data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Two solutions of\n\\[x^4 - 3x^3 + 5x^2 - 27x - 36 = 0\\]are pure imaginary. Enter these solutions, separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "3i,-3i"
} | {
"index": 36600,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two spheres are inscribed in a dihedral angle in such a way that they touch each other. The radius of one sphere is 1.5 times the radius of the other, and the line connecting the centers of the spheres forms an angle of $30^{\\circ}$ with the edge of the dihedral angle. Find th... | math | {
"style": "rule",
"ground_truth": "0.5"
} | {
"index": 36601,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two spheres are inscribed in a dihedral angle such that they touch each other. The radius of one sphere is 4 times that of the other, and the line connecting the centers of the spheres forms an angle of \\(60^\\circ\\) with the edge of the dihedral angle. Find the measure of th... | math | {
"style": "rule",
"ground_truth": "0.04"
} | {
"index": 36602,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two spheres touch the plane of triangle \\(ABC\\) at points \\(A\\) and \\(B\\) and are located on opposite sides of this plane. The sum of the radii of these spheres is 9, and the distance between their centers is \\(\\sqrt{305}\\). The center of a third sphere with a radius o... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{14}"
} | {
"index": 36603,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two spheres touch the plane of triangle \\(ABC\\) at points \\(B\\) and \\(C\\) and are located on opposite sides of this plane. The sum of the radii of these spheres is 11, and the distance between their centers is \\(5 \\sqrt{17}\\). The center of a third sphere with a radius... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{19}"
} | {
"index": 36604,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two spheres touch the plane of triangle \\(ABC\\) at points \\(B\\) and \\(C\\) and are located on opposite sides of this plane. The sum of the radii of these spheres is 12, and the distance between their centers is \\(4 \\sqrt{29}\\). The center of a third sphere with radius 8... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{5}"
} | {
"index": 36605,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two spheres with radii $36$ and one sphere with radius $13$ are each externally tangent to the other two spheres and to two different planes $\\mathcal{P}$ and $\\mathcal{Q}$. The intersection of planes $\\mathcal{P}$ and $\\mathcal{Q}$ is the line $\\ell$. The distance from li... | math | {
"style": "rule",
"ground_truth": "335"
} | {
"index": 36606,
"solution": "The isosceles triangle of centers $O_1 O_2 O$ ($O$ is the center of sphere of radii $13$) has sides $O_1 O = O_2 O = 36 + 13 = 49,$ and $O_1 O_2 = 36 + 36 = 72.$\n Let $N$ be the midpoint $O_1 O_2$.\n The isosceles triangle of points of tangency $T_1 T_2 T$ has sides $T_1 T = T_2 T = ... |
DeepScaleR | [
{
"role": "user",
"content": "Two spinners are divided into fifths and sixths, respectively. If each of these spinners is spun once, what is the probability that the product of the results of the two spins will be an even number? The numbers on the first spinner are 3, 5, 6, 7, and 9. The numbers on the sec... | math | {
"style": "rule",
"ground_truth": "\\frac{11}{15}"
} | {
"index": 36607,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two springs with stiffnesses of $6 \\, \\text{kN} / \\text{m}$ and $12 \\, \\text{kN} / \\text{m}$ are connected in series. How much work is required to stretch this system by 10 cm?"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 36608,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two square napkins with dimensions \\(1 \\times 1\\) and \\(2 \\times 2\\) are placed on a table so that the corner of the larger napkin falls into the center of the smaller napkin. What is the maximum area of the table that the napkins can cover?"
}
] | math | {
"style": "rule",
"ground_truth": "4.75"
} | {
"index": 36609,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two squares of a $7\\times 7$ checkerboard are painted yellow, and the rest are painted green. Two color schemes are equivalent if one can be obtained from the other by applying a rotation in the plane board. How many inequivalent color schemes are possible?"
}
] | math | {
"style": "rule",
"ground_truth": "300"
} | {
"index": 36610,
"solution": "There are 4 cases: 1. The center square is occupied, in which there are $12$ cases. 2. The center square isn't occupied and the two squares that are opposite to each other with respect to the center square, in which there are $12$ cases. 3. The center square isn't occupied and the two... |
DeepScaleR | [
{
"role": "user",
"content": "Two standard 6-sided dice are rolled. What is the probability that the sum rolled is a perfect square?"
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{7}{36}"
} | {
"index": 36611,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two standard 6-sided dice are tossed. What is the probability that the sum of the numbers shown on the dice is a prime number? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{12}"
} | {
"index": 36612,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two standard decks of cards are combined, making a total of 104 cards (each deck contains 13 ranks and 4 suits, with all combinations unique within its own deck). The decks are randomly shuffled together. What is the probability that the top card is an Ace of $\\heartsuit$?"
... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{52}"
} | {
"index": 36613,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two standard dice are rolled. What is the expected number of 1's obtained? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 36614,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two standard six-sided dice are rolled. What is the probability that the product of the two numbers rolled is 12?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{36}"
} | {
"index": 36615,
"solution": "When two standard six-sided dice are rolled, there are $6 \\times 6 = 36$ possibilities for the pair of numbers that are rolled. Of these, the pairs $2 \\times 6, 3 \\times 4, 4 \\times 3$, and $6 \\times 2$ each give 12. (If one of the numbers rolled is 1 or 5, the product cannot be ... |
DeepScaleR | [
{
"role": "user",
"content": "Two stores have warehouses storing millet: the first warehouse has 16 tons more than the second. Every night, exactly at midnight, the owner of each store steals a quarter of their competitor's millet and moves it to their own warehouse. After 10 nights, the thieves were caught... | math | {
"style": "rule",
"ground_truth": "2^{-6}"
} | {
"index": 36616,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two strips of width 2 overlap at an angle of 60 degrees inside a rectangle of dimensions 4 units by 3 units. Find the area of the overlap, considering that the angle is measured from the horizontal line of the rectangle.\nA) $\\frac{2\\sqrt{3}}{3}$\nB) $\\frac{8\\sqrt{3}}{9}$\n... | math | {
"style": "rule",
"ground_truth": "\\frac{4\\sqrt{3}}{3}"
} | {
"index": 36617,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two students are having a pie eating contest. The first student eats $\\frac{6}{7}$ of one pie. The second student eats $\\frac{3}{4}$ of one pie. How much more pie did the first student finish than the second student? Express your answer as a fraction of one pie, reduced to si... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{28}"
} | {
"index": 36618,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two students are preparing to register for the independent admission tests of Zhejiang University, Fudan University, and Shanghai Jiao Tong University, with the requirement that each person can choose at most two schools. Calculate the number of different registration results."... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 36619,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two students were asked to add two positive integers. Alice subtracted the two numbers by mistake and obtained 3. Bob mistakenly multiplied the same two integers and got 63. What was the correct sum of the two integers?"
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 36620,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two students, A and B, are doing rope skipping training. It is known that A takes the same amount of time to skip 300 times as B does to skip 270 times. A skips 19 more times per minute than B. Find out how many times each of them skips per minute."
}
] | math | {
"style": "rule",
"ground_truth": "171"
} | {
"index": 36621,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two students, A and B, are playing table tennis. They have agreed on the following rules: ① Each point won earns 1 point; ② They use a three-point serve system, meaning they switch serving every three points. Assuming that when A serves, the probability of A winning a point is ... | math | {
"style": "rule",
"ground_truth": "\\frac{441}{1000}"
} | {
"index": 36622,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two students, A and B, are preparing to have a table tennis match during their physical education class. Assuming that the probability of A winning against B in each game is $\\frac{1}{3}$, the match follows a best-of-three format (the first player to win two games wins the mat... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{27}"
} | {
"index": 36623,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two students, A and B, each choose 2 out of 6 extracurricular reading materials. Calculate the number of ways in which the two students choose extracurricular reading materials such that they have exactly 1 material in common."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 36624,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two students, A and B, each select two courses from four elective courses. The probability that they share exactly one course in common is ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 36625,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two subsets of the set $S=\\{a, b, c, d, e\\}$ are to be chosen so that their union is $S$ and their intersection contains exactly two elements. In how many ways can this be done, assuming that the order in which the subsets are chosen does not matter?"
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 36626,
"solution": "\nTo solve this problem, we need to choose two subsets of the set $S = \\{a, b, c, d, e\\}$ such that their union is $S$ and their intersection contains exactly two elements. We also need to ensure that the order in which the subsets are chosen does not matter.\n\n#### Step 1: Choose ... |
DeepScaleR | [
{
"role": "user",
"content": "Two subsets of the set $T = \\{w, x, y, z, v\\}$ need to be chosen so that their union is $T$ and their intersection contains exactly three elements. How many ways can this be accomplished, assuming the subsets are chosen without considering the order?"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 36627,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two swimmers, at opposite ends of a $90$-foot pool, start to swim the length of the pool, one at the rate of $3$ feet per second, the other at $2$ feet per second. They swim back and forth for $12$ minutes. Allowing no loss of times at the turns, find the number of times they p... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 36628,
"solution": "1. **Calculate the time taken by each swimmer to swim the length of the pool:**\n - The first swimmer swims at $3$ feet per second, so the time to swim $90$ feet is:\n \\[\n \\frac{90 \\text{ feet}}{3 \\text{ feet/second}} = 30 \\text{ seconds}\n \\]\n - The second swi... |
DeepScaleR | [
{
"role": "user",
"content": "Two symmetrical coins are flipped. What is the probability that both coins show numbers on their upper sides?"
}
] | math | {
"style": "rule",
"ground_truth": "0.25"
} | {
"index": 36629,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two tangents $AT$ and $BT$ touch a circle at $A$ and $B$ , respectively, and meet perpendicularly at $T$ . $Q$ is on $AT$ , $S$ is on $BT$ , and $R$ is on the circle, so that $QRST$ is a rectangle with $QT = 8$ and $ST = 9$ . Determine the radius of the cir... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"index": 36630,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two tangents are drawn to a circle from an exterior point $A$; they touch the circle at points $B$ and $C$ respectively. \nA third tangent intersects segment $AB$ in $P$ and $AC$ in $R$, and touches the circle at $Q$. If $AB=20$, then the perimeter of $\\triangle APR$ is"
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 36631,
"solution": "1. **Identify Tangent Properties**: From the problem, we know that $AB$ and $AC$ are tangents from point $A$ to the circle, touching the circle at points $B$ and $C$ respectively. By the tangent properties, $AB = AC$.\n\n2. **Tangent Lengths**: Since $AB = 20$, we also have $AC = 20$.... |
DeepScaleR | [
{
"role": "user",
"content": "Two tangents are drawn to a circle from an exterior point $A$; they touch the circle at points $B$ and $C$ respectively. A third tangent intersects segment $AB$ in $P$ and $AC$ in $R$, and touches the circle at $Q$. Given that $AB=25$ and $PQ = QR = 2.5$, calculate the perimete... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 36632,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two tangents are drawn to a circle from an exterior point A; they touch the circle at points B and C respectively. A third tangent intersects segment AB in P and AC in R, and touches the circle at Q. If AB = 24, and the lengths BP = PQ = x and QR = CR = y with x + y = 12, find ... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 36633,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two tangents to a circle are drawn from a point $A$. The points of contact $B$ and $C$ divide the circle into arcs with lengths in the ratio $2 : 3$. What is the degree measure of $\\angle{BAC}$?"
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 36634,
"solution": "1. **Identify the Geometry and Key Properties**: \n Let the center of the circle be $O$. The lines $AB$ and $AC$ are tangents to the circle at points $B$ and $C$, respectively. By the property of tangents, the radii $OB$ and $OC$ are perpendicular to $AB$ and $AC$, respectively. Th... |
DeepScaleR | [
{
"role": "user",
"content": "Two teachers and 4 students need to be divided into 2 groups, each consisting of 1 teacher and 2 students. Calculate the number of different arrangements."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 36635,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two teachers are taking a photo with 3 male students and 3 female students lined up in a row. The teachers can only stand at the ends, and the male students cannot be adjacent. How many different ways are there to arrange the photo?"
}
] | math | {
"style": "rule",
"ground_truth": "288"
} | {
"index": 36636,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two teachers, A and B, and four students stand in a row. (Explain the process, list the expressions, and calculate the results, expressing the results in numbers)<br/>$(1)$ The two teachers cannot be adjacent. How many ways are there to arrange them?<br/>$(2)$ A is to the left ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 36637,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two teams of 20 people each participated in a relay race from Moscow to Petushki. Each team divided the distance into 20 segments, not necessarily of equal length, and assigned them to participants such that each member runs exactly one segment (each participant maintains a con... | math | {
"style": "rule",
"ground_truth": "38"
} | {
"index": 36638,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two teams, Team A and Team B, are playing in a basketball finals series that uses a \"best of seven\" format (the first team to win four games wins the series and the finals end). Based on previous game results, Team A's home and away schedule is arranged as \"home, home, away,... | math | {
"style": "rule",
"ground_truth": "0.18"
} | {
"index": 36639,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two teams, Team A and Team B, participate in a table tennis group match. The ratio of their strength is $3:2$. Assuming both teams play to their normal levels, find the probability that Team A wins after playing 4 games in a best-of-5 series (3 wins out of 5 games)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{162}{625}"
} | {
"index": 36640,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two thirds of a pitcher is filled with orange juice and the remaining part is filled with apple juice. The pitcher is emptied by pouring an equal amount of the mixture into each of 6 cups. Calculate the percentage of the total capacity of the pitcher that each cup receives."
... | math | {
"style": "rule",
"ground_truth": "16.67\\%"
} | {
"index": 36641,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two thousand points are given on a circle. Label one of the points $1$. From this point, count $2$ points in the clockwise direction and label this point $2$. From the point labeled $2$, count $3$ points in the clockwise direction and label this point $3$. (See figure.) Continu... | math | {
"style": "rule",
"ground_truth": "118"
} | {
"index": 36642,
"solution": "The label $1993$ will occur on the $\\frac12(1993)(1994) \\pmod{2000}$th point around the circle. (Starting from 1) A number $n$ will only occupy the same point on the circle if $\\frac12(n)(n + 1)\\equiv \\frac12(1993)(1994) \\pmod{2000}$.\n Simplifying this expression, we see that $... |
DeepScaleR | [
{
"role": "user",
"content": "Two three-letter strings, $aaa^{}_{}$ and $bbb^{}_{}$, are transmitted electronically. Each string is sent letter by letter. Due to faulty equipment, each of the six letters has a 1/3 chance of being received incorrectly, as an $a^{}_{}$ when it should have been a $b^{}_{}$, or... | math | {
"style": "rule",
"ground_truth": "532"
} | {
"index": 36643,
"solution": "Consider $n$ letter strings instead. If the first letters all get transmitted correctly, then the $a$ string will be first. Otherwise, the only way is for both of the first letters to be the same, and then we consider the next $n-1$ letter string following the first letter. This easil... |
DeepScaleR | [
{
"role": "user",
"content": "Two tigers, Alice and Betty, run in the same direction around a circular track with a circumference of 400 meters. Alice runs at a speed of \\(10 \\, \\text{m/s}\\) and Betty runs at \\(15 \\, \\text{m/s}\\). Betty gives Alice a 40 meter head start before they both start runnin... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 36644,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two tour guides are leading six tourists. The guides decide to split up. Each tourist must choose one of the guides, but with the stipulation that each guide must take at least one tourist. How many different groupings of guides and tourists are possible?"
}
] | math | {
"style": "rule",
"ground_truth": "62"
} | {
"index": 36645,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two trains are moving towards each other on parallel tracks - one with a speed of 60 km/h and the other with a speed of 80 km/h. A passenger sitting in the second train noticed that the first train passed by him in 6 seconds. What is the length of the first train?"
}
] | math | {
"style": "rule",
"ground_truth": "233.33"
} | {
"index": 36646,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two trains, each composed of 15 identical cars, were moving towards each other at constant speeds. Exactly 28 seconds after their first cars met, a passenger named Sasha, sitting in the compartment of the third car, passed by a passenger named Valera in the opposite train. More... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 36647,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two transformations are applied to the complex number $-3 - 8i$:\n\nA $45^\\circ$ rotation around the origin in the counter-clockwise direction.\nA dilation, centered at the origin, with scale factor $\\sqrt{2}.$\n\nWhat is the resulting complex number?"
}
] | math | {
"style": "rule",
"ground_truth": "5 - 11i"
} | {
"index": 36648,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two triangles are similar. The ratio of their areas is 1:4. If the height of the smaller triangle is 3 cm, how long is the corresponding height of the larger triangle, in centimeters?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 36649,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two trucks are transporting identical sacks of flour from France to Spain. The first truck carries 118 sacks, and the second one carries only 40. Since the drivers of these trucks lack the pesetas to pay the customs duty, the first driver leaves 10 sacks with the customs office... | math | {
"style": "rule",
"ground_truth": "1600"
} | {
"index": 36650,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two types of anti-inflammatory drugs must be selected from $X_{1}$, $X_{2}$, $X_{3}$, $X_{4}$, $X_{5}$, with the restriction that $X_{1}$ and $X_{2}$ must be used together, and one type of antipyretic drug must be selected from $T_{1}$, $T_{2}$, $T_{3}$, $T_{4}$, with the furth... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 36651,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two types of shapes composed of unit squares, each with an area of 3, are placed in an $8 \\times 14$ rectangular grid. It is required that there are no common points between any two shapes. What is the maximum number of these two types of shapes that can be placed in the $8 \\... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 36652,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two types of steel plates need to be cut into three sizes $A$, $B$, and $C$. The number of each size that can be obtained from each type of steel plate is shown in the table:\n\n\\begin{tabular}{|l|c|c|c|}\n\\hline & Size $A$ & Size $B$ & Size $C$ \\\\\n\\hline First type of st... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 36653,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two unit squares $S_{1}$ and $S_{2}$ have horizontal and vertical sides. Let $x$ be the minimum distance between a point in $S_{1}$ and a point in $S_{2}$, and let $y$ be the maximum distance between a point in $S_{1}$ and a point in $S_{2}$. Given that $x=5$, the difference be... | math | {
"style": "rule",
"ground_truth": "472"
} | {
"index": 36654,
"solution": "Consider what must happen in order for the minimum distance to be exactly 5 . Let one square, say $S_{1}$ have vertices of $(0,0),(0,1),(1,0)$, and $(1,1)$. Further, assume WLOG that the center of $S_{2}$ is above the line $y=\\frac{1}{2}$ and to the right of the line $x=\\frac{1}{2}$... |
DeepScaleR | [
{
"role": "user",
"content": "Two unit squares are selected at random without replacement from an $n \\times n$ grid of unit squares. Find the least positive integer $n$ such that the probability that the two selected unit squares are horizontally or vertically adjacent is less than $\\frac{1}{2015}$."
}
... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 36655,
"solution": "There are 3 cases in this problem. Number one, the center squares. Two, the edges, and three, the corners. The probability of getting one center square and an adjacent square is $\\frac{(n-2)(n-2)}{n^2}$ multiplied by $\\frac{4}{n^2 -1}$. Add that to the probability of an edge and an ... |
DeepScaleR | [
{
"role": "user",
"content": "Two vertical towers, \\( AB \\) and \\( CD \\), are located \\( 16 \\mathrm{~m} \\) apart on flat ground. Tower \\( AB \\) is \\( 18 \\mathrm{~m} \\) tall and tower \\( CD \\) is \\( 30 \\mathrm{~m} \\) tall. Ropes are tied from \\( A \\) to \\( C \\) and from \\( B \\) to \\( ... | math | {
"style": "rule",
"ground_truth": "54"
} | {
"index": 36656,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two vertices of a cube are given in space. The locus of points that could be a third vertex of the cube is the union of $n$ circles. Find $n$."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 36657,
"solution": "Let the distance between the two given vertices be 1. If the two given vertices are adjacent, then the other vertices lie on four circles, two of radius 1 and two of radius $\\sqrt{2}$. If the two vertices are separated by a diagonal of a face of the cube, then the locus of possible v... |
DeepScaleR | [
{
"role": "user",
"content": "Two vertices of a square with an area of \\( 256 \\, \\text{cm}^2 \\) lie on a circle, while the other two vertices lie on a tangent to this circle. Find the radius of the circle."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 36658,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two vertices of an obtuse triangle are $(6,4)$ and $(0,0)$. The third vertex is located on the negative branch of the $x$-axis. What are the coordinates of the third vertex if the area of the triangle is 30 square units?"
}
] | math | {
"style": "rule",
"ground_truth": "(-15, 0)"
} | {
"index": 36659,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two walls and the ceiling of a room meet at right angles at point $P.$ A fly is in the air one meter from one wall, eight meters from the other wall, and nine meters from point $P$. How many meters is the fly from the ceiling?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 36660,
"solution": "1. **Identify the coordinates of the fly and point $P$:**\n - Let's assume point $P$ is at the origin $(0,0,0)$ of a three-dimensional coordinate system.\n - The fly is 1 meter from one wall, 8 meters from another wall, and 9 meters from point $P$. We can assume the fly is at poin... |
DeepScaleR | [
{
"role": "user",
"content": "Two water droplets fall freely one after another from a $300 \\mathrm{~m}$ high cliff. The first droplet has already fallen $\\frac{1}{1000} \\mathrm{~mm}$ when the second one starts falling.\n\nHow many millimeters apart will the two droplets be at the moment the first one rea... | math | {
"style": "rule",
"ground_truth": "34.6"
} | {
"index": 36661,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two water particles fall freely in succession from a $300 \\mathrm{~m}$ high cliff. The first one has already fallen $\\frac{1}{1000} \\mathrm{~mm}$ when the second one starts to fall.\n\nHow far apart will the two particles be at the moment when the first particle reaches the ... | math | {
"style": "rule",
"ground_truth": "34.6"
} | {
"index": 36662,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two watermelons and one banana together weigh 8100 grams, and two watermelons and three bananas together weigh 8300 grams, calculate the weight of one watermelon and one banana."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 36663,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two wholesalers, A and B, sell the same brand of teapots and teacups at the same price, with the teapot priced at 30 yuan each and the teacup at 5 yuan each. Both stores are offering a promotional sale: Store A has a 'buy one get one free' promotion (buy a teapot and get a teac... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 36664,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two workers started paving paths in a park simultaneously from point $A$. The first worker paves the section $A-B-C$ and the second worker paves the section $A-D-E-F-C$. They worked at constant speeds and finished at the same time, having spent 9 hours on the job. It is known t... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 36665,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two workers were assigned to produce a batch of identical parts; after the first worked for \\(a\\) hours and the second for \\(0.6a\\) hours, it turned out that they had completed \\(\\frac{5}{n}\\) of the entire job. After working together for another \\(0.6a\\) hours, they f... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 36666,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be $2$ : $1$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 36667,
"solution": "1. **Define Variables:**\n Let $p$ be Pete's current age and $c$ be Claire's current age.\n\n2. **Translate the Problem into Equations:**\n - From the statement \"Two years ago Pete was three times as old as his cousin Claire,\" we have:\n \\[\n p - 2 = 3(c - 2)\n \\]\... |
DeepScaleR | [
{
"role": "user",
"content": "Two years ago there were 20 trailer homes on Elm Street with an average age of 18 years. At that time, a group of brand new trailer homes was then added to Elm Street. Today, the average age of all the trailer homes on Elm Street is 14 years. How many new trailer homes were add... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 36668,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two-thirds of a number plus two times another number equals 20. If one-fourth of the first number minus the second number equals 2, find these two numbers."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{22}{7}"
} | {
"index": 36669,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two-thirds of the people in a room are seated in three-fourths of the chairs. The rest of the people are standing. If there are 6 empty chairs, how many people are in the room?"
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 36670,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Tyler has entered a buffet line in which he chooses one kind of meat, two different vegetables and one dessert. If the order of food items is not important, how many different meals might he choose?\n\n$\\bullet$ Meat: beef, chicken, pork\n\n$\\bullet$ Vegetables: baked bean... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 36671,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Tyler rolls two $ 4025 $ sided fair dice with sides numbered $ 1, \\dots , 4025 $ . Given that the number on the first die is greater than or equal to the number on the second die, what is the probability that the number on the first die is less than or equal to $ 2012 $ ?"
... | math | {
"style": "rule",
"ground_truth": "1006/4025"
} | {
"index": 36672,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Tyrone had $97$ marbles and Eric had $11$ marbles. Tyrone then gave some of his marbles to Eric so that Tyrone ended with twice as many marbles as Eric. How many marbles did Tyrone give to Eric?"
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 36673,
"solution": "1. **Identify the total number of marbles**: Initially, Tyrone has 97 marbles and Eric has 11 marbles. The total number of marbles is:\n \\[\n 97 + 11 = 108\n \\]\n\n2. **Set up the equation after redistribution**: Let's denote the number of marbles Tyrone gives to Eric as $x$. ... |
DeepScaleR | [
{
"role": "user",
"content": "Una rolls $6$ standard $6$-sided dice simultaneously and calculates the product of the $6$ numbers obtained. What is the probability that the product is divisible by $4$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{63}{64}"
} | {
"index": 36674,
"solution": "To solve this problem, we will use complementary counting to determine the probability that the product of the numbers rolled on the six dice is not divisible by $4$. We then subtract this probability from $1$ to find the probability that the product is divisible by $4$.\n\n#### Step ... |
DeepScaleR | [
{
"role": "user",
"content": "Una rolls 8 standard 6-sided dice simultaneously and calculates the product of the 8 numbers obtained. What is the probability that the product is divisible by 8?\nA) $\\frac{1}{4}$\nB) $\\frac{57}{64}$\nC) $\\frac{199}{256}$\nD) $\\frac{57}{256}$\nE) $\\frac{63}{64}$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{199}{256}"
} | {
"index": 36675,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Una rolls 8 standard 6-sided dice simultaneously and calculates the product of the 8 numbers obtained. What is the probability that the product is divisible by 8?\nA) $\\frac{273}{288}$ \nB) $\\frac{275}{288}$ \nC) $\\frac{277}{288}$ \nD) $\\frac{279}{288}$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{277}{288}"
} | {
"index": 36676,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Uncle Wang has some chickens, rabbits, and geese. Geese stand on two legs during the day and on one leg at night; chickens tuck their heads under their wings when sleeping. Careful Yue Yue discovered that the difference between the number of legs and the number of heads is alwa... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 36677,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Unconventional dice are to be designed such that the six faces are marked with numbers from $1$ to $6$ with $1$ and $2$ appearing on opposite faces. Further, each face is colored either red or yellow with opposite faces always of the same color. Two dice are considered ... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 36678,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Undecillion years ago in a galaxy far, far away, there were four space stations in the three-dimensional space, each pair spaced 1 light year away from each other. Admiral Ackbar wanted to establish a base somewhere in space such that the sum of squares of the distances from th... | math | {
"style": "rule",
"ground_truth": "\\frac{27 \\sqrt{6}}{8} \\pi"
} | {
"index": 36679,
"solution": "Solution 1: Set up a coordinate system where the coordinates of the stations are $\\left(\\frac{1}{2 \\sqrt{2}}, \\frac{1}{2 \\sqrt{2}}, \\frac{1}{2 \\sqrt{2}}\\right)$, $\\left(-\\frac{1}{2 \\sqrt{2}},-\\frac{1}{2 \\sqrt{2}}, \\frac{1}{2 \\sqrt{2}}\\right),\\left(\\frac{1}{2 \\sqrt{2... |
DeepScaleR | [
{
"role": "user",
"content": "Under normal circumstances, for people aged between 18 and 38 years old, the regression equation for weight $y$ (kg) based on height $x$ (cm) is $y=0.72x-58.5$. Zhang Honghong, who is neither fat nor thin, has a height of 1.78 meters. His weight should be around \\_\\_\\_\\_\\_... | math | {
"style": "rule",
"ground_truth": "70"
} | {
"index": 36680,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Under normal circumstances, for people aged between 18 and 38, the regression equation of weight $y$ (kg) to height $x$ (cm) is $\\overset{\\land }{y} = 0.72x - 58.2$. Zhang Hong, who is 20 years old and has a height of 178 cm, should have a weight of approximately \\_\\_\\_\\_... | math | {
"style": "rule",
"ground_truth": "69.96"
} | {
"index": 36681,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Under the call for the development of the western region by the country, a certain western enterprise received a $4$ million yuan interest-free loan for equipment renewal. It is predicted that after the equipment renewal, the income of the enterprise in the first month is $200,... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 36682,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Understanding and trying: When calculating $\\left(-4\\right)^{2}-\\left(-3\\right)\\times \\left(-5\\right)$, there are two methods; Method 1, please calculate directly: $\\left(-4\\right)^{2}-\\left(-3\\right)\\times \\left(-5\\right)$, Method 2, use letters to represent numb... | math | {
"style": "rule",
"ground_truth": "-1.35"
} | {
"index": 36683,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Unit circle $\\Omega$ has points $X, Y, Z$ on its circumference so that $X Y Z$ is an equilateral triangle. Let $W$ be a point other than $X$ in the plane such that triangle $W Y Z$ is also equilateral. Determine the area of the region inside triangle $W Y Z$ that lies outside ... | math | {
"style": "rule",
"ground_truth": "$\\frac{3 \\sqrt{3}-\\pi}{3}$"
} | {
"index": 36684,
"solution": "Let $O$ be the center of the circle. Then, we note that since $\\angle W Y Z=60^{\\circ}=\\angle Y X Z$, that $Y W$ is tangent to $\\Omega$. Similarly, $W Z$ is tangent to $\\Omega$. Now, we note that the circular segment corresponding to $Y Z$ is equal to $\\frac{1}{3}$ the area of $... |
DeepScaleR | [
{
"role": "user",
"content": "Unit square $ABCD$ is divided into four rectangles by $EF$ and $GH$ , with $BF = \\frac14$ . $EF$ is parallel to $AB$ and $GH$ parallel to $BC$ . $EF$ and $GH$ meet at point $P$ . Suppose $BF + DH = FH$ , calculate the nearest integer to the degree of $\\ang... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 36685,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Unit squares $A B C D$ and $E F G H$ have centers $O_{1}$ and $O_{2}$ respectively, and are originally situated such that $B$ and $E$ are at the same position and $C$ and $H$ are at the same position. The squares then rotate clockwise about their centers at the rate of one revo... | math | {
"style": "rule",
"ground_truth": "\\frac{2-\\sqrt{3}}{4}"
} | {
"index": 36686,
"solution": "Note that $A E=B F=C G=D H=1$ at all times. Suppose that the squares have rotated $\\theta$ radians. Then $\\angle O_{1} O_{2} H=\\frac{\\pi}{4}-\\theta=\\angle O_{1} D H$, so $\\angle H D C=\\frac{\\pi}{4}-\\angle O_{1} D H=\\theta$. Let $P$ be the intersection of $A B$ and $E H$ and... |
DeepScaleR | [
{
"role": "user",
"content": "Unlucky Emelya was given several metal balls. He broke the 3 largest ones (their mass was 35% of the total mass of all the balls), then lost the 3 smallest ones, and brought home the remaining balls (their mass was \\( \\frac{8}{13} \\) of the unbroken ones). How many balls was... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 36687,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Upon cutting a certain rectangle in half, you obtain two rectangles that are scaled down versions of the original. What is the ratio of the longer side length to the shorter side length?"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 36688,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Uri buys two burgers and a soda for $\\$2.10$, and Gen buys a burger and two sodas for $\\$2.40$. How many cents does a soda cost?"
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 36689,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Urn A contains 4 white balls and 2 red balls. Urn B contains 3 red balls and 3 black balls. An urn is randomly selected, and then a ball inside of that urn is removed. We then repeat the process of selecting an urn and drawing out a ball, without returning the first ball. What ... | math | {
"style": "rule",
"ground_truth": "7/15"
} | {
"index": 36690,
"solution": "This is a case of conditional probability; the answer is the probability that the first ball is red and the second ball is black, divided by the probability that the second ball is black. First, we compute the numerator. If the first ball is drawn from Urn A, we have a probability of ... |
DeepScaleR | [
{
"role": "user",
"content": "Usain runs one lap around the school stadium at a constant speed, and photographers Arina and Marina are positioned near the track. After the start, for 4 seconds, Usain was closer to Arina, then for 21 seconds he was closer to Marina, and then until the finish, he was again cl... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"index": 36691,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Use $ \\log_{10} 2 \\equal{} 0.301,\\ \\log_{10} 3 \\equal{} 0.477,\\ \\log_{10} 7 \\equal{} 0.845$ to find the value of $ \\log_{10} (10!)$ .\r\nNote that you must answer according to the rules:fractional part of $ 0.5$ and higher is rounded up, and everything strictly le... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 36692,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Use Horner's Rule to find the value of $v_2$ when the polynomial $f(x) = x^5 + 4x^4 + x^2 + 20x + 16$ is evaluated at $x = -2$."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"index": 36693,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Use Horner's method to find the value of the polynomial $f(x) = -6x^4 + 5x^3 + 2x + 6$ at $x=3$, denoted as $v_3$."
}
] | math | {
"style": "rule",
"ground_truth": "-115"
} | {
"index": 36694,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Use Horner's method to find the value of the polynomial $f(x) = 5x^5 + 2x^4 + 3.5x^3 - 2.6x^2 + 1.7x - 0.8$ when $x=1$, and find the value of $v_3$."
}
] | math | {
"style": "rule",
"ground_truth": "8.8"
} | {
"index": 36695,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Use \\((a, b)\\) to represent the greatest common divisor of \\(a\\) and \\(b\\). Let \\(n\\) be an integer greater than 2021, and \\((63, n+120) = 21\\) and \\((n+63, 120) = 60\\). What is the sum of the digits of the smallest \\(n\\) that satisfies the above conditions?"
}
... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 36696,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Use all digits from 1 to 9 to form three three-digit numbers such that their product is:\n\na) the smallest; \nb) the largest."
}
] | math | {
"style": "rule",
"ground_truth": "941 \\times 852 \\times 763"
} | {
"index": 36697,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Use each of the digits 3, 4, 6, 8 and 9 exactly once to create the greatest possible five-digit multiple of 6. What is that multiple of 6?"
}
] | math | {
"style": "rule",
"ground_truth": "98,634"
} | {
"index": 36698,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Use each of the five digits $2, 4, 6, 7$ and $9$ only once to form a three-digit integer and a two-digit integer which will be multiplied together. What is the three-digit integer that results in the greatest product?"
}
] | math | {
"style": "rule",
"ground_truth": "762"
} | {
"index": 36699,
"solution": ""
} |
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