data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Ann and Anne are in bumper cars starting 50 meters apart. Each one approaches the other at a constant ground speed of $10 \\mathrm{~km} / \\mathrm{hr}$. A fly starts at Ann, flies to Anne, then back to Ann, and so on, back and forth until it gets crushed when the two bumper car... | math | {
"style": "rule",
"ground_truth": "55"
} | {
"index": 5000,
"solution": "Suppose that at a given instant the fly is at Ann and the two cars are $12 d$ apart. Then, while each of the cars travels $4 d$, the fly travels $8 d$ and meets Anne. Then the fly turns around, and while each of the cars travels $d$, the fly travels $3 d$ and meets Ann again. So, in th... |
DeepScaleR | [
{
"role": "user",
"content": "Ann and Barbara were comparing their ages and found that Barbara is as old as Ann was when Barbara was as old as Ann had been when Barbara was half as old as Ann is. If the sum of their present ages is $44$ years, then Ann's age is"
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 5001,
"solution": "1. **Define Variables:**\n Let $a$ and $b$ be the current ages of Ann and Barbara, respectively. We know that the sum of their ages is $a + b = 44$.\n\n2. **Express Age Difference:**\n Define $y$ as the difference in their ages, so $y = a - b$. Note that $y$ remains constant over t... |
DeepScaleR | [
{
"role": "user",
"content": "Ann and Sue bought identical boxes of stationery. Ann used hers to write $1$-sheet letters and Sue used hers to write $3$-sheet letters. \nAnn used all the envelopes and had $50$ sheets of paper left, while Sue used all of the sheets of paper and had $50$ envelopes left. \nThe ... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"index": 5002,
"solution": "Let $S$ represent the number of sheets of paper in each box, and let $E$ represent the number of envelopes in each box. We can set up the following equations based on the problem statement:\n\n1. Ann used all the envelopes and had 50 sheets of paper left. This implies:\n \\[ S - E = ... |
DeepScaleR | [
{
"role": "user",
"content": "Ann made a $3$-step staircase using $18$ toothpicks as shown in the figure. How many toothpicks does she need to add to complete a $5$-step staircase?\n[asy]\nsize(150);\ndefaultpen(linewidth(0.8));\npath h = ellipse((0.5,0),0.45,0.015), v = ellipse((0,0.5),0.015,0.45);\nfor(in... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 5003,
"solution": "To solve this problem, we first need to understand the pattern in the number of toothpicks required to build each step of the staircase.\n\n1. **Identify the pattern:**\n - A $1$-step staircase requires $4$ toothpicks.\n - A $2$-step staircase requires $10$ toothpicks.\n - A $3$-... |
DeepScaleR | [
{
"role": "user",
"content": "Anna Lisa bought two dozen apples for $\\$$15.60. At that same rate, how much would three dozen of these apples cost?"
}
] | math | {
"style": "rule",
"ground_truth": "\\$23.40"
} | {
"index": 5004,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Anna and Aaron walk along paths formed by the edges of a region of squares. How far did they walk in total?"
}
] | math | {
"style": "rule",
"ground_truth": "640 \\text{ m}"
} | {
"index": 5005,
"solution": "Each square has an area of \\( 400 \\text{ m}^2 \\) and side length 20 m. Anna's path is 400 m and Aaron's path is 240 m. Therefore, the total distance walked is \\( 400 + 240 = 640 \\text{ m} \\)."
} |
DeepScaleR | [
{
"role": "user",
"content": "Anna enjoys dinner at a restaurant in Washington, D.C., where the sales tax on meals is 10%. She leaves a 15% tip on the price of her meal before the sales tax is added, and the tax is calculated on the pre-tip amount. She spends a total of 27.50 dollars for dinner. What is the... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 5006,
"solution": "Let $x$ be the cost of Anna's dinner before tax and tip.\n\n1. **Calculate the tax:** The tax rate is 10%, so the tax amount is $\\frac{10}{100}x = 0.1x$.\n\n2. **Calculate the tip:** The tip is 15% of the pre-tax meal cost, so the tip amount is $\\frac{15}{100}x = 0.15x$.\n\n3. **Tota... |
DeepScaleR | [
{
"role": "user",
"content": "Anna flips an unfair coin 10 times. The coin has a $\\frac{1}{3}$ probability of coming up heads and a $\\frac{2}{3}$ probability of coming up tails. What is the probability that she flips exactly 7 tails?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5120}{19683}"
} | {
"index": 5007,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Anna thinks of an integer that is not a multiple of three, not a perfect square, and the sum of its digits is a prime number. What could the integer be?"
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 5008,
"solution": "Since 12 and 21 are multiples of 3 (12 = 4 \\times 3 and 21 = 7 \\times 3), the answer is not 12 or 21. 16 is a perfect square (16 = 4 \\times 4) so the answer is not 16. The sum of the digits of 26 is 8, which is not a prime number, so the answer is not 26. Since 14 is not a multiple ... |
DeepScaleR | [
{
"role": "user",
"content": "Anna thinks of an integer. It is not a multiple of three. It is not a perfect square. The sum of its digits is a prime number. What could be the integer that Anna is thinking of?"
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 5009,
"solution": "12 and 21 are multiples of 3 (12 = 4 \\times 3 and 21 = 7 \\times 3) so the answer is not (A) or (D). 16 is a perfect square (16 = 4 \\times 4) so the answer is not (C). The sum of the digits of 26 is 8, which is not a prime number, so the answer is not (E). Since 14 is not a multiple ... |
DeepScaleR | [
{
"role": "user",
"content": "Anna walked at a constant rate. If she walked 600 metres in 4 minutes, how far did she walk in 6 minutes?"
}
] | math | {
"style": "rule",
"ground_truth": "900"
} | {
"index": 5010,
"solution": "If Anna walked 600 metres in 4 minutes, then she walked $\\frac{600}{4}=150$ metres each minute. Therefore, in 6 minutes, she walked $6 \\times 150=900$ metres."
} |
DeepScaleR | [
{
"role": "user",
"content": "Anna, Stephanie, and James all start running around a track at 8:00. Anna completes a lap every 4 minutes, Stephanie finishes a lap every 7 minutes, and James finishes a lap every 6 minutes. What is the earliest time when all three meet back at the beginning?"
}
] | math | {
"style": "rule",
"ground_truth": "9:24"
} | {
"index": 5011,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Anne-Marie has a deck of 16 cards, each with a distinct positive factor of 2002 written on it. She shuffles the deck and begins to draw cards from the deck without replacement. She stops when there exists a nonempty subset of the cards in her hand whose numbers multiply to a pe... | math | {
"style": "rule",
"ground_truth": "\\frac{837}{208}"
} | {
"index": 5012,
"solution": "Note that $2002=2 \\cdot 7 \\cdot 11 \\cdot 13$, so that each positive factor of 2002 is included on exactly one card. Each card can identified simply by whether or not it is divisible by each of the 4 primes, and we can uniquely achieve all of the $2^{4}$ possibilities. Also, when con... |
DeepScaleR | [
{
"role": "user",
"content": "Annie and Bonnie are running laps around a $400$-meter oval track. They started together, but Annie has pulled ahead, because she runs $25\\%$ faster than Bonnie. How many laps will Annie have run when she first passes Bonnie?"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 5013,
"solution": "\n#### Step-by-step Analysis:\n\n1. **Understanding the Problem:**\n Annie and Bonnie start running together on a 400-meter track. Annie runs 25% faster than Bonnie. We need to determine how many laps Annie will have run when she first passes Bonnie.\n\n2. **Setting Up the Relationsh... |
DeepScaleR | [
{
"role": "user",
"content": "Another professor enters the same room and says, 'Each of you has to write down an integer between 0 and 200. I will then compute $X$, the number that is 3 greater than half the average of all the numbers that you will have written down. Each student who writes down the number ... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 5014,
"solution": "Use the same logic to get 7. Note 6 and 8 do not work."
} |
DeepScaleR | [
{
"role": "user",
"content": "Another trapezoid \\(ABCD\\) has \\(AD\\) parallel to \\(BC\\). \\(AC\\) and \\(BD\\) intersect at \\(P\\). If \\(\\frac{[ADP]}{[BCP]} = \\frac{1}{2}\\), find \\(\\frac{[ADP]}{[ABCD]}\\). (Here, the notation \\(\\left[P_1 \\cdots P_n\\right]\\) denotes the area of the polygon \... | math | {
"style": "rule",
"ground_truth": "3 - 2\\sqrt{2}"
} | {
"index": 5015,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Antal and Béla start from home on their motorcycles heading towards Cegléd. After traveling one-fifth of the way, Antal for some reason turns back. As a result, he accelerates and manages to increase his speed by one quarter. He immediately sets off again from home. Béla, conti... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 5016,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Anthony made $5$ of his first $12$ free throw attempts. If he makes $2/3$ of his next $24$ attempts, by how many percentage points will he increase his overall success rate percentage? Express your answer to the nearest whole number."
}
] | math | {
"style": "rule",
"ground_truth": "17\\%"
} | {
"index": 5017,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Antoine, Benoît, Claude, Didier, Étienne, and Françoise go to the cinéma together to see a movie. The six of them want to sit in a single row of six seats. But Antoine, Benoît, and Claude are mortal enemies and refuse to sit next to either of the other two. How many different a... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"index": 5018,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Anton colors a cell in a \\(4 \\times 50\\) rectangle. He then repeatedly chooses an uncolored cell that is adjacent to at most one already colored cell. What is the maximum number of cells that can be colored?"
}
] | math | {
"style": "rule",
"ground_truth": "150"
} | {
"index": 5019,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Anton has two species of ants, Species A and Species B, in his ant farm. The two species are identical in appearance, but Anton knows that every day, there are twice as many ants of Species A than before, while there are three times as many ants of Species B. On Day 0, Anton co... | math | {
"style": "rule",
"ground_truth": "608"
} | {
"index": 5020,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Anton makes custom watches for a jewelry store. Each watch consists of a bracelet, a gemstone, and a clasp.\n\nBracelets are available in silver, gold, and steel. Anton has precious stones: cubic zirconia, emerald, quartz, diamond, and agate - and clasps: classic, butterfly, bu... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 5021,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Anton ran down a moving escalator and counted 30 steps. Then he decided to run up the same escalator at the same speed relative to the escalator and counted 150 steps. How many steps did he count while descending with a policeman on a stationary escalator?"
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 5022,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Anton thought of a three-digit number, and Alex is trying to guess it. Alex successively guessed the numbers 109, 704, and 124. Anton observed that each of these numbers matches the thought number exactly in one digit place. What number did Anton think of?"
}
] | math | {
"style": "rule",
"ground_truth": "729"
} | {
"index": 5023,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Anton writes down all positive integers that are divisible by 2. Berta writes down all positive integers that are divisible by 3. Clara writes down all positive integers that are divisible by 4. The orderly Dora notes the numbers written by the others. She arranges these number... | math | {
"style": "rule",
"ground_truth": "3026"
} | {
"index": 5024,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Anton, Boris, Vadim, Gena, Dima, and Egor went to the cinema. They bought 6 consecutive seats in one row. Anton and Boris want to sit next to each other, while Vadim and Gena do not want to sit next to each other. How many ways can they sit in their seats considering these pref... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"index": 5025,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Anton, Vasya, Sasha, and Dima were driving from city A to city B, each taking turns at the wheel. The entire journey was made at a constant speed.\n\nAnton drove the car for half the time Vasya did, and Sasha drove for as long as Anton and Dima together. Dima was at the wheel f... | math | {
"style": "rule",
"ground_truth": "0.4"
} | {
"index": 5026,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Antonette gets $70\\%$ on a 10-problem test, $80\\%$ on a 20-problem test and $90\\%$ on a 30-problem test. If the three tests are combined into one 60-problem test, which percent is her overall score, rounded to the nearest percent?"
}
] | math | {
"style": "rule",
"ground_truth": "83\\%"
} | {
"index": 5027,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Antônio needs to find a code with 3 different digits \\( A, B, C \\). He knows that \\( B \\) is greater than \\( A \\), \\( A \\) is less than \\( C \\), and also:\n\\[\n\\begin{array}{cccc}\n & B & B \\\\\n+ & A & A \\\\\n\\hline\n& C & C \\\\\n\\end{array} = 242\n\\]\n\nWhat... | math | {
"style": "rule",
"ground_truth": "232"
} | {
"index": 5028,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Any five points are taken inside or on a rectangle with dimensions 2 by 1. Let b be the smallest possible number with the property that it is always possible to select one pair of points from these five such that the distance between them is equal to or less than b. What is b?"... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{2}"
} | {
"index": 5029,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Any five points are taken inside or on a square with side length $1$. Let a be the smallest possible number with the property that it is always possible to select one pair of points from these five such that the distance between them is equal to or less than $a$. Then $a$ is:\n... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"index": 5030,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Any seven points are taken inside or on a square with side length $2$. Determine $b$, the smallest possible number with the property that it is always possible to select one pair of points from these seven such that the distance between them is equal to or less than $b$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 5031,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Any six points are taken inside or on a rectangle with dimensions $1 \\times 2$. Let $b$ be the smallest possible value such that it is always possible to select one pair of points from these six such that the distance between them is equal to or less than $b$. Determine the va... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{2}"
} | {
"index": 5032,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Any six points are taken inside or on a rectangle with dimensions $2 \\times 1$. Let $b$ be the smallest possible number with the property that it is always possible to select one pair of points from these six such that the distance between them is equal to or less than $b$. De... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{2}"
} | {
"index": 5033,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Any six points are taken inside or on an equilateral triangle with side length 1. Let $b$ be the smallest possible number with the property that it is always possible to select one pair of points from these six such that the distance between them is equal to or less than $b$. T... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 5034,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Any type of nature use affects at least one of the natural resources, including lithogenic base, soil, water, air, plant world, and animal world. Types that affect the same set of resources belong to the same type. Research has shown that types of nature use developed in the la... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 5035,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Anya wants to buy ice cream that costs 19 rubles. She has two 10-ruble coins, two 5-ruble coins, and one 2-ruble coin in her pocket. Anya randomly picks three coins from her pocket without looking. Find the probability that the selected coins will be enough to pay for the ice c... | math | {
"style": "rule",
"ground_truth": "0.4"
} | {
"index": 5036,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Anya, Vanya, Dania, Sanya, and Tanya were collecting apples. It turned out that each of them collected an integer percentage of the total number of apples, and all these percentages are different and greater than zero. What is the minimum number of apples that could have been c... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 5037,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Anya, Vanya, Danya, and Tanya collected apples. Each of them collected a whole number percentage from the total number of apples, and all these numbers are distinct and more than zero. Then Tanya, who collected the most apples, ate her apples. After that, each of the remaining ... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 5038,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Apartment rentals in Fairview run $\\$$0.90 per square foot. Jillian has determined that she can afford $\\$$630 per month for rent. What is the largest apartment, in square feet, she should consider at the given rate?"
}
] | math | {
"style": "rule",
"ground_truth": "700"
} | {
"index": 5039,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Appending three digits at the end of 2007, one obtains an integer \\(N\\) of seven digits. In order to get \\(N\\) to be the minimal number which is divisible by 3, 5, and 7 simultaneously, what are the three digits that one would append?"
}
] | math | {
"style": "rule",
"ground_truth": "075"
} | {
"index": 5040,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Apples are sold at the farmer's market at a rate of $\\$4$ per five pounds. According to this rate, how many dollars does it cost to buy 15 pounds of apples?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 5041,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Applied to a bill for $\\$10,000$ the difference between a discount of $40\\%$ and two successive discounts of $36\\%$ and $4\\%$, expressed in dollars, is:"
}
] | math | {
"style": "rule",
"ground_truth": "144"
} | {
"index": 5042,
"solution": "1. **Calculate the single discount of 40% on $\\textdollar{10,000}$:**\n - A 40% discount means the remaining amount to be paid is 100% - 40% = 60% of the original price.\n - Calculation: \n \\[\n 10,000 \\times 0.6 = \\textdollar{6,000}\n \\]\n\n2. **Calculate the succ... |
DeepScaleR | [
{
"role": "user",
"content": "Approximate the increase in the volume of a cylinder with a height of \\( H = 40 \\) cm and a base radius of \\( R = 30 \\) cm when the radius is increased by \\( 0.5 \\) cm."
}
] | math | {
"style": "rule",
"ground_truth": "1200\\pi"
} | {
"index": 5043,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Approximate the number $0.00356$ to the nearest ten-thousandth: $0.00356 \\approx$____."
}
] | math | {
"style": "rule",
"ground_truth": "0.0036"
} | {
"index": 5044,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Approximate the reading indicated by the arrow in the diagram of a measuring device."
}
] | math | {
"style": "rule",
"ground_truth": "42.3"
} | {
"index": 5045,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "April has four different basil plants and four different tomato plants. In how many ways can she arrange the plants in a row if she puts all the tomato plants next to each other?"
}
] | math | {
"style": "rule",
"ground_truth": "2880"
} | {
"index": 5046,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Aquatic plants require a specific type of nutrient solution. Given that each time $a (1 \\leqslant a \\leqslant 4$ and $a \\in R)$ units of the nutrient solution are released, its concentration $y (\\text{g}/\\text{L})$ changes over time $x (\\text{days})$ according to the func... | math | {
"style": "rule",
"ground_truth": "24-16\\sqrt{2}"
} | {
"index": 5047,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Ara and Shea were once the same height. Since then Shea has grown 20% while Ara has grown half as many inches as Shea. Shea is now 60 inches tall. How tall, in inches, is Ara now?"
}
] | math | {
"style": "rule",
"ground_truth": "55"
} | {
"index": 5048,
"solution": "1. **Determine Shea's original height**: \n Given that Shea has grown by 20% and is now 60 inches tall, we can set up the equation for her original height $x$:\n \\[\n 1.2x = 60\n \\]\n Solving for $x$, we divide both sides by 1.2:\n \\[\n x = \\frac{60}{1.2} = 50\n \\]... |
DeepScaleR | [
{
"role": "user",
"content": "Arc $AC$ is a quarter-circle with center $B$. The shaded region $ABC$ is \"rolled\" along a straight board $PQ$ until it reaches its original orientation for the first time with point $B$ landing at point $B^{\\prime}$. If $BC = \\frac{2}{\\pi}$ cm, what is the length of the pa... | math | {
"style": "rule",
"ground_truth": "3\\text{ cm}"
} | {
"index": 5049,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Archer Zhang Qiang has the probabilities of hitting the 10-ring, 9-ring, 8-ring, 7-ring, and below 7-ring in a shooting session as 0.24, 0.28, 0.19, 0.16, and 0.13, respectively. Calculate the probability that this archer in a single shot:\n(1) Hits either the 10-ring or the 9-... | math | {
"style": "rule",
"ground_truth": "0.29"
} | {
"index": 5050,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Are there integers $m$ and $n$ such that\n\\[5m^2 - 6mn + 7n^2 = 1985 \\ ?\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\text{No}"
} | {
"index": 5051,
"solution": "\nTo determine whether there are integers \\( m \\) and \\( n \\) such that\n\n\\[\n5m^2 - 6mn + 7n^2 = 1985,\n\\]\n\nwe begin by analyzing the quadratic form. We can rewrite the equation as:\n\n\\[\n5m^2 - 6mn + 7n^2.\n\\]\n\nFirst, let's complete the square with respect to \\( m \\) ... |
DeepScaleR | [
{
"role": "user",
"content": "Aria and Bianca walk at different, but constant speeds. They each begin at 8:00 a.m. from the opposite ends of a road and walk directly toward the other's starting point. They pass each other at 8:42 a.m. Aria arrives at Bianca's starting point at 9:10 a.m. When does Bianca arr... | math | {
"style": "rule",
"ground_truth": "9:45 a.m."
} | {
"index": 5052,
"solution": "Let $A$ be Aria's starting point, $B$ be Bianca's starting point, and $M$ be their meeting point. It takes Aria 42 minutes to walk from $A$ to $M$ and 28 minutes from $M$ to $B$. Since Aria walks at a constant speed, then the ratio of the distance $A M$ to the distance $M B$ is equal t... |
DeepScaleR | [
{
"role": "user",
"content": "Arithmetic sequences $\\left(a_n\\right)$ and $\\left(b_n\\right)$ have integer terms with $a_1=b_1=1<a_2 \\le b_2$ and $a_n b_n = 2010$ for some $n$. What is the largest possible value of $n$?"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 5053,
"solution": "\n1. **Identify the form of the sequences**: Given that $\\left(a_n\\right)$ and $\\left(b_n\\right)$ are arithmetic sequences with integer terms and $a_1 = b_1 = 1$, we can express the $n$-th terms of these sequences as:\n \\[\n a_n = 1 + (n-1)x \\quad \\text{and} \\quad b_n = 1 +... |
DeepScaleR | [
{
"role": "user",
"content": "Arjun and Beth play a game in which they take turns removing one brick or two adjacent bricks from one \"wall\" among a set of several walls of bricks, with gaps possibly creating new walls. The walls are one brick tall. For example, a set of walls of sizes $4$ and $2$ can be c... | math | {
"style": "rule",
"ground_truth": "$(6,2,1)$"
} | {
"index": 5054,
"solution": "To solve this problem, we use the concept of nim-values from combinatorial game theory. The nim-value of a game configuration determines whether a position is winning or losing. A position with a nim-value of $0$ is losing (if both players play optimally), and any other nim-value is wi... |
DeepScaleR | [
{
"role": "user",
"content": "Arnaldo claimed that one billion is the same as one million millions. Professor Piraldo corrected him and said, correctly, that one billion is the same as one thousand millions. What is the difference between the correct value of one billion and Arnaldo's assertion?"
}
] | math | {
"style": "rule",
"ground_truth": "999000000000"
} | {
"index": 5055,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Arnaldo claimed that one billion is the same as one million millions. Professor Piraldo corrected him and said, correctly, that one billion is the same as one thousand millions. What is the difference between the correct value of one billion and Arnaldo's assertion?\n(a) 1000\n... | math | {
"style": "rule",
"ground_truth": "999000000000"
} | {
"index": 5056,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Arnold and Kevin are playing a game in which Kevin picks an integer \\(1 \\leq m \\leq 1001\\), and Arnold is trying to guess it. On each turn, Arnold first pays Kevin 1 dollar in order to guess a number \\(k\\) of Arnold's choice. If \\(m \\geq k\\), the game ends and he pays ... | math | {
"style": "rule",
"ground_truth": "859"
} | {
"index": 5057,
"solution": "We let \\(f(n)\\) denote the smallest amount we can guarantee to pay at most if Arnold's first choice is \\(n\\). For each \\(k<n\\), if Arnold's first choice is \\(k+1\\), in both worst case scenarios, he could end up paying either \\(n-k\\) or \\(11+f(k)\\). It is then clear that \\(... |
DeepScaleR | [
{
"role": "user",
"content": "Arnold is studying the prevalence of three health risk factors, denoted by A, B, and C, within a population of men. For each of the three factors, the probability that a randomly selected man in the population has only this risk factor (and none of the others) is 0.1. For any t... | math | {
"style": "rule",
"ground_truth": "76"
} | {
"index": 5058,
"solution": "We first assume a population of $100$ to facilitate solving. Then we simply organize the statistics given into a Venn diagram.\n[asy] pair A,B,C,D,E,F,G; A=(0,55); B=(60,55); C=(60,0); D=(0,0); draw(A--B--C--D--A); E=(30,35); F=(20,20); G=(40,20); draw(circle(E,15)); draw(circle(F,15))... |
DeepScaleR | [
{
"role": "user",
"content": "Around a circle, an isosceles trapezoid \\(ABCD\\) is described. The side \\(AB\\) touches the circle at point \\(M\\), and the base \\(AD\\) touches the circle at point \\(N\\). The segments \\(MN\\) and \\(AC\\) intersect at point \\(P\\), such that \\(NP: PM=2\\). Find the r... | math | {
"style": "rule",
"ground_truth": "3:1"
} | {
"index": 5059,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Around a circular table, there are 18 girls seated, 11 dressed in blue and 7 dressed in red. Each girl is asked if the girl to her right is dressed in blue, and each one responds with either yes or no. It is known that a girl tells the truth only when both of her neighbors, the... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 5060,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Around the circle, a pentagon is described, the lengths of the sides of which are integers, and the first and third sides are equal to 1. Into which segments does the point of tangency divide the second side?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 5061,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Around the outside of a $4$ by $4$ square, construct four semicircles (as shown in the figure) with the four sides of the square as their diameters. Another square, $ABCD$, has its sides parallel to the corresponding sides of the original square, and each side of $ABCD$ is tang... | math | {
"style": "rule",
"ground_truth": "64"
} | {
"index": 5062,
"solution": "1. **Identify the radius of the semicircles**: The original square has a side length of $4$. Each semicircle is constructed with the side of the square as its diameter. Therefore, the radius of each semicircle is half the side length of the square, which is $\\frac{4}{2} = 2$.\n\n2. **... |
DeepScaleR | [
{
"role": "user",
"content": "Around the outside of a $6$ by $6$ square, construct four semicircles with the four sides of the square as their diameters. Another square, $ABCD$, has its sides parallel to the corresponding sides of the original square, and each side of $ABCD$ is tangent to one of the semicir... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"index": 5063,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Around the outside of a $6$ by $6$ square, construct four semicircles with the four sides of the square as their diameters. Another square, $EFGH$, has its sides parallel to the corresponding sides of the larger square, and each side of $EFGH$ is tangent to one of the semicircl... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"index": 5064,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Arrange 1, 2, 3, a, b, c in a row such that letter 'a' is not at either end and among the three numbers, exactly two are adjacent. The probability is $\\_\\_\\_\\_\\_\\_$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"index": 5065,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Arrange 2002 students numbered from 1 to 2002 in a row from left to right. Counting from left to right from 1 to 11, the student who is counted as 11 remains in place, while the others leave the row. Then, the remaining students count from left to right from 1 to 11 again, and ... | math | {
"style": "rule",
"ground_truth": "1331"
} | {
"index": 5066,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Arrange 3 boys and 4 girls in a row. Calculate the number of different arrangements under the following conditions:\n(1) Person A and Person B must stand at the two ends; \n(2) All boys must stand together; \n(3) No two boys stand next to each other; \n(4) Exactly one person... | math | {
"style": "rule",
"ground_truth": "1200"
} | {
"index": 5067,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Arrange 3 male students and 4 female students in a row. Under the following different requirements, calculate the number of different arrangement methods: \n(1) Person A and Person B must stand at the two ends; \n(2) All male students must be grouped together; \n(3) Male stu... | math | {
"style": "rule",
"ground_truth": "1200"
} | {
"index": 5068,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Arrange 3 volunteer teachers to 4 schools, with at most 2 people per school. How many different distribution plans are there? (Answer with a number)"
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 5069,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Arrange 5 people to be on duty from Monday to Friday, with each person on duty for one day and one person arranged for each day. The conditions are: A and B are not on duty on adjacent days, while B and C are on duty on adjacent days. The number of different arrangements is $\\... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 5070,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Arrange 6 volunteers for 3 different tasks, each task requires 2 people. Due to the work requirements, A and B must work on the same task, and C and D cannot work on the same task. How many different arrangements are there?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 5071,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Arrange 7 staff members to be on duty from May 1st to May 7th. Each person works for one day, among them, person A and person B are not scheduled on May 1st and 2nd. The total number of different arrangements is $\\boxed{\\text{\\_\\_\\_\\_\\_\\_\\_\\_}}$."
}
] | math | {
"style": "rule",
"ground_truth": "2400"
} | {
"index": 5072,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Arrange 8 people, including A and B, to work for 4 days. If 2 people are arranged each day, the probability that A and B are arranged on the same day is ______. (Express the result as a fraction)"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{7}"
} | {
"index": 5073,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Arrange all positive integers whose digits sum to 8 in ascending order to form a sequence $\\{a_n\\}$, called the $P$ sequence. Then identify the position of 2015 within this sequence."
}
] | math | {
"style": "rule",
"ground_truth": "83"
} | {
"index": 5074,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Arrange all the four-digit numbers formed using $1, 2, 3,$ and $4$, each used exactly once, in ascending order. What is the difference between the 23rd number and the 21st number?"
}
] | math | {
"style": "rule",
"ground_truth": "99"
} | {
"index": 5075,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Arrange for 7 staff members to be on duty from May 1st to May 7th. Each person is on duty for one day, with both members A and B not being scheduled on May 1st and 2nd. The total number of different scheduling methods is $\\_\\_\\_\\_\\_\\_\\_$."
}
] | math | {
"style": "rule",
"ground_truth": "2400"
} | {
"index": 5076,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Arrange numbers $ 1,\\ 2,\\ 3,\\ 4,\\ 5$ in a line. Any arrangements are equiprobable. Find the probability such that the sum of the numbers for the first, second and third equal to the sum of that of the third, fourth and fifth. Note that in each arrangement each number are... | math | {
"style": "rule",
"ground_truth": "1/15"
} | {
"index": 5077,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Arrange positive integers that are neither perfect squares nor perfect cubes (excluding 0) in ascending order as 2, 3, 5, 6, 7, 10, ..., and determine the 1000th number in this sequence."
}
] | math | {
"style": "rule",
"ground_truth": "1039"
} | {
"index": 5078,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Arrange the $n$ consecutive positive integers from 1 to $n$ (where $n > 1$) in a sequence such that the sum of each pair of adjacent terms is a perfect square. Find the minimum value of $n$."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 5079,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Arrange the 7 numbers $39, 41, 44, 45, 47, 52, 55$ in a sequence such that the sum of any three consecutive numbers is a multiple of 3. What is the maximum value of the fourth number in all such arrangements?"
}
] | math | {
"style": "rule",
"ground_truth": "47"
} | {
"index": 5080,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Arrange the 9 numbers 12, 13, ..., 20 in a row such that the sum of every three consecutive numbers is a multiple of 3. How many such arrangements are there?"
}
] | math | {
"style": "rule",
"ground_truth": "216"
} | {
"index": 5081,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Arrange the digits \\(1, 2, 3, 4, 5, 6, 7, 8, 9\\) in some order to form a nine-digit number \\(\\overline{\\text{abcdefghi}}\\). If \\(A = \\overline{\\text{abc}} + \\overline{\\text{bcd}} + \\overline{\\text{cde}} + \\overline{\\text{def}} + \\overline{\\text{efg}} + \\overli... | math | {
"style": "rule",
"ground_truth": "4648"
} | {
"index": 5082,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Arrange the letters a, a, b, b, c, c into three rows and two columns, such that in each row and each column, the letters are different. How many different arrangements are there?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 5083,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Arrange the letters a, a, b, b, c, c into three rows and two columns, with the requirement that each row has different letters and each column also has different letters, and find the total number of different arrangements."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 5084,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Arrange the natural numbers according to the following triangular pattern. What is the sum of the numbers in the 10th row?\n\n```\n 1\n 2 3 4\n 5 6 7 8 9\n 10 11 12 13 14 15 16\n17 18 19 20 21 22 23 24 25\n...\n```"
}
] | math | {
"style": "rule",
"ground_truth": "1729"
} | {
"index": 5085,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Arrange the numbers $2011, \\sqrt{2011}, 2011^{2}$ in increasing order."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2011}, 2011, 2011^{2}"
} | {
"index": 5086,
"solution": "Since $2011^{2}=4044121$ and $\\sqrt{2011} \\approx 44.8$, then the list of numbers in increasing order is $\\sqrt{2011}, 2011, 2011^{2}$. (If $n$ is a positive integer with $n>1$, then $n^{2}>n$ and $\\sqrt{n}<n$, so the list $\\sqrt{n}, n, n^{2}$ is always in increasing order.)"
} |
DeepScaleR | [
{
"role": "user",
"content": "Arrange the numbers 1 to 9 in a row from left to right such that every three consecutive numbers form a three-digit number that is a multiple of 3. How many arrangements are there?"
}
] | math | {
"style": "rule",
"ground_truth": "1296"
} | {
"index": 5087,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Arrange the numbers 3, 4, 5, 6 into two natural numbers A and B, so that the product A×B is maximized. Find the value of A×B."
}
] | math | {
"style": "rule",
"ground_truth": "\\left( 3402 \\right)"
} | {
"index": 5088,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Arrange the numbers from 1 to 25 in a random order. Then subtract them sequentially from 1, 2, 3, ..., 25, always subtracting the smaller number from the larger number. What is the maximum number of even differences that can appear among these 25 differences?"
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 5089,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Arrange the numbers in the set \\(\\left\\{2^{x}+2^{y} \\mid x, y\\ \\text{are non-negative integers,}\\ x < y\\right\\}\\) in ascending order. What is the 60th number? (Answer in digits)."
}
] | math | {
"style": "rule",
"ground_truth": "2064"
} | {
"index": 5090,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Arrange the positive integers in the following number matrix:\n\\begin{tabular}{lllll}\n1 & 2 & 5 & 10 & $\\ldots$ \\\\\n4 & 3 & 6 & 11 & $\\ldots$ \\\\\n9 & 8 & 7 & 12 & $\\ldots$ \\\\\n16 & 15 & 14 & 13 & $\\ldots$ \\\\\n$\\ldots$ & $\\ldots$ & $\\ldots$ & $\\ldots$ & $\\ldot... | math | {
"style": "rule",
"ground_truth": "421"
} | {
"index": 5091,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Arrange the positive integers whose digits sum to 4 in ascending order. Which position does the number 2020 occupy in this sequence?"
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 5092,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Arrange the positive odd numbers as shown in the pattern below. What is the 5th number from the left in the 21st row?\n\n$$\n1 \\\\\n3 \\quad 5 \\quad 7 \\\\\n9 \\quad 11 \\quad 13 \\quad 15 \\quad 17 \\\\\n19 \\quad 21 \\quad 23 \\quad 25 \\quad 27 \\quad 29 \\quad 31 \\\\\n\\... | math | {
"style": "rule",
"ground_truth": "809"
} | {
"index": 5093,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Arrange the sequence $\\{2n+1\\} (n\\in\\mathbb{N}^{*})$ sequentially in brackets with one number in the first bracket, two numbers in the second bracket, three numbers in the third bracket, four numbers in the fourth bracket, one number in the fifth bracket, and so on in a cyc... | math | {
"style": "rule",
"ground_truth": "2392"
} | {
"index": 5094,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Arrange the sequence $\\{2n+1\\}$ ($n\\in\\mathbb{N}^*$), sequentially in brackets such that the first bracket contains one number, the second bracket two numbers, the third bracket three numbers, the fourth bracket four numbers, the fifth bracket one number, and so on in a cyc... | math | {
"style": "rule",
"ground_truth": "403"
} | {
"index": 5095,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Art, Bella, Carla, and Dan each bake cookies of different shapes with the same thickness. Art's cookies are rectangles: 4 inches by 6 inches, while Bella's cookies are equilateral triangles with a side length of 5 inches. Carla's cookies are squares with a side length of 4 inch... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 5096,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Arthur walks six blocks east and then 12 blocks north. If each block is one-third of a mile, how many miles did he walk?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 5097,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Arthur, Bob, and Carla each choose a three-digit number. They each multiply the digits of their own numbers. Arthur gets 64, Bob gets 35, and Carla gets 81. Then, they add corresponding digits of their numbers together. The total of the hundreds place is 24, that of the tens ... | math | {
"style": "rule",
"ground_truth": "182"
} | {
"index": 5098,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Arturo has an equal number of $\\$5$ bills, of $\\$10$ bills, and of $\\$20$ bills. The total value of these bills is $\\$700$. How many $\\$5$ bills does Arturo have?"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 5099,
"solution": "Since Arturo has an equal number of $\\$5$ bills, of $\\$10$ bills, and of $\\$20$ bills, then we can divide Arturo's bills into groups, each of which contains one $\\$5$ bill, one $\\$10$ bill, and one $\\$20$ bill. The value of the bills in each group is $\\$5 + \\$10 + \\$20 = \\$35... |
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