data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "28 apples weigh 3 kilograms. If they are evenly divided into 7 portions, each portion accounts for $\\boxed{\\frac{1}{7}}$ of all the apples, and each portion weighs $\\boxed{\\frac{3}{7}}$ kilograms."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{7}"
} | {
"index": 500,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "289. A remarkable number. Find a number such that its fractional part, its integer part, and the number itself form a geometric progression."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1+\\sqrt{5}}{2}"
} | {
"index": 501,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "29 boys and 15 girls attended a ball. Some boys danced with some girls (no more than once with each partner). After the ball, each person told their parents how many times they danced. What is the maximum number of different numbers the children could have mentioned?"
}
] | math | {
"style": "rule",
"ground_truth": "29"
} | {
"index": 502,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "29 boys and 15 girls attended a ball. Some boys danced with some of the girls (no more than once with each pair). After the ball, each person told their parents how many times they danced. What is the maximum number of different numbers the children could have mentioned?"
}
] | math | {
"style": "rule",
"ground_truth": "29"
} | {
"index": 503,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "29 boys and 15 girls came to the ball. Some boys danced with some girls (no more than once per pair). After the ball, each person told their parents how many times they danced. What is the greatest number of distinct counts that the children could report?"
}
] | math | {
"style": "rule",
"ground_truth": "29"
} | {
"index": 504,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "29 boys and 15 girls came to the ball. Some of the boys danced with some of the girls (at most once with each person in the pair). After the ball, each individual told their parents how many times they danced. What is the maximum number of different numbers that the children co... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"index": 505,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "2^{-(2k+1)}-2^{-(2k-1)}+2^{-2k} is equal to"
}
] | math | {
"style": "rule",
"ground_truth": "-2^{-(2k+1)}"
} | {
"index": 506,
"solution": "1. **Rewrite the expression with a common base:** \n We start by expressing each term in the expression $2^{-(2k+1)} - 2^{-(2k-1)} + 2^{-2k}$ using a common base. Let $x = 2^{-2k}$. Then:\n - $2^{-(2k+1)} = 2^{-2k} \\cdot 2^{-1} = x \\cdot \\frac{1}{2} = \\frac{x}{2}$\n - $2^{-(2... |
DeepScaleR | [
{
"role": "user",
"content": "3 points $ O(0,\\ 0),\\ P(a,\\ a^2), Q( \\minus{} b,\\ b^2)\\ (a > 0,\\ b > 0)$ are on the parabpla $ y \\equal{} x^2$ .\r\nLet $ S_1$ be the area bounded by the line $ PQ$ and the parabola and let $ S_2$ be the area of the triangle $ OPQ$ .\r\n\r\nFind the minimum va... | math | {
"style": "rule",
"ground_truth": "4/3"
} | {
"index": 507,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "3 red marbles, 4 blue marbles, and 5 green marbles are distributed to 12 students. Each student gets one and only one marble. In how many ways can the marbles be distributed so that Jamy and Jaren get the same color and Jason gets a green marble?"
}
] | math | {
"style": "rule",
"ground_truth": "3150"
} | {
"index": 508,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "30 beads (blue and green) were arranged in a circle. 26 beads had a neighboring blue bead, and 20 beads had a neighboring green bead. How many blue beads were there?"
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 509,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "30 students from 5 grades participated in answering 40 questions. Each student answered at least 1 question. Every two students from the same grade answered the same number of questions, and students from different grades answered a different number of questions. How many stude... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 510,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "30 students from five courses created 40 problems for the olympiad, with students from the same course creating the same number of problems, and students from different courses creating different numbers of problems. How many students created exactly one problem?"
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 511,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "3000 people each go into one of three rooms randomly. What is the most likely value for the maximum number of people in any of the rooms? Your score for this problem will be 0 if you write down a number less than or equal to 1000. Otherwise, it will be $25-27 \\frac{|A-C|}{\\mi... | math | {
"style": "rule",
"ground_truth": "1019"
} | {
"index": 512,
"solution": "To get a rough approximation, we can use the fact that a sum of identical random variables converges to a Gaussian distribution in this case with a mean of 1000 and a variance of $3000 \\cdot \\frac{2}{9}=667$. Since $\\sqrt{667} \\approx 26,1026$ is a good guess, as Gaussians tend to d... |
DeepScaleR | [
{
"role": "user",
"content": "31 cars simultaneously started from one point on a circular track: the first car at a speed of 61 km/h, the second at 62 km/h, and so on (the 31st at 91 km/h). The track is narrow, and if one car overtakes another on a lap, they collide, both go off the track, and are out of th... | math | {
"style": "rule",
"ground_truth": "76"
} | {
"index": 513,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "31 cars simultaneously started from the same point on a circular track: the first car at a speed of 61 km/h, the second at 62 km/h, and so on up to the 31st car at 91 km/h. The track is narrow, and if one car overtakes another, they collide and both crash out of the race. Event... | math | {
"style": "rule",
"ground_truth": "76"
} | {
"index": 514,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "31 people attended a class club afternoon event, and after the program, they danced. Ági danced with 7 boys, Anikó with 8, Zsuzsa with 9, and so on, with each subsequent girl dancing with one more boy than the previously mentioned one. Finally, Márta danced with all but 3 boys.... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 515,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "35 times 61,000 unit cubes are combined to form a large cube with an edge length of 10 units. After being painted, the large cube is then separated back into the original unit cubes. How many of these unit cubes have at least one face painted?"
}
] | math | {
"style": "rule",
"ground_truth": "488"
} | {
"index": 516,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "4 12-sided dice are rolled. What is the probability that the number of dice showing a two digit number is equal to the number of dice showing a one digit number? Express your answer as a common fraction. (Assume that the numbers on the 12 sides are the numbers from 1 to 12 expr... | math | {
"style": "rule",
"ground_truth": "\\dfrac{27}{128}"
} | {
"index": 517,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "4(299) + 3(299) + 2(299) + 298 ="
}
] | math | {
"style": "rule",
"ground_truth": "2989"
} | {
"index": 518,
"solution": "1. **Identify the common factor and apply the distributive property**: Notice that $299$ appears as a common factor in the first three terms. We can factor out $299$ from these terms:\n \\[\n 4(299) + 3(299) + 2(299) + 298 = (4+3+2)(299) + 298\n \\]\n\n2. **Simplify the expression... |
DeepScaleR | [
{
"role": "user",
"content": "4.\tFind the biggest positive integer $n$ , lesser thar $2012$ , that has the following property:\nIf $p$ is a prime divisor of $n$ , then $p^2 - 1$ is a divisor of $n$ ."
}
] | math | {
"style": "rule",
"ground_truth": "1944"
} | {
"index": 519,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "4. Suppose that $\\overline{A2021B}$ is a six-digit integer divisible by $9$ . Find the maximum possible value of $A \\cdot B$ .\n\n5. In an arbitrary triangle, two distinct segments are drawn from each vertex to the opposite side. What is the minimum possible number of int... | math | {
"style": "rule",
"ground_truth": "143"
} | {
"index": 520,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "4.3535… is a decimal, which can be abbreviated as , and the repeating cycle is ."
}
] | math | {
"style": "rule",
"ground_truth": "35"
} | {
"index": 521,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "40 pikes were released into a pond. A pike is considered well-fed if it has eaten three other pikes (whether well-fed or hungry). What is the maximum number of pikes that can be well-fed?"
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 522,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "48 blacksmiths need to shoe 60 horses. Each blacksmith takes 5 minutes to make one horseshoe. What is the minimum time they should spend on the job? (Note: A horse cannot stand on two legs.)"
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 523,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "5 people are standing in a row for a photo, among them one person must stand in the middle. There are ways to arrange them."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 524,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "5 points in a plane are situated so that no two of the lines joining a pair of points are coincident, parallel, or perpendicular. Through each point, lines are drawn perpendicular to each of the lines through two of the other 4 points. Determine the maximum number of intersecti... | math | {
"style": "rule",
"ground_truth": "315"
} | {
"index": 525,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "5 students stand in a row for a photo, where students A and B must stand next to each other, and A cannot stand at either end. Calculate the total number of possible arrangements."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 526,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "5. Let $S$ denote the set of all positive integers whose prime factors are elements of $\\{2,3,5,7,11\\}$ . (We include 1 in the set $S$ .) If $$ \\sum_{q \\in S} \\frac{\\varphi(q)}{q^{2}} $$ can be written as $\\frac{a}{b}$ for relatively prime positive integers $a$ ... | math | {
"style": "rule",
"ground_truth": "1537"
} | {
"index": 527,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "50 businessmen - Japanese, Koreans, and Chinese - are sitting at a round table. It is known that between any two nearest Japanese, there are exactly as many Chinese as there are Koreans at the table. How many Chinese can be at the table?"
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 528,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "50 people, consisting of 30 people who all know each other, and 20 people who know no one, are present at a conference. Determine the number of handshakes that occur among the individuals who don't know each other."
}
] | math | {
"style": "rule",
"ground_truth": "1170"
} | {
"index": 529,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "50 students from fifth to ninth grade collectively posted 60 photos on Instagram, with each student posting at least one photo. All students in the same grade (parallel) posted an equal number of photos, while students from different grades posted different numbers of photos. H... | math | {
"style": "rule",
"ground_truth": "46"
} | {
"index": 530,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "50 students with blond hair, brown hair, and red hair are sitting around a round table. It is known that in any group of students sitting consecutively, there is at least one student with brown hair between any two students with blond hair, and at least one student with red hai... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 531,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "500 × 3986 × 0.3986 × 5 = ?"
}
] | math | {
"style": "rule",
"ground_truth": "0.25 \\times 3986^2"
} | {
"index": 532,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "58 balls of two colors - red and blue - are arranged in a circle. It is known that the number of consecutive triplets of balls with a majority of red balls is equal to the number of triplets with a majority of blue balls. What is the minimum possible number of red balls?"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 533,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "6 * cos(18°) + 2 * cos(36°) + 4 * cos(54°) + ... + 20 * cos(360°) = ?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 534,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "6 books of different types are to be divided into 3 groups, one group containing 4 books and the other two groups containing 1 book each. Calculate the number of different ways this can be done."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 535,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "6 boys and 4 girls are each assigned as attendants to 5 different buses, with 2 attendants per bus. Assuming that boys and girls are separated, and the buses are distinguishable, how many ways can the assignments be made?"
}
] | math | {
"style": "rule",
"ground_truth": "5400"
} | {
"index": 536,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "6 small circles of equal radius and 1 large circle are arranged as shown in the diagram. The area of the large circle is 120. What is the area of one of the small circles?"
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 537,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "64 people are in a single elimination rock-paper-scissors tournament, which consists of a 6-round knockout bracket. Each person has a different rock-paper-scissors skill level, and in any game, the person with the higher skill level will always win. For how many players $P$ is ... | math | {
"style": "rule",
"ground_truth": "49"
} | {
"index": 538,
"solution": "Note that a sub-bracket, that is, a subset of games of the tournament that themselves constitute a bracket, is always won by the person with the highest skill level. Therefore, a person wins her first four rounds if and only if she has the highest skill level among the people in her 16-... |
DeepScaleR | [
{
"role": "user",
"content": "650 students were surveyed about their pasta preferences. The choices were lasagna, manicotti, ravioli and spaghetti. The results of the survey are displayed in the bar graph. What is the ratio of the number of students who preferred spaghetti to the number of students who pref... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}"
} | {
"index": 539,
"solution": "1. **Identify the number of students who preferred each pasta type:** According to the problem, the number of students who preferred spaghetti is 250 and the number of students who preferred manicotti is 100.\n\n2. **Set up the ratio:** The ratio of the number of students who preferred ... |
DeepScaleR | [
{
"role": "user",
"content": "679 contestants participated in HMMT February 2017. Let \\(N\\) be the number of these contestants who performed at or above the median score in at least one of the three individual tests. Estimate \\(N\\). An estimate of \\(E\\) earns \\(\\left\\lfloor 20-\\frac{|E-N|}{2}\\rig... | math | {
"style": "rule",
"ground_truth": "516"
} | {
"index": 540,
"solution": "Out of the 679 total contestants at HMMT February 2017, 188 contestants scored at least the median on all three tests, 159 contestants scored at least the median on two tests, and 169 contestants scored at least the median on one test, giving a total of 516 contestants"
} |
DeepScaleR | [
{
"role": "user",
"content": "7.61 log₂ 3 + 2 log₄ x = x^(log₉ 16 / log₃ x)."
}
] | math | {
"style": "rule",
"ground_truth": "16/3"
} | {
"index": 541,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "795. Calculate the double integral \\(\\iint_{D} x y \\, dx \\, dy\\), where region \\(D\\) is:\n\n1) A rectangle bounded by the lines \\(x=0, x=a\\), \\(y=0, y=b\\);\n2) An ellipse \\(4x^2 + y^2 \\leq 4\\);\n3) Bounded by the line \\(y=x-4\\) and the parabola \\(y^2=2x\\)."
... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 542,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "8 coins are simultaneously flipped. What is the probability that heads are showing on at most 2 of them?"
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{37}{256}"
} | {
"index": 543,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "8 distinct nonzero natural numbers are arranged in increasing order. The average of the first 3 numbers is 9, the average of all 8 numbers is 19, and the average of the last 3 numbers is 29. What is the maximum possible difference between the second largest number and the secon... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 544,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "8 people are seated in two rows with 4 people in each row. Among them, 2 specific people must sit in the front row, and 1 specific person must sit in the back row."
}
] | math | {
"style": "rule",
"ground_truth": "5760"
} | {
"index": 545,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "8 people are sitting around a circular table for a meeting, including one leader, one deputy leader, and one recorder. If the recorder is sitting between the leader and the deputy leader, how many different seating arrangements are possible (seating arrangements that can be mad... | math | {
"style": "rule",
"ground_truth": "240"
} | {
"index": 546,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "8 people are sitting around a circular table for a meeting, including one leader, one vice leader, and one recorder. If the recorder is seated between the leader and vice leader, how many different seating arrangements are possible (considering that arrangements that can be obt... | math | {
"style": "rule",
"ground_truth": "240"
} | {
"index": 547,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "8 people attend a party. During the party everyone shakes hands with everyone else. How many handshakes take place at the party?"
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 548,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "8 singers are participating in a festival and are scheduled to perform \\( m \\) times, with 4 singers performing in each show. Design a plan to minimize the number of performances \\( m \\) so that any two singers perform together the same number of times."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 549,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "8 students are practicing for a math contest, and they divide into pairs to take a practice test. In how many ways can they be split up?"
}
] | math | {
"style": "rule",
"ground_truth": "105"
} | {
"index": 550,
"solution": "We create the pairs one at a time. The first person has 7 possible partners. Set this pair aside. Of the remaining six people, pick a person. He or she has 5 possible partners. Set this pair aside. Of the remaining four people, pick a person. He or she has 3 possible partners. Set this ... |
DeepScaleR | [
{
"role": "user",
"content": "8. Andrey likes all numbers that are not divisible by 3, and Tanya likes all numbers that do not contain digits that are divisible by 3.\n\na) How many four-digit numbers are liked by both Andrey and Tanya?\n\nb) Find the total sum of the digits of all such four-digit numbers."... | math | {
"style": "rule",
"ground_truth": "14580"
} | {
"index": 551,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "8. Shortening. There is a sequence of 2015 digits. All digits are chosen randomly from the set {0, 9} independently of each other. The following operation is performed on the resulting sequence. If several identical digits go in a row, they are replaced by one such digit. For e... | math | {
"style": "rule",
"ground_truth": "1813.6"
} | {
"index": 552,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "9 judges each award 20 competitors a rank from 1 to 20. The competitor's score is the sum of the ranks from the 9 judges, and the winner is the competitor with the lowest score. For each competitor, the difference between the highest and lowest ranking (from different judges) i... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 553,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "9 judges score a gymnast in artistic gymnastics, with each giving an integer score. One highest score and one lowest score are removed, and the average of the remaining scores determines the gymnast's score. If the score is rounded to one decimal place using the rounding method... | math | {
"style": "rule",
"ground_truth": "8.43"
} | {
"index": 554,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "9 kg of toffees cost less than 10 rubles, and 10 kg of the same toffees cost more than 11 rubles. How much does 1 kg of these toffees cost?"
}
] | math | {
"style": "rule",
"ground_truth": "1.11"
} | {
"index": 555,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "9 pairs of table tennis players participate in a doubles match, their jersey numbers are 1, 2, …, 18. The referee is surprised to find that the sum of the jersey numbers of each pair of players is exactly a perfect square. The player paired with player number 1 is ."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 556,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "9 people are arranged in a 3×3 matrix (3 rows, 3 columns). Choose 3 people from them to serve as the team leader, deputy team leader, and discipline officer, respectively. The requirement is that at least two of these three people must be in the same row or column. The number o... | math | {
"style": "rule",
"ground_truth": "468"
} | {
"index": 557,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "9 seeds are divided among three pits, labeled A, B, and C, with each pit containing 3 seeds. Each seed has a 0.5 probability of germinating. If at least one seed in a pit germinates, then that pit does not need to be replanted; if no seeds in a pit germinate, then that pit need... | math | {
"style": "rule",
"ground_truth": "0.330"
} | {
"index": 558,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "9. The real quartic $P x^{4}+U x^{3}+M x^{2}+A x+C$ has four different positive real roots. Find the square of the smallest real number $z$ for which the expression $M^{2}-2 U A+z P C$ is always positive, regardless of what the roots of the quartic are."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 559,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "90 + 91 + 92 + 93 + 94 + 95 + 96 + 97 + 98 + 99 ="
}
] | math | {
"style": "rule",
"ground_truth": "945"
} | {
"index": 560,
"solution": "We are given the sum of consecutive integers from $90$ to $99$. We can calculate this sum using the formula for the sum of an arithmetic series, or by simplifying the expression directly.\n\n1. **Identify the series**: The series is $90 + 91 + 92 + 93 + 94 + 95 + 96 + 97 + 98 + 99$.\n\n... |
DeepScaleR | [
{
"role": "user",
"content": "900 cards are inscribed with all natural numbers from 1 to 900. Cards inscribed with squares of integers are removed, and the remaining cards are renumbered starting from 1.\n\nThen, the operation of removing the squares is repeated. How many times must this operation be repeat... | math | {
"style": "rule",
"ground_truth": "59"
} | {
"index": 561,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "99 dwarfs stand in a circle, some of them wear hats. There are no adjacent dwarfs in hats and no dwarfs in hats with exactly 48 dwarfs standing between them. What is the maximal possible number of dwarfs in hats?"
}
] | math | {
"style": "rule",
"ground_truth": "33"
} | {
"index": 562,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "<u>Set 1</u>**p1.** Assume the speed of sound is $343$ m/s. Anastasia and Bananastasia are standing in a field in front of you. When they both yell at the same time, you hear Anastasia’s yell $5$ seconds before Bananastasia’s yell. If Bananastasia yells first, and then Anas... | math | {
"style": "rule",
"ground_truth": "1715"
} | {
"index": 563,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "<u>Set 4</u> **p10.** Eve has nine letter tiles: three $C$ ’s, three $M$ ’s, and three $W$ ’s. If she arranges them in a random order, what is the probability that the string “ $CMWMC$ ” appears somewhere in the arrangement?**p11.** Bethany’s Batteries sells two kinds of bat... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{28}"
} | {
"index": 564,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A $1 \\times n$ rectangle ( $n \\geq 1 $ ) is divided into $n$ unit ( $1 \\times 1$ ) squares. Each square of this rectangle is colored red, blue or green. Let $f(n)$ be the number of colourings of the rectangle in which there are an even number of red squares. What is th... | math | {
"style": "rule",
"ground_truth": "37"
} | {
"index": 565,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A $3$ by $3$ determinant has three entries equal to $2$ , three entries equal to $5$ , and three entries equal to $8$ . Find the maximum possible value of the determinant."
}
] | math | {
"style": "rule",
"ground_truth": "405"
} | {
"index": 566,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A $30\\times30$ table is given. We want to color some of it's unit squares such that any colored square has at most $k$ neighbors. ( Two squares $(i,j)$ and $(x,y)$ are called neighbors if $i-x,j-y\\equiv0,-1,1 \\pmod {30}$ and $(i,j)\\neq(x,y)$ . Therefore, each squ... | math | {
"style": "rule",
"ground_truth": "300"
} | {
"index": 567,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A $40$ feet high screen is put on a vertical wall $10$ feet above your eye-level. How far should you stand to maximize the angle subtended by the screen (from top to bottom) at your eye?"
}
] | math | {
"style": "rule",
"ground_truth": "10\\sqrt{5}"
} | {
"index": 568,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A $6$ -inch-wide rectangle is rotated $90$ degrees about one of its corners, sweeping out an area of $45\\pi$ square inches, excluding the area enclosed by the rectangle in its starting position. Find the rectangle’s length in inches."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 569,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A $9 \\times 9$ square consists of $81$ unit squares. Some of these unit squares are painted black, and the others are painted white, such that each $2 \\times 3$ rectangle and each $3 \\times 2$ rectangle contain exactly 2 black unit squares and 4 white unit squares. D... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 570,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle with perimeter $7$ has integer sidelengths. What is the maximum possible area of such a triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{7}}{4}"
} | {
"index": 571,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A \"Kiwi\" business owner has two types of \"Kiwi\" for promotion, type \\\\(A\\\\) and type \\\\(B\\\\). If you buy 2 pieces of type \\\\(A\\\\) \"Kiwi\" and 1 piece of type \\\\(B\\\\) \"Kiwi\", the total cost is 120 yuan; if you buy 3 pieces of type \\\\(A\\\\) \"Kiwi\" and ... | math | {
"style": "rule",
"ground_truth": "1125"
} | {
"index": 572,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A \"clearance game\" has the following rules: in the $n$-th round, a die is rolled $n$ times. If the sum of the points from these $n$ rolls is greater than $2^n$, then the player clears the round. Questions:\n(1) What is the maximum number of rounds a player can clear in this g... | math | {
"style": "rule",
"ground_truth": "\\frac{100}{243}"
} | {
"index": 573,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A \"double-single\" number is a three-digit number made up of two identical digits followed by a different digit. For example, 553 is a double-single number. How many double-single numbers are there between 100 and 1000?"
}
] | math | {
"style": "rule",
"ground_truth": "81"
} | {
"index": 574,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A \"fifty percent mirror\" is a mirror that reflects half the light shined on it back and passes the other half of the light onward. Two \"fifty percent mirrors\" are placed side by side in parallel, and a light is shined from the left of the two mirrors. How much of the light ... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 575,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A \"pass level game\" has the following rules: On the \\( n \\)-th level, a die is thrown \\( n \\) times. If the sum of the points that appear in these \\( n \\) throws is greater than \\( 2^{n} \\), then the player passes the level. \n\n1. What is the maximum number of levels... | math | {
"style": "rule",
"ground_truth": "\\frac{100}{243}"
} | {
"index": 576,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A \"progressive number\" refers to a positive integer in which each digit is greater than the digit to its left (such as 1 458). If the four-digit \"progressive numbers\" are arranged in ascending order, then the 30th number is \\_\\_\\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "1359"
} | {
"index": 577,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A \"progressive number\" refers to a positive integer in which, except for the highest digit, each digit is greater than the digit to its left (for example, 13456 and 35678 are both five-digit \"progressive numbers\").\n\n(I) There are _______ five-digit \"progressive numbers\"... | math | {
"style": "rule",
"ground_truth": "34579"
} | {
"index": 578,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A \"stair-step\" figure is made of alternating black and white squares in each row. Rows $1$ through $4$ are shown. All rows begin and end with a white square. The number of black squares in the $37\\text{th}$ row is\n[asy]\ndraw((0,0)--(7,0)--(7,1)--(0,1)--cycle);\ndraw((1,0)-... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 579,
"solution": "1. **Identify the pattern in the number of squares per row**: \n - Observing the given diagram, each row starts and ends with a white square and has alternating black and white squares in between.\n - From row 1 to row 2, we notice that two squares are added (one on each side). This... |
DeepScaleR | [
{
"role": "user",
"content": "A $ 4\\times 4$ table is divided into $ 16$ white unit square cells. Two cells are called neighbors if they share a common side. A [i]move[/i] consists in choosing a cell and the colors of neighbors from white to black or from black to white. After exactly $ n$ moves all the $ ... | math | {
"style": "rule",
"ground_truth": "6, 8, 10, 12, 14, 16, \\ldots"
} | {
"index": 580,
"solution": "\nTo solve this problem, we must determine the number of moves, \\( n \\), necessary to change all 16 cells of a \\( 4 \\times 4 \\) grid from white to black. The transformation involves a series of operations, each toggling the color (from white to black or black to white) of a chosen ... |
DeepScaleR | [
{
"role": "user",
"content": "A $1 \\times 2$ rectangle is inscribed in a semicircle with the longer side on the diameter. What is the area of the semicircle?"
}
] | math | {
"style": "rule",
"ground_truth": "\\pi"
} | {
"index": 581,
"solution": "1. **Visualize the Problem**: We start by visualizing a $1 \\times 2$ rectangle inscribed in a semicircle, with the longer side of the rectangle lying along the diameter of the semicircle.\n\n2. **Double the Figure**: To simplify the problem, we consider doubling the semicircle along it... |
DeepScaleR | [
{
"role": "user",
"content": "A $1 \\times 3$ rectangle is inscribed in a semicircle with the longer side on the diameter. What is the area of the semicircle?\nA) $\\frac{9\\pi}{8}$\nB) $\\frac{12\\pi}{8}$\nC) $\\frac{13\\pi}{8}$\nD) $\\frac{15\\pi}{8}$\nE) $\\frac{16\\pi}{8}$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13\\pi}{8}"
} | {
"index": 582,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A $1.4 \\mathrm{~m}$ long rod has $3 \\mathrm{~kg}$ masses at both ends. Where should the rod be pivoted so that, when released from a horizontal position, the mass on the left side passes under the pivot with a speed of $1.6 \\mathrm{~m} /\\mathrm{s}$?"
}
] | math | {
"style": "rule",
"ground_truth": "0.8"
} | {
"index": 583,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A $10 \\times 10$ table consists of 100 unit cells. A block is a $2 \\times 2$ square consisting of 4 unit cells of the table. A set $C$ of $n$ blocks covers the table (i.e. each cell of the table is covered by some block of $C$ ) but no $n-1$ blocks of $C$ cover the table. Fin... | math | {
"style": "rule",
"ground_truth": "39"
} | {
"index": 584,
"solution": "Consider an infinite table divided into unit cells. Any $2 \\times 2$ square consisting of 4 unit cells of the table we also call a block. Fix arbitrary finite set $M$ of blocks lying on the table. Now we will consider arbitrary finite sets of unit cells of the table covered by $M$. For... |
DeepScaleR | [
{
"role": "user",
"content": "A $10$ by $4$ rectangle has the same center as a circle of radius $5$. Calculate the area of the region common to both the rectangle and the circle.\nA) $40 + 2\\pi$\nB) $36 + 4\\pi$\nC) $40 + 4\\pi$\nD) $44 + 4\\pi$\nE) $48 + 2\\pi$"
}
] | math | {
"style": "rule",
"ground_truth": "40 + 4\\pi"
} | {
"index": 585,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A $100$-gon $P_1$ is drawn in the Cartesian plane. The sum of the $x$-coordinates of the $100$ vertices equals 2009. The midpoints of the sides of $P_1$ form a second $100$-gon, $P_2$. Finally, the midpoints of the sides of $P_2$ form a third $100$-gon, $P_3$. Find the sum ... | math | {
"style": "rule",
"ground_truth": "2009"
} | {
"index": 586,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A $10\\times10\\times10$ grid of points consists of all points in space of the form $(i,j,k)$, where $i$, $j$, and $k$ are integers between $1$ and $10$, inclusive. Find the number of different lines that contain exactly $8$ of these points."
}
] | math | {
"style": "rule",
"ground_truth": "168"
} | {
"index": 587,
"solution": "Considered the cases $(1, 2, ..., 8), (2, 3, ...,9), (3, 4, ..., 10)$ and reverse. Also, consider the constant subsequences of length 8 $(1, 1, ..., 1), (2, 2, ..., 2), ..., (10, 10, ..., 10)$. Of all the triplets that work they cannot be extended to form another point on the line in th... |
DeepScaleR | [
{
"role": "user",
"content": "A $150$-gon $Q_1$ is drawn in the Cartesian plane. The sum of the $x$-coordinates of the $150$ vertices equals $3000$. The midpoints of the sides of $Q_1$ form a second $150$-gon, $Q_2$. Finally, the midpoints of the sides of $Q_2$ form a third $150$-gon, $Q_3$. Find the sum of... | math | {
"style": "rule",
"ground_truth": "3000"
} | {
"index": 588,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A $150\\times 324\\times 375$ rectangular solid is made by gluing together $1\\times 1\\times 1$ cubes. An internal diagonal of this solid passes through the interiors of how many of the $1\\times 1\\times 1$ cubes?"
}
] | math | {
"style": "rule",
"ground_truth": "768"
} | {
"index": 589,
"solution": "Consider a point travelling across the internal diagonal, and let the internal diagonal have a length of $d$. The point enters a new unit cube in the $x,y,z$ dimensions at multiples of $\\frac{d}{150}, \\frac{d}{324}, \\frac{d}{375}$ respectively. We proceed by using PIE.\n The point en... |
DeepScaleR | [
{
"role": "user",
"content": "A $16$-quart radiator is filled with water. Four quarts are removed and replaced with pure antifreeze liquid. Then four quarts of the mixture are removed and replaced with pure antifreeze. This is done a third and a fourth time. The fractional part of the final mixture that is ... | math | {
"style": "rule",
"ground_truth": "\\frac{81}{256}"
} | {
"index": 590,
"solution": "1. **Initial Setup**: The radiator starts with 16 quarts of water.\n\n2. **First Replacement**: \n - 4 quarts of water are removed, leaving $16 - 4 = 12$ quarts of water.\n - 4 quarts of antifreeze are added, making the total still 16 quarts.\n - The fraction of water remaining is... |
DeepScaleR | [
{
"role": "user",
"content": "A $180^\\circ$ rotation around the origin in the counter-clockwise direction is applied to $-6 - 3i.$ What is the resulting complex number?"
}
] | math | {
"style": "rule",
"ground_truth": "6 + 3i"
} | {
"index": 591,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A $2$ by $2$ square is divided into four $1$ by $1$ squares. Each of the small squares is to be painted either green or red. In how many different ways can the painting be accomplished so that no green square shares its top or right side with any red square? There may be as ... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 592,
"solution": "To solve this problem, we need to consider all possible configurations of green and red squares in the $2 \\times 2$ grid such that no green square shares its top or right side with a red square. We will analyze each case based on the number of green squares.\n\n#### Case 1: No green sq... |
DeepScaleR | [
{
"role": "user",
"content": "A $20$-quart radiator initially contains a mixture of $18$ quarts of water and $2$ quarts of antifreeze. Six quarts of the mixture are removed and replaced with pure antifreeze liquid. This process is repeated three more times. Calculate the fractional part of the final mixture... | math | {
"style": "rule",
"ground_truth": "\\frac{4.322}{20}"
} | {
"index": 593,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A $2018 \\times 2018$ square was cut into rectangles with integer side lengths. Some of these rectangles were used to form a $2000 \\times 2000$ square, and the remaining rectangles were used to form a rectangle whose length differs from its width by less than 40. Find the peri... | math | {
"style": "rule",
"ground_truth": "1076"
} | {
"index": 594,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A $25$ foot ladder is placed against a vertical wall of a building. The foot of the ladder is $7$ feet from the base of the building. If the top of the ladder slips $4$ feet, then the foot of the ladder will slide:"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 595,
"solution": "1. **Identify the initial setup**: A $25$ foot ladder is placed against a vertical wall, with the foot of the ladder $7$ feet from the base of the building. This forms a right triangle where the ladder acts as the hypotenuse.\n\n2. **Calculate the initial height of the ladder on the wal... |
DeepScaleR | [
{
"role": "user",
"content": "A $2\\times 3$ rectangle and a $3\\times 4$ rectangle are contained within a square without overlapping at any interior point, and the sides of the square are parallel to the sides of the two given rectangles. What is the smallest possible area of the square?"
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 596,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A $3 \\times 3$ square is partitioned into $9$ unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is then rotated $90^{\\circ}$ clockwise about its center, and every white square i... | math | {
"style": "rule",
"ground_truth": "\\frac{49}{512}"
} | {
"index": 597,
"solution": "1. **Identify the invariant**: The center square remains unchanged after a $90^\\circ$ rotation. Therefore, for the entire grid to be black after the process, the center square must initially be black. The probability of the center square being black is $\\frac{1}{2}$.\n\n2. **Consider ... |
DeepScaleR | [
{
"role": "user",
"content": "A $3 \\times 3$ table is initially filled with zeros. In one move, any $2 \\times 2$ square in the table is chosen, and all zeros in it are replaced with crosses, and all crosses with zeros. Let's call a \"pattern\" any arrangement of crosses and zeros in the table. How many di... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 598,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A $3 \\times 3$ table starts with every entry equal to 0 and is modified using the following steps: (i) adding 1 to all three numbers in any row; (ii) adding 2 to all three numbers in any column. After step (i) has been used a total of $a$ times and step (ii) has been used a to... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 599,
"solution": "Since the second column includes the number 1, then step (ii) was never used on the second column, otherwise each entry would be at least 2 . To generate the 1,3 and 2 in the second column, we thus need to have used step (i) 1 time on row 1,3 times on row 2 , and 2 times on row 3 . This... |
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