data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Allen and Yang want to share the numbers \\(1,2,3,4,5,6,7,8,9,10\\). How many ways are there to split all ten numbers among Allen and Yang so that each person gets at least one number, and either Allen's numbers or Yang's numbers sum to an even number?"
}
] | math | {
"style": "rule",
"ground_truth": "1022"
} | {
"index": 4500,
"solution": "Since the sum of all of the numbers is odd, exactly one of Allen's sum and Yang's sum must be odd. Therefore any way of splitting the numbers up where each person receives at least one number is valid, so the answer is \\(2^{10}-2=1022\\)."
} |
DeepScaleR | [
{
"role": "user",
"content": "Alli rolls a fair $8$-sided die twice. What is the probability of rolling numbers that differ by $3$ in her first two rolls? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"index": 4501,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alli rolls a standard $6$-sided die twice. What is the probability of rolling integers that differ by $2$ on her first two rolls? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{2}{9}"
} | {
"index": 4502,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alli rolls a standard $8$-sided die twice. What is the probability of rolling integers that differ by $3$ on her first two rolls? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"index": 4503,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alli rolls a standard 8-sided die twice. What is the probability of rolling integers that differ by 3 on her first two rolls? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{8}"
} | {
"index": 4504,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Allie and Betty play a game where they take turns rolling a standard die. If a player rolls $n$, she is awarded $f(n)$ points, where \\[f(n) = \\left\\{\n\\begin{array}{cl} 6 & \\text{ if }n\\text{ is a multiple of 2 and 3}, \\\\\n2 & \\text{ if }n\\text{ is only a multiple of... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 4505,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Allie and Betty play a game where they take turns rolling a standard die. If a player rolls $n$, she is awarded $g(n)$ points, where \\[g(n) = \\left\\{\n\\begin{array}{cl}\n8 & \\text{ if } n \\text{ is a multiple of 3 and 4}, \\\\\n3 & \\text{ if } n \\text{ is only a multipl... | math | {
"style": "rule",
"ground_truth": "84"
} | {
"index": 4506,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Allison, Brian and Noah each have a 6-sided cube. All of the faces on Allison's cube have a 5. The faces on Brian's cube are numbered 1, 2, 3, 4, 5 and 6. Three of the faces on Noah's cube have a 2 and three of the faces have a 6. All three cubes are rolled. What is the probabi... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 4507,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Allison, Brian, and Noah each have a die. All of the faces on Allison's die have a 5. The faces on Brian's die are numbered 1, 2, 3, 4, 4, 5, 5, and 6. Noah's die has an 8-sided die with the numbers 2, 2, 6, 6, 3, 3, 7, and 7. All three dice are rolled. What is the probability ... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{16}"
} | {
"index": 4508,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alloy $A$ of two metals has a mass of 6 kg, with the first metal being twice as abundant as the second metal. When placed in a container of water, it exerts a force of $30\\ \\mathrm{N}$ on the bottom. Alloy $B$ of the same metals has a mass of 3 kg, with the first metal being ... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 4509,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Almondine has a bag with $N$ balls, each of which is red, white, or blue. If Almondine picks three balls from the bag without replacement, the probability that she picks one ball of each color is larger than 23 percent. Compute the largest possible value of $\\left\\lfloor\\fra... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"index": 4510,
"solution": "If $k=\\left\\lfloor\\frac{N}{3}\\right\\rfloor$, then the maximum possible probability is $\\frac{6 k^{3}}{(3 k)(3 k-1)(3 k-2)}$. with equality when there are $k$ balls of each of the three colors. Going from $3 k \\rightarrow 3 k+1$ replaces $\\frac{k}{3 k-2} \\rightarrow \\frac{k+1}... |
DeepScaleR | [
{
"role": "user",
"content": "Along a road, there are 25 poles in a single row. Sometimes a siskin lands on one of the poles, and immediately a siskin flies off from one of the neighboring poles (if there was at least one siskin sitting on the neighboring poles at that moment). Also, each pole can hold no m... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 4511,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Along a straight alley at equal intervals, there are 400 lampposts, numbered sequentially from 1 to 400. At the same time, Alla and Boris started walking towards each other from opposite ends of the alley with different constant speeds (Alla started from the first lamppost and ... | math | {
"style": "rule",
"ground_truth": "163"
} | {
"index": 4512,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Along a straight alley, there are 400 streetlights placed at equal intervals, numbered consecutively from 1 to 400. Alla and Boris start walking towards each other from opposite ends of the alley at the same time but with different constant speeds (Alla from the first streetlig... | math | {
"style": "rule",
"ground_truth": "163"
} | {
"index": 4513,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Along an alley, 75 trees consisting of maples and larches were planted in a single row. It is known that there are no two maples with exactly 5 trees between them. What is the maximum number of maples that could have been planted along the alley?"
}
] | math | {
"style": "rule",
"ground_truth": "39"
} | {
"index": 4514,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Along the school corridor hangs a Christmas garland consisting of red and blue bulbs. Next to each red bulb, there must be a blue bulb. What is the maximum number of red bulbs that can be in this garland if there are a total of 50 bulbs?"
}
] | math | {
"style": "rule",
"ground_truth": "33"
} | {
"index": 4515,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Along the southern shore of a boundless sea stretches an archipelago of an infinite number of islands. The islands are connected by an endless chain of bridges, and each island is connected by a bridge to the shore. In the event of a strong earthquake, each bridge independently... | math | {
"style": "rule",
"ground_truth": "2/3"
} | {
"index": 4516,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alpha and Beta both took part in a two-day problem-solving competition. At the end of the second day, each had attempted questions worth a total of 500 points. Alpha scored 160 points out of 300 points attempted on the first day, and scored 140 points out of 200 points attempte... | math | {
"style": "rule",
"ground_truth": "849"
} | {
"index": 4517,
"solution": "Let $q$ be the number of questions Beta takes on day 1 and $a$ be the number he gets right. Let $b$ be the number he gets right on day 2.\n These inequalities follow: \\[\\frac{a}{q} < \\frac{160}{300} = \\frac{8}{15}\\] \\[\\frac{b}{500-q} < \\frac{140}{200} = \\frac{7}{10}\\] Solving... |
DeepScaleR | [
{
"role": "user",
"content": "Alpha and Beta both took part in a two-day problem-solving competition. At the end of the second day, each had attempted questions worth a total of 500 points. Alpha scored 160 points out of 300 points attempted on the first day, and scored 140 points out of 200 points attempte... | math | {
"style": "rule",
"ground_truth": "\\frac{349}{500}"
} | {
"index": 4518,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Although I am certain that my clock is 5 minutes fast, it is actually 10 minutes slow. On the other hand, my friend's clock is really 5 minutes fast, even though he thinks it is correct. We scheduled a meeting for 10 o'clock and plan to arrive on time. Who will arrive first? Af... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 4519,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Altitudes $B E$ and $C F$ of acute triangle $A B C$ intersect at $H$. Suppose that the altitudes of triangle $E H F$ concur on line $B C$. If $A B=3$ and $A C=4$, then $B C^{2}=\\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100 a+b$."
}
] | math | {
"style": "rule",
"ground_truth": "33725"
} | {
"index": 4520,
"solution": "Let $P$ be the orthocenter of $\\triangle E H F$. Then $E H \\perp F P$ and $E H \\perp A C$, so $F P$ is parallel to $A C$. Similarly, $E P$ is parallel to $A B$. Using similar triangles gives $$1=\\frac{B P}{B C}+\\frac{C P}{B C}=\\frac{A E}{A C}+\\frac{A F}{A B}=\\frac{A B \\cos A}{... |
DeepScaleR | [
{
"role": "user",
"content": "Altitudes $\\overline{AD}$ and $\\overline{BE}$ of $\\triangle ABC$ intersect at $H$. If $\\angle BAC = 46^\\circ$ and $\\angle ABC = 71^\\circ$, then what is $\\angle AHB$?\n\n[asy]\nsize(150); defaultpen(linewidth(0.8));\npair B = (0,0), C = (3,0), A = (1.8,2), P = foot(A,B,... | math | {
"style": "rule",
"ground_truth": "117^\\circ"
} | {
"index": 4521,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Altitudes $\\overline{AP}$ and $\\overline{BQ}$ of an acute triangle $\\triangle ABC$ intersect at point $H$. If $HP=5$ while $HQ=2$, then calculate $(BP)(PC)-(AQ)(QC)$. [asy]\nsize(150); defaultpen(linewidth(0.8));\npair B = (0,0), C = (3,0), A = (2,2), P = foot(A,B,C), Q = f... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 4522,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Altitudes $\\overline{AP}$ and $\\overline{BQ}$ of an acute triangle $\\triangle ABC$ intersect at point $H$. If $HP=3$ and $HQ=4$, then calculate $(BP)(PC)-(AQ)(QC)$."
}
] | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"index": 4523,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Altitudes $\\overline{AP}$ and $\\overline{BQ}$ of an acute triangle $\\triangle ABC$ intersect at point $H$. If $HP=8$ and $HQ=3$, then calculate $(BP)(PC)-(AQ)(QC)$."
}
] | math | {
"style": "rule",
"ground_truth": "55"
} | {
"index": 4524,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Altitudes $\\overline{AX}$ and $\\overline{BY}$ of acute triangle $ABC$ intersect at $H$. If $\\angle BAC = 61^\\circ$ and $\\angle ABC = 73^\\circ$, then what is $\\angle CHX$?"
}
] | math | {
"style": "rule",
"ground_truth": "73^\\circ"
} | {
"index": 4525,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alvin rides on a flat road at $25$ kilometers per hour, downhill at $35$ kph, and uphill at $10$ kph. Benny rides on a flat road at $35$ kph, downhill at $45$ kph, and uphill at $15$ kph. Alvin goes from town $X$ to town $Y$, a distance of $15$ km all uphill, then from town $Y$... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"index": 4526,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alvin, Bingyi, and Cheska play a two-player game that never ends in a tie. In a recent tournament between the three players, a total of 60 games were played and each pair of players played the same number of games. When Alvin and Bingyi played, Alvin won \\(20\\%\\) of the game... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 4527,
"solution": "Since 60 games are played and each of the 3 pairs plays the same number of games, each pair plays \\(60 \\div 3 = 20\\) games. Alvin wins \\(20\\%\\) of the 20 games that Alvin and Bingyi play, so Alvin wins \\(\\frac{20}{100} \\times 20 = \\frac{1}{5} \\times 20 = 4\\) of these 20 gam... |
DeepScaleR | [
{
"role": "user",
"content": "Amanda, Ben, and Carlos share a sum of money. Their portions are in the ratio of 1:2:7, respectively. If Amanda's portion is $\\$$20, what is the total amount of money shared?"
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"index": 4528,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Amelia and Blaine are playing a modified game where they toss their respective coins. Amelia's coin lands on heads with a probability of $\\frac{3}{7}$, and Blaine's lands on heads with a probability of $\\frac{1}{3}$. They begin their game only after observing at least one hea... | math | {
"style": "rule",
"ground_truth": "\\frac{21609}{64328}"
} | {
"index": 4529,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Amelia has a coin that lands heads with a probability of $\\frac{3}{7}$, and Blaine has a coin that lands on heads with a probability of $\\frac{1}{4}$. Initially, they simultaneously toss their coins once. If both get heads, they stop; otherwise, if either gets a head and the ... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{14}"
} | {
"index": 4530,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Amelia has a coin that lands heads with probability $\\frac{1}{3}$, and Blaine has a coin that lands on heads with probability $\\frac{2}{5}$. Amelia and Blaine alternately toss their coins until someone gets a head; the first one to get a head wins. All coin tosses are indepen... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 4531,
"solution": "To find the probability that Amelia wins, we consider the following cases:\n\n1. **Amelia wins on her first turn.**\n2. **Amelia wins on her second turn.**\n3. **Amelia wins on her third turn.**\n4. **And so on.**\n\n#### Case Analysis:\n- **Case 1:** Amelia wins on her first turn. The... |
DeepScaleR | [
{
"role": "user",
"content": "Amerigo Vespucci has a map of America drawn on the complex plane. The map does not distort distances. Los Angeles corresponds to $0$ on this complex plane, and Boston corresponds to $2600i$. Meanwhile, Knoxville corresponds to the point $780+1040i$. With these city-point corres... | math | {
"style": "rule",
"ground_truth": "1300"
} | {
"index": 4532,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Amina and Bert alternate turns tossing a fair coin. Amina goes first and each player takes three turns. The first player to toss a tail wins. If neither Amina nor Bert tosses a tail, then neither wins. What is the probability that Amina wins?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{21}{32}"
} | {
"index": 4533,
"solution": "If Amina wins, she can win on her first turn, on her second turn, or on her third turn. If she wins on her first turn, then she went first and tossed tails. This occurs with probability $\\frac{1}{2}$. If she wins on her second turn, then she tossed heads, then Bert tossed heads, then ... |
DeepScaleR | [
{
"role": "user",
"content": "Amir is 8 kg heavier than Ilnur, and Daniyar is 4 kg heavier than Bulat. The sum of the weights of the heaviest and lightest boys is 2 kg less than the sum of the weights of the other two boys. All four boys together weigh 250 kg. How many kilograms does Amir weigh?"
}
] | math | {
"style": "rule",
"ground_truth": "67"
} | {
"index": 4534,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among 100 young men, if at least one of the height or weight of person A is greater than that of person B, then A is considered not inferior to B. Determine the maximum possible number of top young men among these 100 young men."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 4535,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among 150 schoolchildren, only boys collect stamps. 67 people collect USSR stamps, 48 people collect African stamps, and 32 people collect American stamps. 11 people collect only USSR stamps, 7 people collect only African stamps, 4 people collect only American stamps, and only ... | math | {
"style": "rule",
"ground_truth": "66"
} | {
"index": 4536,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among 5 table tennis players, there are 2 veteran players and 3 new players. Now, 3 players are to be selected to form a team and arranged in positions 1, 2, and 3 to participate in a team competition. The arrangement must include at least 1 veteran player, and among players 1 ... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 4537,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among 50 school teams participating in the HKMO, no team answered all four questions correctly. The first question was solved by 45 teams, the second by 40 teams, the third by 35 teams, and the fourth by 30 teams. How many teams solved both the third and the fourth questions?"
... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 4538,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among 51 consecutive odd numbers $1, 3, 5, \\cdots, 101$, select $\\mathrm{k}$ numbers such that their sum is 1949. What is the maximum value of $\\mathrm{k}$?"
}
] | math | {
"style": "rule",
"ground_truth": "44"
} | {
"index": 4539,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among 6 courses, if person A and person B each choose 3 courses, the number of ways in which exactly 1 course is chosen by both A and B is \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "180"
} | {
"index": 4540,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among 6 internists and 4 surgeons, there is one chief internist and one chief surgeon. Now, a 5-person medical team is to be formed to provide medical services in rural areas. How many ways are there to select the team under the following conditions? \n(1) The team includes 3 ... | math | {
"style": "rule",
"ground_truth": "191"
} | {
"index": 4541,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among 7 students, 6 are to be arranged to participate in social practice activities in two communities on Saturday, with 3 people in each community. How many different arrangements are there? (Answer with a number)"
}
] | math | {
"style": "rule",
"ground_truth": "140"
} | {
"index": 4542,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among A, B, C, and D comparing their heights, the sum of the heights of two of them is equal to the sum of the heights of the other two. The average height of A and B is 4 cm more than the average height of A and C. D is 10 cm taller than A. The sum of the heights of B and C is... | math | {
"style": "rule",
"ground_truth": "139"
} | {
"index": 4543,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among a group of 120 people, some pairs are friends. A [i]weak quartet[/i] is a set of four people containing exactly one pair of friends. What is the maximum possible number of weak quartets ?"
}
] | math | {
"style": "rule",
"ground_truth": "4769280"
} | {
"index": 4544,
"solution": "\nGiven a group of 120 people, where some pairs are friends, we need to determine the maximum possible number of weak quartets. A weak quartet is defined as a set of four people containing exactly one pair of friends. \n\nTo solve this, we need to analyze the structure of weak quartets... |
DeepScaleR | [
{
"role": "user",
"content": "Among all integers that alternate between 1 and 0, starting and ending with 1 (e.g., 101, 10101, 10101…), how many are prime numbers? Why? And list all the prime numbers."
}
] | math | {
"style": "rule",
"ground_truth": "101"
} | {
"index": 4545,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among all natural numbers not greater than 200, how many numbers are coprime to both 2 and 3 and are not prime numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"index": 4546,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among all pairs of real numbers $(x, y)$ such that $\\cos \\sin x = \\cos \\sin y$ with $-\\frac{15\\pi}{2} \\le x, y \\le \\frac{15\\pi}{2}$, Ana randomly selects a pair $(X, Y)$. Compute the probability that $X = Y$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 4547,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among all pairs of real numbers $(x, y)$ such that $\\sin \\sin x = \\sin \\sin y$ with $-10 \\pi \\le x, y \\le 10 \\pi$, Oleg randomly selected a pair $(X, Y)$. Compute the probability that $X = Y$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{20}"
} | {
"index": 4548,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among all polynomials $P(x)$ with integer coefficients for which $P(-10)=145$ and $P(9)=164$, compute the smallest possible value of $|P(0)|$."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 4549,
"solution": "Since $a-b \\mid P(a)-P(b)$ for any integer polynomial $P$ and integers $a$ and $b$, we require that $10 \\mid P(0)-P(-10)$ and $9 \\mid P(0)-P(9)$. So, we are looking for an integer $a$ near 0 for which $$a \\equiv 5 \\bmod 10, a \\equiv 2 \\bmod 9$$ The smallest such positive integer... |
DeepScaleR | [
{
"role": "user",
"content": "Among all proper fractions whose numerator and denominator are two-digit numbers, find the smallest fraction that is greater than \\(\\frac{4}{9}\\). Provide the numerator of this fraction in your answer."
}
] | math | {
"style": "rule",
"ground_truth": "41"
} | {
"index": 4550,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among all right triangles \\(ABC\\) with \\( \\angle C = 90^\\circ\\), find the maximum value of \\( \\sin A + \\sin B + \\sin^2 A \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2} + \\frac{1}{2}"
} | {
"index": 4551,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among all the 5-digit numbers composed of the digits 1, 2, 3, 4, and 5 without repetition, how many are greater than 23145 and less than 43521?"
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 4552,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among all the factors of $11!$ (where $11! = 11 \\times 10 \\times \\cdots \\times 1$), the largest factor that can be expressed in the form $6k + 1$ (where $k$ is a natural number) is $\\qquad$."
}
] | math | {
"style": "rule",
"ground_truth": "385"
} | {
"index": 4553,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among all the five-digit numbers formed without repeating any of the digits 0, 1, 2, 3, 4, if they are arranged in ascending order, determine the position of the number 12340."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 4554,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among all the five-digit numbers formed without repeating digits using 0, 1, 2, 3, and 4, if arranged in ascending order, what position would the number 12340 occupy?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 4555,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among all the four-digit numbers without repeated digits, how many numbers have the digit in the thousandth place 2 greater than the digit in the unit place?"
}
] | math | {
"style": "rule",
"ground_truth": "448"
} | {
"index": 4556,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among all the roots of\n\\[z^8 - z^6 + z^4 - z^2 + 1 = 0,\\]the maximum imaginary part of a root can be expressed as $\\sin \\theta,$ where $-90^\\circ \\le \\theta \\le 90^\\circ.$ Find $\\theta.$"
}
] | math | {
"style": "rule",
"ground_truth": "54^\\circ"
} | {
"index": 4557,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among all the simple fractions where both the numerator and the denominator are two-digit numbers, find the smallest fraction that is greater than $\\frac{3}{5}$. Provide the numerator of this fraction in your answer."
}
] | math | {
"style": "rule",
"ground_truth": "59"
} | {
"index": 4558,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among all the simple fractions with a numerator and denominator that are two-digit numbers, find the smallest fraction greater than $\\frac{3}{4}$. Provide its numerator in the answer."
}
] | math | {
"style": "rule",
"ground_truth": "73"
} | {
"index": 4559,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among all triangles $ABC$, find the maximum value of $\\cos A + \\cos B \\cos C$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 4560,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among all triangles $ABC,$ find the maximum value of $\\cos A + \\cos B \\cos C.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}"
} | {
"index": 4561,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among all triangles $ABC,$ find the maximum value of $\\sin A + \\sin B \\sin C.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1 + \\sqrt{5}}{2}"
} | {
"index": 4562,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among all victims of zombie bites, 10% are prescribed the experimental drug Undetenin to treat them. Overall, 4% of the human population suffer an adverse reaction to Undetenin. Out of all the patients being treated with Undetenin, 2% suffer an adverse reaction to the drug. Wha... | math | {
"style": "rule",
"ground_truth": "0.05"
} | {
"index": 4563,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among five numbers, if we take the average of any four numbers and add the remaining number, the sums will be 74, 80, 98, 116, and 128, respectively. By how much is the smallest number less than the largest number among these five numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 4564,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among four people, A, B, C, and D, they pass a ball to each other. The first pass is from A to either B, C, or D, and the second pass is from the receiver to any of the other three. This process continues for several passes. Calculate the number of ways the ball can be passed s... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 4565,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the 100 natural numbers from 1 to 100, how many numbers can be represented as \\( m \\cdot n + m + n \\) where \\( m \\) and \\( n \\) are natural numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "74"
} | {
"index": 4566,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the 200 natural numbers from 1 to 200, list the numbers that are neither multiples of 3 nor multiples of 5 in ascending order. What is the 100th number in this list?"
}
] | math | {
"style": "rule",
"ground_truth": "187"
} | {
"index": 4567,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the 2019 natural numbers from 1 to 2019, how many of them, when added to the four-digit number 8866, result in at least one carry?"
}
] | math | {
"style": "rule",
"ground_truth": "1956"
} | {
"index": 4568,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the 8 vertices, the midpoints of the 12 edges, the centers of the 6 faces, and the center of a cube (totaling 27 points), how many groups of three collinear points are there?"
}
] | math | {
"style": "rule",
"ground_truth": "49"
} | {
"index": 4569,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the 9 natural numbers $1,2,3, \\cdots, 9$, if 3 numbers are chosen, let $x$ be the number of pairs of adjacent numbers among the chosen 3 numbers (for example, if the 3 chosen numbers are $1,2,3$, there are 2 pairs of adjacent numbers: 1,2 and 2,3, so the value of $x$ is ... | math | {
"style": "rule",
"ground_truth": "2/3"
} | {
"index": 4570,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the 95 numbers $1^2, 2^2, 3^2, \\ldots, 95^2$, how many of them have an odd digit in the tens place?"
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 4571,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the digits 0, 1, ..., 9, calculate the number of three-digit numbers that can be formed using repeated digits."
}
] | math | {
"style": "rule",
"ground_truth": "252"
} | {
"index": 4572,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the first 1500 positive integers, there are n whose hexadecimal representation contains only numeric digits. What is the sum of the digits of n?"
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"index": 4573,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the five-digit numbers formed using the digits 0, 1, 2, 3, 4, how many have the first and last digits the same, and the three middle digits all different?"
}
] | math | {
"style": "rule",
"ground_truth": "240"
} | {
"index": 4574,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the following 4 propositions, the correct one is: \n(1) If a solid's three views are completely identical, then the solid is a cube; \n(2) If a solid's front view and top view are both rectangles, then the solid is a cuboid; \n(3) If a solid's three views are all recta... | math | {
"style": "rule",
"ground_truth": "(3)"
} | {
"index": 4575,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the following four propositions:\n\n(1) The domain of the function $y=\\tan (x+ \\frac {π}{4})$ is $\\{x|x\\neq \\frac {π}{4}+kπ,k\\in Z\\}$;\n\n(2) Given $\\sin α= \\frac {1}{2}$, and $α\\in[0,2π]$, the set of values for $α$ is $\\{\\frac {π}{6}\\}$;\n\n(3) The graph of ... | math | {
"style": "rule",
"ground_truth": "(1)(3)(4)"
} | {
"index": 4576,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the following four propositions: \n(1) If line $a$ is parallel to line $b$, and line $a$ is parallel to plane $\\alpha$, then line $b$ is parallel to plane $\\alpha$. \n(2) If line $a$ is parallel to plane $\\alpha$, and line $b$ is contained in plane $\\alpha$, then pl... | math | {
"style": "rule",
"ground_truth": "(4)"
} | {
"index": 4577,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the following functions, identify which pairs represent the same function.\n\n1. $f(x) = |x|, g(x) = \\sqrt{x^2}$;\n2. $f(x) = \\sqrt{x^2}, g(x) = (\\sqrt{x})^2$;\n3. $f(x) = \\frac{x^2 - 1}{x - 1}, g(x) = x + 1$;\n4. $f(x) = \\sqrt{x + 1} \\cdot \\sqrt{x - 1}, g(x) = \\s... | math | {
"style": "rule",
"ground_truth": "(1)"
} | {
"index": 4578,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the following numbers\n\n① $111111_{(2)}$ \n② $210_{(6)}$ \n③ $1000_{(4)}$ \n④ $81_{(8)}$\n\nThe largest number is \\_\\_\\_\\_\\_\\_\\_\\_, and the smallest number is \\_\\_\\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "111111_{(2)}"
} | {
"index": 4579,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the following propositions, the correct ones are __________.\n\n(1) The regression line $\\hat{y}=\\hat{b}x+\\hat{a}$ always passes through the center of the sample points $(\\bar{x}, \\bar{y})$, and at least through one sample point;\n\n(2) After adding the same constant... | math | {
"style": "rule",
"ground_truth": "(2)(6)(7)"
} | {
"index": 4580,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the following propositions, the true one is __________ \n (1) In a plane, the locus of points whose sum of distances from two fixed points $F_{1}$ and $F_{2}$ is a constant is an ellipse; \n (2) If vectors $\\overrightarrow{e_{1}}$, $\\overrightarrow{e_{2}}$, $\\overrig... | math | {
"style": "rule",
"ground_truth": "(3)"
} | {
"index": 4581,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the following propositions, the true one is marked by \\_\\_\\_\\_\\_\\_.\n\\\\((1)\\\\) The negation of the proposition \"For all \\\\(x > 0\\\\), \\\\(x^{2}-x \\leqslant 0\\\\)\" is \"There exists an \\\\(x > 0\\\\) such that \\\\(x^{2}-x > 0\\\\).\"\n\\\\((2)\\\\) If \... | math | {
"style": "rule",
"ground_truth": "(1)"
} | {
"index": 4582,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the following propositions, the true one is numbered \\_\\_\\_\\_\\_\\_. \n(1) The negation of the proposition \"For all $x>0$, $x^2-x\\leq0$\" is \"There exists an $x>0$ such that $x^2-x>0$.\" \n(2) If $A>B$, then $\\sin A > \\sin B$. \n(3) Given a sequence $\\{a_n\\}... | math | {
"style": "rule",
"ground_truth": "(1)"
} | {
"index": 4583,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the following statements, the correct one is: \n(1) The probability of event A or B happening is definitely greater than the probability of exactly one of A or B happening; \n(2) The probability of events A and B happening simultaneously is definitely less than the prob... | math | {
"style": "rule",
"ground_truth": "(4)"
} | {
"index": 4584,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the four students A, B, C, and D participating in competitions in mathematics, writing, and English, each subject must have at least one participant (and each participant can only choose one subject). If students A and B cannot participate in the same competition, the tot... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 4585,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the four-digit numbers composed of the digits $0$, $1$, $2$, $3$, $4$, $5$ without repetition, there are a total of \\_\\_\\_\\_\\_ numbers that are not divisible by $5$."
}
] | math | {
"style": "rule",
"ground_truth": "192"
} | {
"index": 4586,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the four-digit numbers formed by the digits 0, 1, 2, ..., 9 without repetition, determine the number of cases where the absolute difference between the units digit and the hundreds digit equals 8."
}
] | math | {
"style": "rule",
"ground_truth": "210"
} | {
"index": 4587,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the four-digit numbers, the number of four-digit numbers that have exactly 2 digits repeated is."
}
] | math | {
"style": "rule",
"ground_truth": "3888"
} | {
"index": 4588,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the integers from 1 to 100, how many integers can be divided by exactly two of the following four numbers: 2, 3, 5, 7?"
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 4589,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the natural numbers from 1 to 100, find the total number of numbers that contain a digit 7 or are multiples of 7."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 4590,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the natural numbers from 1 to 1000, there are a total of number 7s."
}
] | math | {
"style": "rule",
"ground_truth": "300"
} | {
"index": 4591,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the natural numbers not exceeding 10,000, calculate the number of odd numbers with distinct digits."
}
] | math | {
"style": "rule",
"ground_truth": "2605"
} | {
"index": 4592,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the nine fractions\n\n$$\n\\frac{5}{4}, \\frac{17}{6}, \\frac{-5}{4}, \\frac{10}{7}, \\frac{2}{3}, \\frac{14}{8}, \\frac{-1}{3}, \\frac{5}{3} \\text { and } \\frac{-3}{2}\n$$\n\nwe have eight with the following properties:\n\n- 2 fractions whose sum is $\\frac{2}{5}$\n- 2... | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{2}"
} | {
"index": 4593,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the non-empty subsets of the set \\( A = \\{1, 2, \\cdots, 10\\} \\), how many subsets have the sum of their elements being a multiple of 10?"
}
] | math | {
"style": "rule",
"ground_truth": "103"
} | {
"index": 4594,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the numbers $1, 2, 3, \\cdots, 50$, if 10 consecutive numbers are selected, what is the probability that exactly 3 of them are prime numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "22/41"
} | {
"index": 4595,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the numbers $1^{2}, 2^{2}, 3^{2}, \\cdots, 95^{2}$, how many of these 95 numbers have an odd tens digit?"
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 4596,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the numbers $210_{(6)}$, $1000_{(4)}$, and $111111_{(2)}$, the smallest number is \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "111111_{(2)}"
} | {
"index": 4597,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the numbers $85_{(9)}$, $210_{(6)}$, $1000_{(4)}$, and $111111_{(2)}$, the smallest number is __________."
}
] | math | {
"style": "rule",
"ground_truth": "111111_{(2)}"
} | {
"index": 4598,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, draw one at random. The probability of drawing a prime number is ____, and the probability of drawing a composite number is ____."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{9}"
} | {
"index": 4599,
"solution": ""
} |
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