data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "A tetrahedron with four equilateral triangular faces has a sphere inscribed within it and a sphere circumscribed about it. For each of the four faces, there is a sphere tangent externally to the face at its center and to the circumscribed sphere. A point $P$ is selected at rand... | math | {
"style": "rule",
"ground_truth": ".2"
} | {
"index": 3900,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A thief on a bus gets off at a bus stop and walks in the direction opposite to the bus’s travel direction. The bus continues its journey, and a passenger realizes they have been robbed. The passenger gets off at the next stop and starts chasing the thief. If the passenger's spe... | math | {
"style": "rule",
"ground_truth": "440"
} | {
"index": 3901,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A thin diverging lens with an optical power of $D_{p} = -6$ diopters is illuminated by a beam of light with a diameter $d_{1} = 10$ cm. On a screen positioned parallel to the lens, a light spot with a diameter $d_{2} = 20$ cm is observed. After replacing the thin diverging lens... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 3902,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A thousand integer divisions are made: $2018$ is divided by each of the integers from $ 1$ to $1000$ . Thus, a thousand integer quotients are obtained with their respective remainders. Which of these thousand remainders is the bigger?"
}
] | math | {
"style": "rule",
"ground_truth": "672"
} | {
"index": 3903,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A thousand points form the vertices of a convex polygon with 1000 sides. Inside this polygon, there are another 500 points placed such that no three of these 500 points are collinear. The polygon is triangulated in such a way that all of these 1500 points are vertices of the tr... | math | {
"style": "rule",
"ground_truth": "1998"
} | {
"index": 3904,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A three-digit $\\overline{abc}$ number is called *Ecuadorian* if it meets the following conditions: $\\bullet$ $\\overline{abc}$ does not end in $0$ . $\\bullet$ $\\overline{abc}$ is a multiple of $36$ . $\\bullet$ $\\overline{abc} - \\overline{cba}$ is positive an... | math | {
"style": "rule",
"ground_truth": "864"
} | {
"index": 3905,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A three-digit natural number $abc$ is termed a \"convex number\" if and only if the digits $a$, $b$, and $c$ (representing the hundreds, tens, and units place, respectively) satisfy $a < b$ and $c < b$. Given that $a$, $b$, and $c$ belong to the set $\\{5, 6, 7, 8, 9\\}$ and ar... | math | {
"style": "rule",
"ground_truth": "\\frac {1}{3}"
} | {
"index": 3906,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A three-digit natural number with digits in the hundreds, tens, and units places denoted as $a$, $b$, $c$ is called a \"concave number\" if and only if $a > b$, $b < c$, such as $213$. If $a$, $b$, $c \\in \\{1,2,3,4\\}$, and $a$, $b$, $c$ are all different, then the probabilit... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 3907,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A three-digit number \\( X \\) was composed of three different digits, \\( A, B, \\) and \\( C \\). Four students made the following statements:\n\n- Petya: \"The largest digit in the number \\( X \\) is \\( B \\).\"\n- Vasya: \"\\( C = 8 \\).\"\n- Tolya: \"The largest digit is... | math | {
"style": "rule",
"ground_truth": "798"
} | {
"index": 3908,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A three-digit number \\( \\mathrm{abc} \\) divided by the sum of its digits leaves a remainder of 1. The three-digit number \\( \\mathrm{cba} \\) divided by the sum of its digits also leaves a remainder of 1. If different letters represent different digits and \\( a > c \\), th... | math | {
"style": "rule",
"ground_truth": "452"
} | {
"index": 3909,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A three-digit number has a remainder of 2 when divided by 4, 5, and 6. If three digits are appended to this three-digit number to make it a six-digit number divisible by 4, 5, and 6, what is the smallest six-digit number that meets this condition?"
}
] | math | {
"style": "rule",
"ground_truth": "122040"
} | {
"index": 3910,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A three-digit number has different digits in each position. By writing a 2 to the left of this three-digit number, we get a four-digit number; and by writing a 2 to the right of this three-digit number, we get another four-digit number. The difference between these two four-dig... | math | {
"style": "rule",
"ground_truth": "327"
} | {
"index": 3911,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A three-digit number has digits a, b, and c in the hundreds, tens, and units place respectively. If a < b and b > c, then the number is called a \"convex number\". If you randomly select three digits from 1, 2, 3, and 4 to form a three-digit number, what is the probability that... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 3912,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A three-digit number has distinct digits. By arbitrarily swapping the positions of its digits, five other three-digit numbers can be obtained. If the sum of these six three-digit numbers equals 2220, then among all the numbers that meet this condition, the smallest three-digit ... | math | {
"style": "rule",
"ground_truth": "127"
} | {
"index": 3913,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A three-digit number is called a \"concave number\" if the digit in the tens place is smaller than both the digit in the hundreds place and the digit in the units place. For example, 504 and 746 are concave numbers. How many three-digit concave numbers are there if all the digi... | math | {
"style": "rule",
"ground_truth": "240"
} | {
"index": 3914,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A three-digit number is composed of three different non-zero digits in base ten. When divided by the sum of these three digits, the smallest quotient value is what?"
}
] | math | {
"style": "rule",
"ground_truth": "10.5"
} | {
"index": 3915,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A three-digit number is formed by the digits $0$, $1$, $2$, $3$, $4$, $5$, with exactly two digits being the same. There are a total of \\_\\_\\_\\_\\_ such numbers."
}
] | math | {
"style": "rule",
"ground_truth": "75"
} | {
"index": 3916,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A three-digit positive integer \\( n \\) has digits \\( a, b, c \\). (That is, \\( a \\) is the hundreds digit of \\( n \\), \\( b \\) is the tens digit of \\( n \\), and \\( c \\) is the ones (units) digit of \\( n \\).) Determine the largest possible value of \\( n \\) for wh... | math | {
"style": "rule",
"ground_truth": "870"
} | {
"index": 3917,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A three-dimensional rectangular box with dimensions $X$, $Y$, and $Z$ has faces whose surface areas are $24$, $24$, $48$, $48$, $72$, and $72$ square units. What is $X$ + $Y$ + $Z$?"
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 3918,
"solution": "1. **Identify the relationships between dimensions and surface areas**: Given that the surface areas of the faces of the box are $24$, $24$, $48$, $48$, $72$, and $72$ square units, we can infer that each pair of opposite faces has the same area. Let's denote the dimensions of the box ... |
DeepScaleR | [
{
"role": "user",
"content": "A three-meter gas pipe has rusted in two places. Determine the probability that all three resulting parts can be used as connections to gas stoves, given that regulations require a stove to be no closer than 75 cm from the main gas pipeline."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 3919,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A three-meter gas pipe has rusted in two places. Determine the probability that all three resulting segments can be used as connections to gas stoves, given that regulations require the stove to be no closer than 75 cm to the main gas pipe."
}
] | math | {
"style": "rule",
"ground_truth": "1/4"
} | {
"index": 3920,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A three-quarter sector of a circle of radius $4$ inches together with its interior can be rolled up to form the lateral surface area of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches?"
}
] | math | {
"style": "rule",
"ground_truth": "$3 \\pi \\sqrt{7}$"
} | {
"index": 3921,
"solution": "\n1. **Understanding the Geometry of the Problem**:\n The problem states that a three-quarter sector of a circle with radius $4$ inches is rolled up to form a cone. The arc length of this sector becomes the circumference of the base of the cone, and the radius of the sector becomes t... |
DeepScaleR | [
{
"role": "user",
"content": "A three-wheeled vehicle travels 100 km. Two spare wheels are available. Each of the five wheels is used for the same distance during the trip. For how many kilometers is each wheel used?"
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 3922,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A ticket contains six digits \\(a, b, c, d, e, f\\). This ticket is said to be \"lucky\" if \\(a + b + c = d + e + f\\). How many lucky tickets are there (including the ticket 000000)?"
}
] | math | {
"style": "rule",
"ground_truth": "55252"
} | {
"index": 3923,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A ticket to a school play cost $x$ dollars, where $x$ is a whole number. A group of 9th graders buys tickets costing a total of $48, and a group of 10th graders buys tickets costing a total of $64. How many values for $x$ are possible?"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 3924,
"solution": "To determine the possible values of $x$, the cost of each ticket, we need to consider the conditions given in the problem:\n\n1. The total cost for the 9th graders is $48$ dollars.\n2. The total cost for the 10th graders is $64$ dollars.\n3. $x$ must be a whole number.\n\nWe start by n... |
DeepScaleR | [
{
"role": "user",
"content": "A token starts at the point $(0,0)$ of an $xy$-coordinate grid and then makes a sequence of six moves. Each move is 1 unit in a direction parallel to one of the coordinate axes. Each move is selected randomly from the four possible directions and independently of the other move... | math | {
"style": "rule",
"ground_truth": "391"
} | {
"index": 3925,
"solution": "Perform the coordinate transformation $(x, y)\\rightarrow (x+y, x-y)$. Then we can see that a movement up, right, left, or down in the old coordinates adds the vectors $\\langle 1, -1 \\rangle$, $\\langle 1, 1 \\rangle$, $\\langle -1, -1 \\rangle$, $\\langle -1, 1 \\rangle$ respectivel... |
DeepScaleR | [
{
"role": "user",
"content": "A top hat contains 3 red chips and 2 green chips. Chips are drawn randomly, one at a time without replacement, until all 3 of the reds are drawn or until both green chips are drawn. What is the probability that the 3 reds are drawn?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"index": 3926,
"solution": "\nTo solve this problem, we need to determine the probability that all three red chips are drawn before both green chips are drawn. We can approach this problem by considering the order in which the chips are drawn.\n\n#### Step-by-step Analysis:\n\n1. **Total Arrangements**: The total... |
DeepScaleR | [
{
"role": "user",
"content": "A torus (donut) having inner radius $2$ and outer radius $4$ sits on a flat table. What is the radius of the largest spherical ball that can be placed on top of the center torus so that the ball still touches the horizontal plane? (If the $xy$-plane is the table, the torus is... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{4}"
} | {
"index": 3927,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A total area of \\( 2500 \\, \\mathrm{m}^2 \\) will be used to build identical houses. The construction cost for a house with an area \\( a \\, \\mathrm{m}^2 \\) is the sum of the material cost \\( 100 p_{1} a^{\\frac{3}{2}} \\) yuan, labor cost \\( 100 p_{2} a \\) yuan, and ot... | math | {
"style": "rule",
"ground_truth": "156"
} | {
"index": 3928,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A total of $731$ objects are put into $n$ nonempty bags where $n$ is a positive integer. These bags can be distributed into $17$ red boxes and also into $43$ blue boxes so that each red and each blue box contain $43$ and $17$ objects, respectively. Find the minimu... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 3929,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A total of $n$ people compete in a mathematical match which contains $15$ problems where $n>12$ . For each problem, $1$ point is given for a right answer and $0$ is given for a wrong answer. Analysing each possible situation, we find that if the sum of points every gro... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 3930,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A total of $960$ people are randomly numbered from $1$ to $960$. Using systematic sampling, $32$ people are selected for a survey. Find the number of people to be selected from those with numbers falling within $[450,750]$."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 3931,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A total of $n$ points are equally spaced around a circle and are labelled with the integers 1 to $n$, in order. Two points are called diametrically opposite if the line segment joining them is a diameter of the circle. If the points labelled 7 and 35 are diametrically opposite,... | math | {
"style": "rule",
"ground_truth": "56"
} | {
"index": 3932,
"solution": "The number of points on the circle equals the number of spaces between the points around the circle. Moving from the point labelled 7 to the point labelled 35 requires moving $35-7=28$ points and so 28 spaces around the circle. Since the points labelled 7 and 35 are diametrically oppos... |
DeepScaleR | [
{
"role": "user",
"content": "A total of 1000 senior high school students from a certain school participated in a mathematics exam. The scores in this exam follow a normal distribution N(90, σ²). If the probability of a score being within the interval (70, 110] is 0.7, estimate the number of students with s... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"index": 3933,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A total of 120 is divided into four parts proportional to 3, 2, 4, and 5. What is the second largest part?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{240}{7}"
} | {
"index": 3934,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A total of 17 teams play in a single-elimination tournament. (A single-elimination tournament is one where once a team has lost, it is removed from the competition.) How many total games must be played before a winner can be declared, assuming there is no possibility of ties?... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 3935,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A total of 6 letters are used to spell the English word \"theer\". Calculate the probability that the person spells this English word incorrectly."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{59}{60}"
} | {
"index": 3936,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A tour group has three age categories of people, represented in a pie chart. The central angle of the sector corresponding to older people is $9^{\\circ}$ larger than the central angle for children. The percentage of total people who are young adults is $5\\%$ higher than the p... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 3937,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A tourist attraction estimates that the number of tourists $p(x)$ (in ten thousand people) from January 2013 onwards in the $x$-th month is approximately related to $x$ as follows: $p(x)=-3x^{2}+40x (x \\in \\mathbb{N}^{*}, 1 \\leqslant x \\leqslant 12)$. The per capita consump... | math | {
"style": "rule",
"ground_truth": "3125"
} | {
"index": 3938,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A tourist city was surveyed, and it was found that the number of tourists per day $f(t)$ (in ten thousand people) and the time $t$ (in days) within the past month (calculated as $30$ days) approximately satisfy the function relationship $f(t)=4+ \\frac {1}{t}$. The average cons... | math | {
"style": "rule",
"ground_truth": "403 \\frac {1}{3}"
} | {
"index": 3939,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A tourist is learning an incorrect way to sort a permutation $(p_{1}, \\ldots, p_{n})$ of the integers $(1, \\ldots, n)$. We define a fix on two adjacent elements $p_{i}$ and $p_{i+1}$, to be an operation which swaps the two elements if $p_{i}>p_{i+1}$, and does nothing otherwi... | math | {
"style": "rule",
"ground_truth": "1009! \\cdot 1010!"
} | {
"index": 3940,
"solution": "Note that the given algorithm is very similar to the well-known Bubble Sort algorithm for sorting an array. The exception is that in the $i$-th round through the array, the first $i-1$ pairs are not checked. We claim a necessary and sufficient condition for the array to be sorted after... |
DeepScaleR | [
{
"role": "user",
"content": "A tourist spends half of their money and an additional $100 \\mathrm{Ft}$ each day. By the end of the fifth day, all their money is gone. How much money did they originally have?"
}
] | math | {
"style": "rule",
"ground_truth": "6200"
} | {
"index": 3941,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A tournament among 2021 ranked teams is played over 2020 rounds. In each round, two teams are selected uniformly at random among all remaining teams to play against each other. The better ranked team always wins, and the worse ranked team is eliminated. Let $p$ be the probabili... | math | {
"style": "rule",
"ground_truth": "674"
} | {
"index": 3942,
"solution": "In any given round, the second-best team is only eliminated if it plays against the best team. If there are $k$ teams left and the second-best team has not been eliminated, the second-best team plays the best team with probability $\\frac{1}{\\binom{k}{2}}$, so the second-best team sur... |
DeepScaleR | [
{
"role": "user",
"content": "A tower is $45 \\mathrm{~m}$ away from the bank of a river. If the width of the river is seen at an angle of $20^{\\circ}$ from a height of $18 \\mathrm{~m}$ in the tower, how wide is the river?"
}
] | math | {
"style": "rule",
"ground_truth": "16.38"
} | {
"index": 3943,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A town's population increased by $1,200$ people, and then this new population decreased by $11\\%$. The town now had $32$ less people than it did before the $1,200$ increase. What is the original population?"
}
] | math | {
"style": "rule",
"ground_truth": "10000"
} | {
"index": 3944,
"solution": "1. **Define the variables**: Let $n$ be the original population of the town.\n\n2. **Calculate the population after the increase**: When $1,200$ people are added to the town, the new population becomes $n + 1,200$.\n\n3. **Calculate the population after the decrease**: The new populati... |
DeepScaleR | [
{
"role": "user",
"content": "A town's reservoir has a usable water volume of 120 million cubic meters. Assuming the annual precipitation remains unchanged, it can sustain a water supply for 160,000 people for 20 years. After urbanization and the migration of 40,000 new residents, the reservoir will only be... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 3945,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A toy factory has a total of 450 labor hours and 400 units of raw materials for production. Producing a bear requires 15 labor hours and 20 units of raw materials, with a selling price of 80 yuan; producing a cat requires 10 labor hours and 5 units of raw materials, with a sell... | math | {
"style": "rule",
"ground_truth": "2200"
} | {
"index": 3946,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A toy store manager received a large order of Mr. Slinkums just in time for the holidays. The manager places $20\\%$ of them on the shelves, leaving the other 120 Mr. Slinkums in storage. How many Mr. Slinkums were in this order?"
}
] | math | {
"style": "rule",
"ground_truth": "150"
} | {
"index": 3947,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A toy store sells a type of building block set: each starship is priced at 8 yuan, and each mech is priced at 26 yuan. A starship and a mech can be combined to form an ultimate mech, which sells for 33 yuan per set. If the store owner sold a total of 31 starships and mechs in o... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 3948,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A track and field team has three sprinters. According to the statistics from their usual training sessions, the probabilities of sprinters A, B, and C finishing a 100m dash within 13 seconds (considered qualified) are $\\frac{2}{5}$, $\\frac{3}{4}$, and $\\frac{1}{3}$, respecti... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{10}"
} | {
"index": 3949,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A tractor is dragging a very long pipe on sleds. Gavrila walked along the entire pipe in the direction of the tractor's movement and counted 210 steps. When he walked in the opposite direction, the number of steps was 100. What is the length of the pipe if Gavrila's step is 80 ... | math | {
"style": "rule",
"ground_truth": "108"
} | {
"index": 3950,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A traffic light at an intersection has a red light that stays on for $40$ seconds, a yellow light that stays on for $5$ seconds, and a green light that stays on for $50$ seconds (no two lights are on simultaneously). What is the probability of encountering each of the following... | math | {
"style": "rule",
"ground_truth": "\\frac{11}{19}"
} | {
"index": 3951,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A traffic light cycles as follows: green for 45 seconds, then yellow for 5 seconds, and then red for 50 seconds. Mark chooses a random five-second interval to observe the light. What is the probability that the color changes during his observation?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{20}"
} | {
"index": 3952,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A traffic light cycles as follows: green for 45 seconds, yellow for 5 seconds, then red for 50 seconds. Felix chooses a random five-second interval to observe the light. What is the probability that the color changes while he is observing?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{20}"
} | {
"index": 3953,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A traffic light cycles repeatedly in the following order: green for 45 seconds, then yellow for 5 seconds, and red for 50 seconds. Cody picks a random five-second time interval to observe the light. What is the probability that the color changes while he is watching?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{20}"
} | {
"index": 3954,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A traffic light follows a cycle of green for 45 seconds, yellow for 5 seconds, and red for 40 seconds. Sam observes the light for a random five-second interval. What is the probability that the light changes from one color to another during his observation?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"index": 3955,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A traffic light runs repeatedly through the following cycle: green for 30 seconds, then yellow for 3 seconds, and then red for 30 seconds. Leah picks a random three-second time interval to watch the light. What is the probability that the color changes while she is watching?"... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{7}"
} | {
"index": 3956,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A traffic light runs repeatedly through the following cycle: green for 45 seconds, then yellow for 5 seconds, and then red for 50 seconds. Mark picks a random five-second time interval to watch the light. What is the probability that the color changes while he is watching?"
}... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{20}"
} | {
"index": 3957,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A traffic light runs repeatedly through the following cycle: green for 45 seconds, then yellow for 5 seconds, then blue for 10 seconds, and finally red for 40 seconds. Peter picks a random five-second time interval to observe the light. What is the probability that the color ch... | math | {
"style": "rule",
"ground_truth": "0.15"
} | {
"index": 3958,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A train arrives randomly some time between 2:00 and 4:00 PM, waits for 30 minutes, and then leaves. If Maria also arrives randomly between 2:00 and 4:00 PM, what is the probability that the train will be there when Maria arrives?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{32}"
} | {
"index": 3959,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A train arrives randomly sometime between 3:00 and 4:00 PM and waits for 15 minutes before leaving. If John arrives randomly between 3:15 and 4:15 PM, what is the probability that the train will still be there when John arrives?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{16}"
} | {
"index": 3960,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A train departs from point $A$ and travels towards point $B$ at a uniform speed. 11 minutes later, another train departs from point $B$ traveling towards point $A$ at a constant speed on the same route. After their meeting point, it takes the trains 20 minutes and 45 minutes to... | math | {
"style": "rule",
"ground_truth": "9/5"
} | {
"index": 3961,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A train has 18 identical cars. In some of the cars, half of the seats are free, in others, one third of the seats are free, and in the remaining cars, all the seats are occupied. In the entire train, exactly one ninth of all seats are free. How many cars have all seats occupied... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 3962,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A train has five carriages, each containing at least one passenger. Two passengers are said to be 'neighbours' if either they are in the same carriage or they are in adjacent carriages. Each passenger has exactly five or exactly ten neighbours. How many passengers are there on ... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 3963,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A train is made up of 18 carriages. There are 700 passengers traveling on the train. In any block of five adjacent carriages, there are 199 passengers in total. How many passengers in total are in the middle two carriages of the train?"
}
] | math | {
"style": "rule",
"ground_truth": "96"
} | {
"index": 3964,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A train is scheduled to arrive at a station randomly between 1:00 PM and 3:00 PM, and it waits for 15 minutes before leaving. If Alex arrives at the station randomly between 1:00 PM and 3:00 PM as well, what is the probability that he will find the train still at the station wh... | math | {
"style": "rule",
"ground_truth": "\\frac{105}{1920}"
} | {
"index": 3965,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A train leaves station K for station L at 09:30, while another train leaves station L for station K at 10:00. The first train arrives at station L 40 minutes after the trains pass each other. The second train arrives at station K 1 hour and 40 minutes after the trains pass each... | math | {
"style": "rule",
"ground_truth": "10:50"
} | {
"index": 3966,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A train moves along a straight track, and from the moment it starts braking to the moment it stops, the distance $S$ in meters that the train travels in $t$ seconds after braking is given by $S=27t-0.45t^2$. Find the time in seconds after braking when the train stops, and the d... | math | {
"style": "rule",
"ground_truth": "405"
} | {
"index": 3967,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A train moves at a speed of 60 kilometers per hour, making stops every 48 kilometers. The duration of each stop, except for the fifth stop, is 10 minutes. The fifth stop lasts half an hour. What distance did the train travel if it departed at noon on September 29 and arrived at... | math | {
"style": "rule",
"ground_truth": "2870"
} | {
"index": 3968,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A train passenger knows that the speed of their train is 40 km/h. As soon as a passing train started to go by the window, the passenger started a stopwatch and noted that the passing train took 3 seconds to pass completely. Determine the speed of the passing train, given that i... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 3969,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A train starts its journey, then stops after 1 hour due to an incident and remains halted for half an hour. After that, it continues at $\\frac{3}{4}$ of its original speed, resulting in a delay of $3 \\frac{1}{2}$ hours upon reaching its destination. If the incident had occurr... | math | {
"style": "rule",
"ground_truth": "600"
} | {
"index": 3970,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A train takes 60 seconds to pass through a 1260-meter-long bridge and 90 seconds to pass through a 2010-meter-long tunnel. What is the speed of the train in meters per second, and what is the length of the train?"
}
] | math | {
"style": "rule",
"ground_truth": "240"
} | {
"index": 3971,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A train took $X$ minutes ($0 < X < 60$) to travel from platform A to platform B. Find $X$ if it's known that at both the moment of departure from A and the moment of arrival at B, the angle between the hour and minute hands of the clock was $X$ degrees."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 3972,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A train travelling at constant speed takes five seconds to pass completely through a tunnel which is $85 \\mathrm{~m}$ long, and eight seconds to pass completely through a second tunnel which is $160 \\mathrm{~m}$ long. What is the speed of the train?"
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 3973,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A train, an hour after starting, meets with an accident which detains it a half hour, after which it proceeds at $\\frac{3}{4}$ of its former rate and arrives $3\\tfrac{1}{2}$ hours late. Had the accident happened $90$ miles farther along the line, it would have arrived only $3... | math | {
"style": "rule",
"ground_truth": "600"
} | {
"index": 3974,
"solution": "1. **Define Variables:**\n Let the speed of the train be $x$ miles per hour, and let $D$ miles be the total distance of the trip.\n\n2. **Analyze the First Scenario:**\n - The train travels for 1 hour before the accident, covering $x$ miles.\n - After the accident, it is detained... |
DeepScaleR | [
{
"role": "user",
"content": "A trainee cook took two buckets of unpeeled potatoes and peeled everything in an hour. Meanwhile, $25\\%$ of the potatoes went into peels. How much time did it take for him to collect exactly one bucket of peeled potatoes?"
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 3975,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A tram ticket is called \"lucky in Leningrad style\" if the sum of its first three digits is equal to the sum of its last three digits. A tram ticket is called \"lucky in Moscow style\" if the sum of its digits in even positions is equal to the sum of its digits in odd position... | math | {
"style": "rule",
"ground_truth": "6700"
} | {
"index": 3976,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A transformation of the first quadrant of the coordinate plane maps each point $(x,y)$ to the point $(\\sqrt{x},\\sqrt{y}).$ The vertices of quadrilateral $ABCD$ are $A=(900,300), B=(1800,600), C=(600,1800),$ and $D=(300,900).$ Let $k_{}$ be the area of the region enclosed by t... | math | {
"style": "rule",
"ground_truth": "314"
} | {
"index": 3977,
"solution": "\\begin{eqnarray*}A' = & (\\sqrt {900}, \\sqrt {300})\\\\ B' = & (\\sqrt {1800}, \\sqrt {600})\\\\ C' = & (\\sqrt {600}, \\sqrt {1800})\\\\ D' = & (\\sqrt {300}, \\sqrt {900}) \\end{eqnarray*}\n First we see that lines passing through $AB$ and $CD$ have equations $y = \\frac {1}{3}x$ a... |
DeepScaleR | [
{
"role": "user",
"content": "A transgalactic ship encountered an astonishing meteor stream. Some meteors fly along a straight line at the same speed, equally spaced from each other. Another group of meteors flies in the exact same manner along another straight line, parallel to the first, but in the opposi... | math | {
"style": "rule",
"ground_truth": "4.6"
} | {
"index": 3978,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A translation of the plane takes $-3 + 2i$ to $-7 - i.$ Find the complex number that the translation takes $-4 + 5i$ to."
}
] | math | {
"style": "rule",
"ground_truth": "-8 + 2i"
} | {
"index": 3979,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A trapezium is given with parallel bases having lengths $1$ and $4$ . Split it into two trapeziums by a cut, parallel to the bases, of length $3$ . We now want to divide the two new trapeziums, always by means of cuts parallel to the bases, in $m$ and $n$ trapeziums, re... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 3980,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A trapezoid $ABCD$ lies on the $xy$ -plane. The slopes of lines $BC$ and $AD$ are both $\\frac 13$ , and the slope of line $AB$ is $-\\frac 23$ . Given that $AB=CD$ and $BC< AD$ , the absolute value of the slope of line $CD$ can be expressed as $\\frac mn$ , wh... | math | {
"style": "rule",
"ground_truth": "1706"
} | {
"index": 3981,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A trapezoid $ABCD$ has bases $AD$ and $BC$. If $BC = 60$ units, and altitudes from $B$ and $C$ to line $AD$ divide it into segments of lengths $AP = 20$ units and $DQ = 19$ units, with the length of the altitude itself being $30$ units, what is the perimeter of trapezoid $ABCD$... | math | {
"style": "rule",
"ground_truth": "\\sqrt{1300} + \\sqrt{1261} + 159"
} | {
"index": 3982,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A trapezoid \\(ABCD\\) (\\(AD \\parallel BC\\)) and a rectangle \\(A_{1}B_{1}C_{1}D_{1}\\) are inscribed in a circle \\(\\Omega\\) with a radius of 13 such that \\(AC \\parallel B_{1}D_{1}\\) and \\(BD \\parallel A_{1}C_{1}\\). Find the ratio of the areas of \\(ABCD\\) and \\(A... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 3983,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A trapezoid \\(ABCD\\) is inscribed in a circle, with bases \\(AB = 1\\) and \\(DC = 2\\). Let \\(F\\) denote the intersection point of the diagonals of this trapezoid. Find the ratio of the sum of the areas of triangles \\(ABF\\) and \\(CDF\\) to the sum of the areas of triang... | math | {
"style": "rule",
"ground_truth": "5/4"
} | {
"index": 3984,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A trapezoid \\(AEFG\\) (\\(EF \\parallel AG\\)) is positioned inside a square \\(ABCD\\) with a side length of 14, such that points \\(E\\), \\(F\\), and \\(G\\) lie on sides \\(AB\\), \\(BC\\), and \\(CD\\) respectively. The diagonals \\(AF\\) and \\(EG\\) are perpendicular, a... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 3985,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A trapezoid field is uniformly planted with wheat. The trapezoid has one pair of parallel sides measuring 80 m and 160 m, respectively, with the longer side on the bottom. The other two non-parallel sides each measure 120 m. The angle between a slanted side and the longer base ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 3986,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A trapezoid has one base equal to twice its height, $x$, and the other base is three times as long as the height. Write the expression for the area of the trapezoid as a common fraction in terms of the height $x$."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{5x^2}{2}"
} | {
"index": 3987,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A trapezoid has side lengths 3, 5, 7, and 11. The sum of all the possible areas of the trapezoid can be written in the form of $r_1\\sqrt{n_1}+r_2\\sqrt{n_2}+r_3$, where $r_1$, $r_2$, and $r_3$ are rational numbers and $n_1$ and $n_2$ are positive integers not divisible by the ... | math | {
"style": "rule",
"ground_truth": "63"
} | {
"index": 3988,
"solution": "To solve this problem, we need to consider all possible configurations of the trapezoid with sides 3, 5, 7, and 11. We will use Heron's formula to calculate the area of triangles formed by these sides and then determine the area of the trapezoid.\n\n#### Step 1: Identify possible confi... |
DeepScaleR | [
{
"role": "user",
"content": "A trapezoid has side lengths 4, 6, 8, and 10. The trapezoid can be rearranged to form different configurations with sides 4 and 8 as the parallel bases. Calculate the total possible area of the trapezoid with its different configurations.\nA) $24\\sqrt{2}$\nB) $36\\sqrt{2}$\nC)... | math | {
"style": "rule",
"ground_truth": "48\\sqrt{2}"
} | {
"index": 3989,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A trapezoid inscribed in a circle with a radius of $13 \\mathrm{~cm}$ has its diagonals located $5 \\mathrm{~cm}$ away from the center of the circle. What is the maximum possible area of the trapezoid?"
}
] | math | {
"style": "rule",
"ground_truth": "288"
} | {
"index": 3990,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A trapezoid is divided into seven strips of equal width. What fraction of the trapezoid's area is shaded? Explain why your answer is correct."
}
] | math | {
"style": "rule",
"ground_truth": "4/7"
} | {
"index": 3991,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A trapezoid is given where the ratio of the lengths of its bases is 2. If the trapezoid is rotated $360^\\circ$ around the longer base, the resulting solid is $\\Phi_1$. If the trapezoid is rotated $360^\\circ$ around the shorter base, the resulting solid is $\\Phi_2$. Find the... | math | {
"style": "rule",
"ground_truth": "5/4"
} | {
"index": 3992,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A trauma hospital uses a rectangular piece of white cloth that is 60m long and 0.8m wide to make triangular bandages with both base and height of 0.4m. How many bandages can be made in total?"
}
] | math | {
"style": "rule",
"ground_truth": "600"
} | {
"index": 3993,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A travel agency conducted a promotion: \"Buy a trip to Egypt, bring four friends who also buy trips, and get your trip cost refunded.\" During the promotion, 13 customers came on their own, and the rest were brought by friends. Some of these customers each brought exactly four ... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"index": 3994,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A travel agency has 120 rooms, with a daily rental rate of 50 yuan per room, and they are fully booked every day. After renovation, the agency plans to increase the rental rate. Market research indicates that for every 5 yuan increase in the daily rental rate per room, the numb... | math | {
"style": "rule",
"ground_truth": "750"
} | {
"index": 3995,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A traveler visited a village where each person either always tells the truth or always lies. The villagers stood in a circle, and each person told the traveler whether the neighbor to their right was truthful or deceitful. Based on these statements, the traveler was able to det... | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"index": 3996,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A tree doubled its height every year until it reached a height of 32 feet at the end of 6 years. What was the height of the tree, in feet, at the end of 3 years?"
}
] | math | {
"style": "rule",
"ground_truth": "4 \\text{ feet}"
} | {
"index": 3997,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A tree grows in a rather peculiar manner. Lateral cross-sections of the trunk, leaves, branches, twigs, and so forth are circles. The trunk is 1 meter in diameter to a height of 1 meter, at which point it splits into two sections, each with diameter .5 meter. These sections are... | math | {
"style": "rule",
"ground_truth": "\\pi / 2"
} | {
"index": 3998,
"solution": "If we count the trunk as level 0, the two sections emerging from it as level 1, and so forth, then the $n$th level consists of $2^{n}$ sections each with diameter $1 / 2^{n}$, for a volume of $2^{n}(\\pi / 4 \\cdot 2^{-2 n})=(\\pi / 4) \\cdot 2^{-n}$. So the total volume is given by a ... |
DeepScaleR | [
{
"role": "user",
"content": "A tree has 10 pounds of apples at dawn. Every afternoon, a bird comes and eats x pounds of apples. Overnight, the amount of food on the tree increases by 10%. What is the maximum value of x such that the bird can sustain itself indefinitely on the tree without the tree runni... | math | {
"style": "rule",
"ground_truth": "10/11"
} | {
"index": 3999,
"solution": ""
} |
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