": "rule",
"ground_truth": "32.94"
} | {
"index": 36416,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two identical cylindrical vessels are connected by a small tube with a valve at the bottom. Initially, the valve is closed, and water is poured into the first vessel while oil is poured into the second vessel, such that the liquid levels are equal and are $h=40$ cm. At what lev... | math | {
"style": "rule",
"ground_truth": "32.94"
} | {
"index": 36417,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two identical rectangular crates are packed with cylindrical pipes, using different methods. Each pipe has a diameter of 8 cm. In Crate A, the pipes are packed directly on top of each other in 25 rows of 8 pipes each across the width of the crate. In Crate B, pipes are packed i... | math | {
"style": "rule",
"ground_truth": "200 - 96\\sqrt{3}"
} | {
"index": 36418,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two identical rectangular crates are packed with cylindrical pipes, using different methods. Each pipe has diameter $10\\text{ cm}.$ A side view of the first four rows of each of the two different methods of packing is shown below.\n\n[asy]\ndraw(circle((1,1),1),black+linewidth... | math | {
"style": "rule",
"ground_truth": "10 + 5\\sqrt{3}"
} | {
"index": 36419,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two identical rectangular crates are packed with cylindrical pipes, using different methods. Each pipe has diameter 10 cm. A side view of the first four rows of each of the two different methods of packing is shown below.\n\n[asy]\ndraw(circle((1,1),1),black+linewidth(1));\ndr... | math | {
"style": "rule",
"ground_truth": "190-100\\sqrt{3}"
} | {
"index": 36420,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two identical resistors $R_{0}$ are connected in series and connected to a DC voltage source. An ideal voltmeter is connected in parallel with one of the resistors. Its reading is $U=2 \\text{V}$. If the voltmeter is replaced with an ideal ammeter, its reading will be $I=4 \\te... | math | {
"style": "rule",
"ground_truth": "0.5"
} | {
"index": 36421,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two identical rulers are placed together. Each ruler is exactly 10 cm long and is marked in centimeters from 0 to 10. The 3 cm mark on each ruler is aligned with the 4 cm mark on the other. The overall length is \\( L \\) cm. What is the value of \\( L \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 36422,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two identical smaller cubes are stacked next to a larger cube. Each of the two smaller cubes has a volume of 8. What is the volume of the larger cube?"
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"index": 36423,
"solution": "Since $2 \\times 2 \\times 2=8$, a cube with edge length 2 has volume 8. Therefore, each of the cubes with volume 8 have a height of 2. This means that the larger cube has a height of $2+2=4$, which means that its volume is $4^{3}=4 \\times 4 \\times 4=64$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Two identical squares, \\(A B C D\\) and \\(P Q R S\\), have side length 12. They overlap to form the 12 by 20 rectangle \\(A Q R D\\). What is the area of the shaded rectangle \\(P B C S\\)?"
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 36424,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two identical test tubes were filled with 200 ml of a liquid substance each. From the first test tube, $1 / 4$ of the content was poured out and replaced with the same amount of water. This procedure was repeated 3 more times, each time pouring out a quarter of the content and ... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 36425,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two individuals undertake a certain task and work for an equal amount of time. $A$ misses 2 days and earns 80 forints in total, while $B$ misses 5 days and earns 63 forints. If $A$ had missed 5 days and $B$ had missed 2 days, then $A$ would earn 2 forints more than $B$. How man... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 36426,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two individuals, A and B, start traveling towards each other from points A and B, respectively, at the same time. They meet at point C, after which A continues to point B and B rests for 14 minutes before continuing to point A. Both A and B, upon reaching points B and A, immedi... | math | {
"style": "rule",
"ground_truth": "1680"
} | {
"index": 36427,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two individuals, A and B, travel from point $A$ to point $B$. A departs 48 minutes before B. When A has traveled $\\frac{2}{3}$ of the total distance, B catches up to A. If B immediately returns to point $A$ after arriving at point $B$ at the same speed, B meets A 6 minutes aft... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 36428,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two integers are relatively prime if they don't share any common factors, i.e. if their greatest common divisor is 1. Define $\\varphi(n)$ as the number of positive integers that are less than $n$ and relatively prime to $n$. Define $\\varphi_{d}(n)$ as the number of positive i... | math | {
"style": "rule",
"ground_truth": "41"
} | {
"index": 36429,
"solution": "For fixed $n$, the pattern of integers relatively prime to $n$ repeats every $n$ integers, so $\\varphi_{d}(n)=d \\varphi(n)$. Therefore the expression in the problem equals $\\varphi(n)^{3}$. The cube root of 64000 is $40 . \\varphi(p)=p-1$ for any prime $p$. Since 40 is one less tha... |
DeepScaleR | [
{
"role": "user",
"content": "Two integers are relatively prime if they have no common factors other than 1 or -1. What is the probability that a positive integer less than or equal to 30 is relatively prime to 30? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{15}"
} | {
"index": 36430,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two integers have a sum of $26$. When two more integers are added to the first two, the sum is $41$. Finally, when two more integers are added to the sum of the previous $4$ integers, the sum is $57$. What is the minimum number of even integers among the $6$ integers?"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 36431,
"solution": "1. **Identify the sums at each stage:**\n - Let the first two integers be $x$ and $y$. We know $x + y = 26$.\n - Let the next two integers added be $a$ and $b$. Then, $x + y + a + b = 41$.\n - Let the final two integers added be $m$ and $n$. Then, $x + y + a + b + m + n = 57$.\n... |
DeepScaleR | [
{
"role": "user",
"content": "Two integers have a sum of 26. When two more integers are added to the first two integers the sum is 41. Finally when two more integers are added to the sum of the previous four integers the sum is 57. What is the minimum number of odd integers among the 6 integers?"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 36432,
"solution": "Let's denote the six integers as $a, b, c, d, e, f$. We are given the following conditions:\n1. $a + b = 26$\n2. $a + b + c + d = 41$\n3. $a + b + c + d + e + f = 57$\n\nFrom these conditions, we can derive the sums of the additional integers:\n- From 1 and 2, $c + d = 41 - 26 = 15$\n... |
DeepScaleR | [
{
"role": "user",
"content": "Two interior angles $A$ and $B$ of pentagon $ABCDE$ are $60^{\\circ}$ and $85^{\\circ}$. Two of the remaining angles, $C$ and $D$, are equal and the fifth angle $E$ is $15^{\\circ}$ more than twice $C$. Find the measure of the largest angle."
}
] | math | {
"style": "rule",
"ground_truth": "205^\\circ"
} | {
"index": 36433,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two irreducible fractions have their denominators equal to 600 and 700. Find the minimum value for the denominator of the sum of the fractions."
}
] | math | {
"style": "rule",
"ground_truth": "168"
} | {
"index": 36434,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two is $10 \\%$ of $x$ and $20 \\%$ of $y$. What is $x - y$?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 36435,
"solution": "1. **Translate the percentages into equations:**\n - Given that two is $10\\%$ of $x$, we can write this as:\n \\[\n 2 = 0.10 \\times x\n \\]\n - Similarly, two is $20\\%$ of $y$, which can be written as:\n \\[\n 2 = 0.20 \\times y\n \\]\n\n2. **Solve for $... |
DeepScaleR | [
{
"role": "user",
"content": "Two isosceles triangles are given with equal perimeters. The base of the second triangle is 15% larger than the base of the first, and the leg of the second triangle is 5% smaller than the leg of the first triangle. Find the ratio of the sides of the first triangle."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 36436,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two isosceles triangles each have at least one angle that measures $70^{\\circ}$. In the first triangle, the measure in degrees of each of the remaining two angles is even. In the second triangle, the measure in degrees of each of the remaining two angles is odd. Let $S$ be the... | math | {
"style": "rule",
"ground_truth": "250"
} | {
"index": 36437,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two isosceles triangles with sidelengths $x,x,a$ and $x,x,b$ ($a \\neq b$) have equal areas. Find $x$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{a^2 + b^2}}{2}"
} | {
"index": 36438,
"solution": "\nWe are given two isosceles triangles with side lengths \\( x, x, a \\) and \\( x, x, b \\), where \\( a \\neq b \\), and they have equal areas. We need to find the value of \\( x \\).\n\n### Step-by-Step Solution\n\n1. **Area of an Isosceles Triangle:**\n\n For an isosceles triang... |
DeepScaleR | [
{
"role": "user",
"content": "Two jars each contain the same number of marbles, and every marble is either blue or green. In Jar $1$ the ratio of blue to green marbles is $9:1$, and the ratio of blue to green marbles in Jar $2$ is $8:1$. There are $95$ green marbles in all. How many more blue marbles are in... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 36439,
"solution": "1. **Set up the ratios for each jar:** \n For Jar $1$, the ratio of blue to green marbles is $9:1$. This means for every $10$ marbles in Jar $1$, $9$ are blue and $1$ is green. \n For Jar $2$, the ratio of blue to green marbles is $8:1$. This means for every $9$ marbles in Jar $... |
DeepScaleR | [
{
"role": "user",
"content": "Two joggers each run at their own constant speed and in opposite directions from one another around an oval track. They meet every 36 seconds. The first jogger completes one lap of the track in a time that, when measured in seconds, is a number (not necessarily an integer) betw... | math | {
"style": "rule",
"ground_truth": "3705"
} | {
"index": 36440,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two jokers are added to a 52 card deck and the entire stack of 54 cards is shuffled randomly. What is the expected number of cards that will be between the two jokers?"
}
] | math | {
"style": "rule",
"ground_truth": "52 / 3"
} | {
"index": 36441,
"solution": "Each card has an equal likelihood of being either on top of the jokers, in between them, or below the jokers. Thus, on average, $1 / 3$ of them will land between the two jokers."
} |
DeepScaleR | [
{
"role": "user",
"content": "Two linear functions \\( f(x) \\) and \\( g(x) \\) satisfy the properties that for all \\( x \\),\n- \\( f(x) + g(x) = 2 \\)\n- \\( f(f(x)) = g(g(x)) \\)\nand \\( f(0) = 2022 \\). Compute \\( f(1) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "2021"
} | {
"index": 36442,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two lines are perpendicular and intersect at point $O$. Points $A$ and $B$ move along these two lines at a constant speed. When $A$ is at point $O$, $B$ is 500 yards away from point $O$. After 2 minutes, both points $A$ and $B$ are equidistant from $O$. After another 8 minutes,... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 36443,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two lines are perpendicular and intersect at point $O$. Points $A$ and $B$ move along these two lines at a constant speed. When $A$ is at point $O$, $B$ is 500 yards away from point $O$. After 2 minutes, both points $A$ and $B$ are equidistant from $O$. After another 8 minutes,... | math | {
"style": "rule",
"ground_truth": "2: 3"
} | {
"index": 36444,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two lines are perpendicular. One line has a direction vector of $\\begin{pmatrix} 3 \\\\ -7 \\end{pmatrix}.$ The other line has a direction vector of $\\begin{pmatrix} a \\\\ 2 \\end{pmatrix}.$ Find $a.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{14}{3}"
} | {
"index": 36445,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two lines defined by the equations $y = mx + 4$ and $y = 3x + b$, where $m$ and $b$ are constants, intersect at the point $(6, 10)$. What is the value of $b + m$?"
}
] | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"index": 36446,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two lines have the same non-zero $y$-intercept. The first line has a slope of 10 and an $x$-intercept of $(s, 0)$. The second line has a slope of 6 and an $x$-intercept of $(t, 0)$. What is the ratio of $s $ to $t$? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac 35"
} | {
"index": 36447,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two lines passing through point \\( M \\), which lies outside the circle with center \\( O \\), touch the circle at points \\( A \\) and \\( B \\). Segment \\( OM \\) is divided in half by the circle. In what ratio is segment \\( OM \\) divided by line \\( AB \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "1:3"
} | {
"index": 36448,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two lines touch a circle with center \\(O\\) at points \\(A\\) and \\(B\\) and intersect at point \\(C\\). Find the angle between these lines if \\(\\angle ABO = 40^{\\circ}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"index": 36449,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two lines with slopes $-\\frac{1}{3}$ and $3$ intersect at $(3,3)$. Find the area of the triangle enclosed by these two lines and the line $x+y=12$."
}
] | math | {
"style": "rule",
"ground_truth": "22.5"
} | {
"index": 36450,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two lines with slopes $\\dfrac{1}{3}$ and $3$ intersect at $(3,3)$. Find the area of the triangle enclosed by these two lines and the line $x+y=12$."
}
] | math | {
"style": "rule",
"ground_truth": "8.625"
} | {
"index": 36451,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two lines with slopes $\\frac{1}{2}$ and $2$ intersect at $(2,2)$. What is the area of the triangle enclosed by these two lines and the line $x+y=10$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 36452,
"solution": "1. **Identify the equations of the lines**: \n - The line with slope $\\frac{1}{2}$ passing through $(2,2)$ has the equation $y - 2 = \\frac{1}{2}(x - 2)$, which simplifies to $y = \\frac{1}{2}x + 1$.\n - The line with slope $2$ passing through $(2,2)$ has the equation $y - 2 = 2(... |
DeepScaleR | [
{
"role": "user",
"content": "Two lines with slopes 3 and -1 intersect at the point $(3, 1)$. What is the area of the triangle enclosed by these two lines and the horizontal line $y = 8$?\n- **A)** $\\frac{25}{4}$\n- **B)** $\\frac{98}{3}$\n- **C)** $\\frac{50}{3}$\n- **D)** 36\n- **E)** $\\frac{200}{9}$"
... | math | {
"style": "rule",
"ground_truth": "\\frac{98}{3}"
} | {
"index": 36453,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two long cylindrical tubes of the same length but different diameters lie parallel to each other on a flat surface. The larger tube has radius $72$ and rolls along the surface toward the smaller tube, which has radius $24$. It rolls over the smaller tube and continues rolling a... | math | {
"style": "rule",
"ground_truth": "312"
} | {
"index": 36454,
"solution": "1. **Understanding the Problem:**\n Two cylindrical tubes, one with radius 72 and the other with radius 24, are lying parallel on a flat surface. The larger tube rolls over the smaller tube and completes one full revolution, returning to the same point on its circumference, without ... |
DeepScaleR | [
{
"role": "user",
"content": "Two machine tools, A and B, produce the same product. The products are divided into first-class and second-class according to quality. In order to compare the quality of the products produced by the two machine tools, each machine tool produced 200 products. The quality of the ... | math | {
"style": "rule",
"ground_truth": "99\\%"
} | {
"index": 36455,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two machine tools, A and B, produce the same product. The products are divided into first-class and second-class according to quality. In order to compare the quality of the products produced by the two machine tools, each machine tool produced 200 products. The quality of the ... | math | {
"style": "rule",
"ground_truth": "99\\%"
} | {
"index": 36456,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two mathematicians take a morning coffee break each day. They arrive at the cafeteria independently, at random times between 9 a.m. and 10 a.m., and stay for exactly $m$ minutes. The probability that either one arrives while the other is in the cafeteria is $40 \\%,$ and $m = a... | math | {
"style": "rule",
"ground_truth": "87"
} | {
"index": 36457,
"solution": "Case 1:\n Case 2:\n We draw a number line representing the time interval. If mathematician $M_1$ comes in at the center of the time period, then the two mathematicions will meet if $M_2$ comes in somewhere between $m$ minutes before and after $M_1$ comes (a total range of $2m$ minutes... |
DeepScaleR | [
{
"role": "user",
"content": "Two mathematicians were both born in the last 500 years. Each lives (or will live) to be 100 years old, then dies. Each mathematician is equally likely to be born at any point during those 500 years. What is the probability that they were contemporaries for any length of time?"... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{25}"
} | {
"index": 36458,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two mathematicians, Kelly and Jason, play a cooperative game. The computer selects some secret positive integer $n<60$ (both Kelly and Jason know that $n<60$, but that they don't know what the value of $n$ is). The computer tells Kelly the unit digit of $n$, and it tells Jason ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 36459,
"solution": "The only way in which Kelly can know that $n$ is divisible by at least two different primes is if she is given 0 as the unit digit of $n$, since if she received anything else, then there is some number with that unit digit and not divisible by two primes (i.e., $1,2,3,4,5,16,7,8,9$ ).... |
DeepScaleR | [
{
"role": "user",
"content": "Two mathematics teachers administer a geometry test, assessing the ability to solve problems and knowledge of theory for each 10th-grade student. The first teacher spends 5 and 7 minutes per student, and the second teacher spends 3 and 4 minutes per student. What is the minimum... | math | {
"style": "rule",
"ground_truth": "110"
} | {
"index": 36460,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two medians of a triangle with unequal sides are $3$ inches and $6$ inches. Its area is $3 \\sqrt{15}$ square inches. The length of the third median in inches, is:"
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{6}"
} | {
"index": 36461,
"solution": "1. **Understanding the Problem**: We are given a triangle with two medians of lengths 3 inches and 6 inches, and the area of the triangle is $3 \\sqrt{15}$ square inches. We need to find the length of the third median.\n\n2. **Labeling the Triangle**: Let's denote the triangle as $\\t... |
DeepScaleR | [
{
"role": "user",
"content": "Two men at points $R$ and $S$, $76$ miles apart, set out at the same time to walk towards each other.\nThe man at $R$ walks uniformly at the rate of $4\\tfrac{1}{2}$ miles per hour; the man at $S$ walks at the constant\nrate of $3\\tfrac{1}{4}$ miles per hour for the first hour... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 36462,
"solution": "1. **Calculate the distance each person walks**: \n - The man starting from $R$ walks at a rate of $4.5$ miles per hour. Therefore, in $h$ hours, he walks a distance of $4.5h$ miles.\n - The man starting from $S$ walks at a rate that increases by $0.5$ miles per hour every hour, s... |
DeepScaleR | [
{
"role": "user",
"content": "Two more than three times $B$ is equal to 20. What is the value of $B$?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 36463,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two motorcyclists departed simultaneously from points \\( A \\) and \\( B \\) towards each other and met 50 km from point \\( B \\). After arriving at points \\( A \\) and \\( B \\), the motorcyclists immediately turned back and met for the second time 25 km from point \\( A \\... | math | {
"style": "rule",
"ground_truth": "125"
} | {
"index": 36464,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two mutually perpendicular chords \\( AB \\) and \\( CD \\) are drawn in a circle. Determine the distance between the midpoint of segment \\( AD \\) and the line \\( BC \\), given that \\( BD = 6 \\), \\( AC = 12 \\), and \\( BC = 10 \\). If necessary, round your answer to two ... | math | {
"style": "rule",
"ground_truth": "2.5"
} | {
"index": 36465,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two non-collinear vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ are given, and $\\overrightarrow{a}+2\\overrightarrow{b}$ is perpendicular to $2\\overrightarrow{a}-\\overrightarrow{b}$, $\\overrightarrow{a}-\\overrightarrow{b}$ is perpendicular to $\\overrightarrow{a}... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{10}}{5}"
} | {
"index": 36466,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two noncongruent integer-sided isosceles triangles have the same perimeter and the same area. The ratio of the lengths of the bases of the two triangles is $8: 7$. Find the minimum possible value of their common perimeter."
}
] | math | {
"style": "rule",
"ground_truth": "676"
} | {
"index": 36467,
"solution": "Let the first triangle have sides $16n,a,a$, so the second has sides $14n,a+n,a+n$. The height of the first triangle is $\\frac{7}{8}$ the height of the second triangle. Therefore, we have \\[a^2-64n^2=\\frac{49}{64}((a+n)^2-49n^2).\\] Multiplying this, we get \\[64a^2-4096n^2=49a^2+9... |
DeepScaleR | [
{
"role": "user",
"content": "Two noncongruent integer-sided isosceles triangles have the same perimeter and the same area. The ratio of the lengths of the bases of the two triangles is \\(5:4\\). Find the minimum possible value of their common perimeter."
}
] | math | {
"style": "rule",
"ground_truth": "524"
} | {
"index": 36468,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two nonhorizontal, non vertical lines in the $xy$-coordinate plane intersect to form a $45^{\\circ}$ angle. One line has slope equal to $6$ times the slope of the other line. What is the greatest possible value of the product of the slopes of the two lines?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 36469,
"solution": "1. **Identify the relationship between the slopes and the angle**: Given that two lines intersect at a $45^\\circ$ angle, we can use the tangent addition formula to relate the slopes of the lines. Let the slope of one line be $m_1$ and the other be $m_2$. The tangent of the angle betw... |
DeepScaleR | [
{
"role": "user",
"content": "Two numbers $90$ and $m$ share exactly three positive divisors. What is the greatest of these three common divisors?"
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 36470,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two numbers $a$ and $b$ with $0 \\leq a \\leq 1$ and $0 \\leq b \\leq 1$ are chosen at random. The number $c$ is defined by $c=2a+2b$. The numbers $a, b$ and $c$ are each rounded to the nearest integer to give $A, B$ and $C$, respectively. What is the probability that $2A+2B=C$... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{16}"
} | {
"index": 36471,
"solution": "By definition, if $0 \\leq a<\\frac{1}{2}$, then $A=0$ and if $\\frac{1}{2} \\leq a \\leq 1$, then $A=1$. Similarly, if $0 \\leq b<\\frac{1}{2}$, then $B=0$ and if $\\frac{1}{2} \\leq b \\leq 1$, then $B=1$. We keep track of our information on a set of axes, labelled $a$ and $b$. The ... |
DeepScaleR | [
{
"role": "user",
"content": "Two numbers \\( x \\) and \\( y \\) satisfy the equation \\( 26x^2 + 23xy - 3y^2 - 19 = 0 \\) and are respectively the sixth and eleventh terms of a decreasing arithmetic progression consisting of integers. Find the common difference of this progression."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 36472,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two numbers \\( x \\) and \\( y \\) satisfy the equation \\( 280x^{2} - 61xy + 3y^{2} - 13 = 0 \\) and are respectively the fourth and ninth terms of a decreasing arithmetic progression consisting of integers. Find the common difference of this progression."
}
] | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"index": 36473,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two numbers are independently selected from the set of positive integers less than or equal to 5. What is the probability that the sum of the two numbers is less than their product? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"index": 36474,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two numbers are independently selected from the set of positive integers less than or equal to 6. Exactly one of the numbers must be even. What is the probability that the sum of the two numbers is less than their product? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 36475,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two numbers are independently selected from the set of positive integers less than or equal to 6. What is the probability that the sum of the two numbers is less than their product by at least 2? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{9}"
} | {
"index": 36476,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two numbers are independently selected from the set of positive integers less than or equal to 6. What is the probability that the sum of the two numbers is less than their product? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{9}"
} | {
"index": 36477,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two numbers are independently selected from the set of positive integers less than or equal to 7. What is the probability that the sum of the two numbers is less than their product? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{36}{49}"
} | {
"index": 36478,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two numbers are such that their difference, their sum, and their product are to one another as $1:7:24$. The product of the two numbers is:"
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 36479,
"solution": "1. Let the two numbers be $x$ and $y$. According to the problem, their difference, sum, and product are in the ratio $1:7:24$. This can be expressed as:\n \\[\n x - y : x + y : xy = 1 : 7 : 24\n \\]\n This implies:\n \\[\n x+y = 7(x-y) \\quad \\text{and} \\quad xy = 24(x-y... |
DeepScaleR | [
{
"role": "user",
"content": "Two numbers have a difference of 3 and a sum of 31. What is the larger of the two numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 36480,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two numbers in the $4 \\times 4$ grid can be swapped to create a Magic Square (in which all rows, all columns and both main diagonals add to the same total).\nWhat is the sum of these two numbers?\nA 12\nB 15\nC 22\nD 26\nE 28\n\n\\begin{tabular}{|c|c|c|c|}\n\\hline \n9 & 6 & 3... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 36481,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two numbers need to be inserted between $4$ and $16$ such that the first three numbers are in arithmetic progression and the last three numbers are in geometric progression. What is the sum of those two numbers?\nA) $6\\sqrt{3} + 8$\nB) $10\\sqrt{3} + 6$\nC) $6 + 10\\sqrt{3}$\n... | math | {
"style": "rule",
"ground_truth": "6\\sqrt{3} + 8"
} | {
"index": 36482,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two numbers whose sum is $6$ and the absolute value of whose difference is $8$ are roots of the equation:"
}
] | math | {
"style": "rule",
"ground_truth": "x^2-6x-7=0"
} | {
"index": 36483,
"solution": "Let the roots of the equation be $x$ and $y$. Given the conditions:\n1. \\(x + y = 6\\)\n2. \\(|x - y| = 8\\)\n\nFrom the first equation, we can express \\(y\\) in terms of \\(x\\):\n\\[ y = 6 - x \\]\n\nSubstituting \\(y\\) into the second equation:\n\\[ |x - (6 - x)| = 8 \\]\n\\[ |2... |
DeepScaleR | [
{
"role": "user",
"content": "Two of the altitudes of an acute triangle divide the sides into segments of lengths $5,3,2$ and $x$ units, as shown. What is the value of $x$? [asy]\ndefaultpen(linewidth(0.7)); size(75);\npair A = (0,0);\npair B = (1,0);\npair C = (74/136,119/136);\npair D = foot(B, A, C);\npa... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 36484,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two of the altitudes of an acute triangle divide the sides into segments of lengths $7, 4, 3$, and $y$ units, as shown. Calculate the value of $y$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{12}{7}"
} | {
"index": 36485,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two of the altitudes of the scalene triangle $ABC$ have length $4$ and $12$. If the length of the third altitude is also an integer, what is the biggest it can be?\n$\\textbf{(A)}\\ 4\\qquad \\textbf{(B)}\\ 5\\qquad \\textbf{(C)}\\ 6\\qquad \\textbf{(D)}\\ 7\\qquad \\textbf{(E)... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 36486,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two of the roots of\n\\[ax^3 + (a + 2b) x^2 + (b - 3a) x + (8 - a) = 0\\]are $-2$ and 3. Find the third root."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"index": 36487,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two of the roots of the equation \\[ax^3+bx^2+cx+d=0\\]are $3$ and $-2.$ Given that $a \\neq 0,$ compute $\\frac{b+c}{a}.$"
}
] | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"index": 36488,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two of the vertices of a regular octahedron are to be chosen at random. What is the probability that they will be the endpoints of an edge of the octahedron? Express your answer as a common fraction. [asy]\nsize(150);\npair A, B, C, D, E, F;\nA=(1,1);\nB=(-1,-1);\nC=(0,6);\nD=(... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{5}"
} | {
"index": 36489,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two opposite sides of a rectangle are each divided into $n$ congruent segments, and the endpoints of one segment are joined to the center to form triangle $A$. The other sides are each divided into $m$ congruent segments, and the endpoints of one of these segments are joined to... | math | {
"style": "rule",
"ground_truth": "\\frac{m}{n}"
} | {
"index": 36490,
"solution": "1. **Positioning the Rectangle**: Place the rectangle on a coordinate grid with diagonal vertices at $(0,0)$ and $(x,y)$. This sets the rectangle's length along the x-axis as $x$ and its height along the y-axis as $y$.\n\n2. **Dividing the Sides**: \n - Each horizontal side of the r... |
DeepScaleR | [
{
"role": "user",
"content": "Two out of three independently operating components of a computing device have failed. Find the probability that the first and second components have failed, given that the failure probabilities of the first, second, and third components are 0.2, 0.4, and 0.3, respectively."
... | math | {
"style": "rule",
"ground_truth": "0.3"
} | {
"index": 36491,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two painters are painting a fence that surrounds garden plots. They come every other day and paint one plot (there are 100 plots) in either red or green. The first painter is colorblind and mixes up the colors; he remembers which plots he painted, but cannot distinguish the col... | math | {
"style": "rule",
"ground_truth": "49"
} | {
"index": 36492,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two parabolas are the graphs of the equations $y=2x^2-10x-10$ and $y=x^2-4x+6$. Find all points where they intersect. List the points in order of increasing $x$-coordinate, separated by semicolons."
}
] | math | {
"style": "rule",
"ground_truth": "(8,38)"
} | {
"index": 36493,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two parabolas are the graphs of the equations $y=2x^2-7x+1$ and $y=8x^2+5x+1$. Give all points where they intersect. List the points in order of increasing $x$-coordinate, separated by semicolons."
}
] | math | {
"style": "rule",
"ground_truth": "(0, 1)"
} | {
"index": 36494,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two parabolas are the graphs of the equations $y=3x^2+4x-5$ and $y=x^2+11$. Give all points where they intersect. List the points in order of increasing $x$-coordinate, separated by semicolons."
}
] | math | {
"style": "rule",
"ground_truth": "(-4, 27);(2, 15)"
} | {
"index": 36495,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two parabolas have equations $y= x^2 + ax +b$ and $y= x^2 + cx +d$, where $a, b, c,$ and $d$ are integers, each chosen independently by rolling a fair six-sided die. What is the probability that the parabolas will have at least one point in common?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{31}{36}"
} | {
"index": 36496,
"solution": "\n1. **Set the equations equal**: Given the equations of the parabolas $y = x^2 + ax + b$ and $y = x^2 + cx + d$, set them equal to find the condition for intersection:\n \\[\n x^2 + ax + b = x^2 + cx + d.\n \\]\n Simplifying this, we get:\n \\[\n ax + b = cx + d.\n \\]\... |
DeepScaleR | [
{
"role": "user",
"content": "Two parallel chords of a circle have lengths 24 and 32 respectively, and the distance between them is 14. What is the length of another parallel chord midway between the two chords?"
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{249}"
} | {
"index": 36497,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two particles move along the edges of a square $ABCD$ with \\[A \\Rightarrow B \\Rightarrow C \\Rightarrow D \\Rightarrow A,\\] starting simultaneously and moving at the same speed. One starts at vertex $A$, and the other starts at the midpoint of side $CD$. The midpoint of the... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 36498,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two particles move along the edges of equilateral $\\triangle ABC$ in the direction $A\\Rightarrow B\\Rightarrow C\\Rightarrow A,$ starting simultaneously and moving at the same speed. One starts at $A$, and the other starts at the midpoint of $\\overline{BC}$. The midpoint of ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{16}"
} | {
"index": 36499,
"solution": "1. **Setup and Coordinate Assignment**:\n Without loss of generality (WLOG), let's place point $A$ at the origin of a coordinate system. Assume $\\overline{AB}$ lies on the $x$-axis, and let point $B$ be at $(1, 0)$. Since $\\triangle ABC$ is equilateral, point $C$ will be at $\\lef... |
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