data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "For which value of \\( x \\) is \\( x^3 < x^2 \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 12100,
"solution": "If \\( x = 1 \\), then \\( x^2 = 1 \\) and \\( x^3 = 1 \\) and so \\( x^3 = x^2 \\). If \\( x > 1 \\), then \\( x^3 \\) equals \\( x \\) times \\( x^2 \\); since \\( x > 1 \\), then \\( x \\) times \\( x^2 \\) is greater than \\( x^2 \\) and so \\( x^3 > x^2 \\). Therefore, if \\( x \... |
DeepScaleR | [
{
"role": "user",
"content": "For which values of \\( a \\) do the quadratic trinomials \\( x^{2} + a x + 1 \\) and \\( x^{2} + x + a \\) have a common root?"
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 12101,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For which values of \\( x \\) and \\( y \\) the number \\(\\overline{x x y y}\\) is a square of a natural number?"
}
] | math | {
"style": "rule",
"ground_truth": "7744"
} | {
"index": 12102,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For which values of \\(a\\) does the equation \\(|x-3| = a x - 1\\) have two solutions? Enter the midpoint of the interval of parameter \\(a\\) in the provided field. Round the answer to three significant digits according to rounding rules and enter it in the provided field."
... | math | {
"style": "rule",
"ground_truth": "0.667"
} | {
"index": 12103,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For which values of the parameter \\( a \\) does the equation \\( x^{3}-11x^{2}+ax-8=0 \\) have three distinct real roots that form a geometric progression?"
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 12104,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For which values of the parameter \\(a\\) does the equation \\(x^{3} + 16x^{2} + ax + 64 = 0\\) have three distinct real roots that form a geometric progression?"
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"index": 12105,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For which values of the parameter \\(a\\) does the equation \\(x^{4} - 40 x^{2} + 144 = a(x^{2} + 4x - 12)\\) have exactly three distinct solutions?"
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 12106,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Form a five-digit number with no repeated digits using the numbers 0, 1, 2, 3, and 4, where exactly one even number is sandwiched between two odd numbers. The total number of such five-digit numbers is"
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 12107,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Form a five-digit number without repeating digits using the numbers 0, 1, 2, 3, 4, where exactly one even number is sandwiched between two odd numbers. How many such five-digit numbers are there?"
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 12108,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Form a four-digit number using the digits 0, 1, 2, 3, 4, 5 without repetition.\n(I) How many different four-digit numbers can be formed?\n(II) How many of these four-digit numbers have a tens digit that is larger than both the units digit and the hundreds digit?\n(III) Arrange ... | math | {
"style": "rule",
"ground_truth": "2301"
} | {
"index": 12109,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Form a four-digit number without repeating digits using the numbers 0, 1, 2, 3, 4, 5, 6, where the sum of the digits in the units, tens, and hundreds places is even. How many such four-digit numbers are there? (Answer with a number)"
}
] | math | {
"style": "rule",
"ground_truth": "324"
} | {
"index": 12110,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Form a six-digit number using the digits 1, 2, 3, 4, 5, 6 without repetition, where both 5 and 6 are on the same side of 3. How many such six-digit numbers are there?"
}
] | math | {
"style": "rule",
"ground_truth": "480"
} | {
"index": 12111,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Form a three-digit number using the digits 0, 1, 2, 3. Repeating digits is not allowed. \n① How many three-digit numbers can be formed? \n② If the three-digit numbers from ① are sorted in ascending order, what position does 230 occupy? \n③ If repeating digits is allowed, how... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 12112,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Form a three-digit number using the digits 4, 5, and 6. The probability that the number formed is a common multiple of 3 and 5 is ____."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 12113,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Form a three-digit number without repeating digits using 1, 2, 3, 4, where the number of odd numbers is a certain number."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 12114,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Form five-digit numbers without repeating digits using the numbers \\\\(0\\\\), \\\\(1\\\\), \\\\(2\\\\), \\\\(3\\\\), and \\\\(4\\\\).\n\n\\\\((\\\\)I\\\\()\\\\) How many of these five-digit numbers are even?\n\n\\\\((\\\\)II\\\\()\\\\) How many of these five-digit numbers are... | math | {
"style": "rule",
"ground_truth": "54"
} | {
"index": 12115,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Form natural numbers without repeating digits using the digits 0, 1, and 2, and calculate the total number of such natural numbers."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 12116,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Formulas for shortened multiplication (other). \nCommon fractions"
}
] | math | {
"style": "rule",
"ground_truth": "198719871987"
} | {
"index": 12117,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Formulate the Taylor series expansion for \\( n=2 \\) of the function \\( f(x, y) = x^y \\) near the point \\( M_0(1,1) \\) and approximately calculate \\( 1.1^{1.02} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "1.102"
} | {
"index": 12118,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Forty cards are placed into a box, each bearing a number 1, 2, 3, 4, 5, 6, 7, 8, 9, or 10, with each number entered on four cards. Four cards are drawn from the box at random and without replacement. Let $p$ be the probability that all four cards bear the same number. Let $q$ b... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"index": 12119,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Forty mice were walking, carrying forty coins. Two less-capable mice carried two coins each. Several mice did not carry any coins at all. The larger ones carried seven coins each. The rest carried four coins each. How many mice were walking without any coins?"
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 12120,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Forty slips are placed into a hat, each bearing a number 1, 2, 3, 4, 5, 6, 7, 8, 9, or 10, with each number entered on four slips. Four slips are drawn from the hat at random and without replacement. Let $p$ be the probability that all four slips bear the same number. Let $q$ b... | math | {
"style": "rule",
"ground_truth": "162"
} | {
"index": 12121,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Forty slips of paper numbered $1$ to $40$ are placed in a hat. Alice and Bob each draw one number from the hat without replacement, keeping their numbers hidden from each other. Alice says, \"I can't tell who has the larger number.\" Then Bob says, \"I know who has the larger n... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 12122,
"solution": "1. **Alice's Statement Analysis**:\n - Alice says she cannot determine who has the larger number. This implies that Alice's number, $A$, must be such that it is not the smallest number (1) because if Alice had 1, she would know for sure that Bob has a larger number.\n\n2. **Bob's St... |
DeepScaleR | [
{
"role": "user",
"content": "Forty teams play a tournament in which every team plays every other team exactly once. No ties occur, and each team has a $50 \\%$ chance of winning any game it plays. The probability that no two teams win the same number of games is $\\frac mn,$ where $m$ and $n$ are relativel... | math | {
"style": "rule",
"ground_truth": "742"
} | {
"index": 12123,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Forty teams play a tournament in which every team plays every other team exactly once. No ties occur, and each team has a $50 \\%$ chance of winning any game it plays. The probability that no two teams win the same number of games is $\\frac mn,$ where $m_{}$ and $n_{}$ are rel... | math | {
"style": "rule",
"ground_truth": "742"
} | {
"index": 12124,
"solution": "There are ${40 \\choose 2} = 780$ total pairings of teams, and thus $2^{780}$ possible outcomes. In order for no two teams to win the same number of games, they must each win a different number of games. Since the minimum and maximum possible number of games won are 0 and 39 respectiv... |
DeepScaleR | [
{
"role": "user",
"content": "Forty two cards are labeled with the natural numbers 1 through 42 and randomly shuffled into a stack. One by one, cards are taken off of the top of the stack until a card labeled with a prime number is removed. How many cards are removed on average?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{43}{14}"
} | {
"index": 12125,
"solution": "Note that there are 13 prime numbers amongst the cards. We may view these as separating the remaining 29 cards into 14 groups of nonprimes - those appearing before the first prime, between the first and second, etc. Each of these groups is equally likely to appear first, so 29/14 nonp... |
DeepScaleR | [
{
"role": "user",
"content": "Four A's, four B's, four C's, and four D's are placed in the sixteen spaces of a 4x4 grid so that each row and column contains one of each letter. If A is placed in the upper left corner, determine the number of possible arrangements."
}
] | math | {
"style": "rule",
"ground_truth": "144"
} | {
"index": 12126,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four ambassadors and one advisor for each of them are to be seated at a round table with $12$ chairs numbered in order $1$ to $12$. Each ambassador must sit in an even-numbered chair. Each advisor must sit in a chair adjacent to his or her ambassador. There are $N$ ways for the... | math | {
"style": "rule",
"ground_truth": "520"
} | {
"index": 12127,
"solution": "We see that for every 2 adjacent spots on the table, there is exactly one way for an ambassador and his or her partner to sit. There are 2 cases:\nThere is an ambassador at the 12th chair and his or her partner at the 1st chair\nThere is no pair that has their chairs numbered as 12 an... |
DeepScaleR | [
{
"role": "user",
"content": "Four balls numbered $1, 2, 3$, and $4$ are placed in an urn. One ball is drawn, its number noted, and then returned to the urn. This process is repeated three times. If the sum of the numbers noted is $9$, determine the probability that the ball numbered $3$ was drawn all three... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{10}"
} | {
"index": 12128,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four balls of radius $1$ are placed in space so that each of them touches the other three. What is the radius of the smallest sphere containing all of them?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6} + 2}{2}"
} | {
"index": 12129,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four balls of radius $1$ are mutually tangent, three resting on the floor and the fourth resting on the others.\nA tetrahedron, each of whose edges have length $s$, is circumscribed around the balls. Then $s$ equals"
}
] | math | {
"style": "rule",
"ground_truth": "2+2\\sqrt{6}"
} | {
"index": 12130,
"solution": "To solve this problem, we need to find the edge length $s$ of the tetrahedron that circumscribes four mutually tangent balls of radius $1$. The configuration described is such that three balls are resting on the floor and the fourth ball is resting on top of these three.\n\n1. **Under... |
DeepScaleR | [
{
"role": "user",
"content": "Four boxes with ball capacity 3, 5, 7, and 8 are given. Find the number of ways to distribute 19 identical balls into these boxes."
}
] | math | {
"style": "rule",
"ground_truth": "34"
} | {
"index": 12131,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four boys bought a boat for $60. The first boy paid one half of the sum of the amounts paid by the other boys; the second boy paid one third of the sum of the amounts paid by the other boys; and the third boy paid one fourth of the sum of the amounts paid by the other boys. How... | math | {
"style": "rule",
"ground_truth": "$13"
} | {
"index": 12132,
"solution": "Let the amounts paid by the first, second, third, and fourth boys be $a$, $b$, $c$, and $d$ respectively. According to the problem, we have the following equations:\n\n1. $a = \\frac{1}{2}(b+c+d)$\n2. $b = \\frac{1}{3}(a+c+d)$\n3. $c = \\frac{1}{4}(a+b+d)$\n4. $a + b + c + d = 60$\n\n... |
DeepScaleR | [
{
"role": "user",
"content": "Four boys, four girls, and a coach are positioned on a circular track. Each girl is diametrically opposite to one of the boys. The length of the track is 50 meters. On the coach's signal, they all run towards the coach by the shortest path along the track. What is the total dis... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 12133,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four brothers have together forty-eight Kwanzas. If the first brother's money were increased by three Kwanzas, if the second brother's money were decreased by three Kwanzas, if the third brother's money were triplicated and if the last brother's money were reduced by a third, t... | math | {
"style": "rule",
"ground_truth": "6, 12, 3, 27"
} | {
"index": 12134,
"solution": "\nLet \\( x_1, x_2, x_3, \\) and \\( x_4 \\) be the amounts of money that the first, second, third, and fourth brothers have, respectively. According to the problem, we have the following equation describing their total amount of money:\n\n\\[\nx_1 + x_2 + x_3 + x_4 = 48\n\\]\n\nWe ar... |
DeepScaleR | [
{
"role": "user",
"content": "Four buddies bought a ball. First one paid half of the ball price. Second one gave one third of money that other three gave. Third one paid a quarter of sum paid by other three. Fourth paid $5\\$ $. How much did the ball cost?"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 12135,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is an Ace, the second card is a 2, the third card is a 3, and the fourth card is a 4? Assume that the dealing is done without replacement."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{16}{405525}"
} | {
"index": 12136,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four carpenters were hired by a guest to build a yard. The first carpenter said: \"If only I alone were to build the yard, I would complete it in one year.\" The second carpenter said: \"If only I alone were to build the yard, I would complete it in two years.\" The third carpe... | math | {
"style": "rule",
"ground_truth": "175.2"
} | {
"index": 12137,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four cars $A$, $B$, $C$, and $D$ start simultaneously from the same point on a circular track. Cars $A$ and $B$ travel clockwise, while cars $C$ and $D$ travel counterclockwise. All cars move at constant but distinct speeds. Exactly 7 minutes after the race starts, $A$ meets $C... | math | {
"style": "rule",
"ground_truth": "53"
} | {
"index": 12138,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four cars $A, B, C$ and $D$ start simultaneously from the same point on a circular track. $A$ and $B$ drive clockwise, while $C$ and $D$ drive counter-clockwise. All cars move at constant (but pairwise different) speeds. Exactly 7 minutes after the race begins, $A$ meets $C$ fo... | math | {
"style": "rule",
"ground_truth": "371"
} | {
"index": 12139,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four cars $A, B, C,$ and $D$ start simultaneously from the same point on a circular track. $A$ and $B$ travel clockwise, while $C$ and $D$ travel counterclockwise. All cars move at constant (but pairwise different) speeds. Exactly 7 minutes after the start of the race, $A$ meet... | math | {
"style": "rule",
"ground_truth": "53"
} | {
"index": 12140,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four cars, \\( A, B, C, \\) and \\( D \\) start simultaneously from the same point on a circular track. \\( A \\) and \\( B \\) drive clockwise, while \\( C \\) and \\( D \\) drive counterclockwise. All cars move at constant (but pairwise different) speeds. Exactly 7 minutes af... | math | {
"style": "rule",
"ground_truth": "53"
} | {
"index": 12141,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four cats, four dogs, and four mice are placed in 12 cages. If a cat and a mouse are in the same column, the cat will meow non-stop; if a mouse is surrounded by two cats on both sides, the mouse will squeak non-stop; if a dog is flanked by a cat and a mouse, the dog will bark n... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 12142,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four chess players - Ivanov, Petrov, Vasiliev, and Kuznetsov - played a round-robin tournament (each played one game against each of the others). A victory awards 1 point, a draw awards 0.5 points to each player. It was found that the player in first place scored 3 points, and ... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 12143,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four circles $\\omega,$ $\\omega_{A},$ $\\omega_{B},$ and $\\omega_{C}$ with the same radius are drawn in the interior of triangle $ABC$ such that $\\omega_{A}$ is tangent to sides $AB$ and $AC$, $\\omega_{B}$ to $BC$ and $BA$, $\\omega_{C}$ to $CA$ and $CB$, and $\\omega$ is e... | math | {
"style": "rule",
"ground_truth": "389"
} | {
"index": 12144,
"solution": "Consider a 13-14-15 triangle. $A=84.$ [By Heron's Formula or by 5-12-13 and 9-12-15 right triangles.]\n The inradius is $r=\\frac{A}{s}=\\frac{84}{21}=4$, where $s$ is the semiperimeter. Scale the triangle with the inradius by a linear scale factor, $u.$\n The circumradius is $R=\\fra... |
DeepScaleR | [
{
"role": "user",
"content": "Four circles are inscribed such that each circle touches the midpoint of each side of a square. The side of the square is 10 cm, and the radius of each circle is 5 cm. Determine the area of the square not covered by any circle."
}
] | math | {
"style": "rule",
"ground_truth": "100 - 50\\pi"
} | {
"index": 12145,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four circles of radius 1 are externally tangent to each other and internally tangent to a larger circle. What is the radius of the large circle? Express your answer as a common fraction in simplest radical form."
}
] | math | {
"style": "rule",
"ground_truth": "1 + \\sqrt{2}"
} | {
"index": 12146,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four circles, each with a radius of 7 inches, are arranged such that they touch each other and the sides of a square as shown. Determine the area, in square inches, of the square."
}
] | math | {
"style": "rule",
"ground_truth": "784"
} | {
"index": 12147,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four circles, no two of which are congruent, have centers at $A$, $B$, $C$, and $D$, and points $P$ and $Q$ lie on all four circles. The radius of circle $A$ is $\\frac{5}{8}$ times the radius of circle $B$, and the radius of circle $C$ is $\\frac{5}{8}$ times the radius of cir... | math | {
"style": "rule",
"ground_truth": "192"
} | {
"index": 12148,
"solution": "1. **Understanding the Problem**: We are given four circles with centers at $A$, $B$, $C$, and $D$. Points $P$ and $Q$ lie on all four circles. The radius of circle $A$ is $\\frac{5}{8}$ times the radius of circle $B$, and similarly for circles $C$ and $D$. The distances $AB$ and $CD$... |
DeepScaleR | [
{
"role": "user",
"content": "Four complex numbers lie at the vertices of a square in the complex plane. Three of the numbers are $1+2i, -2+i$, and $-1-2i$. The fourth number is"
}
] | math | {
"style": "rule",
"ground_truth": "$2-i$"
} | {
"index": 12149,
"solution": "1. **Identify the given points in the complex plane**: The complex numbers given are $1+2i, -2+i$, and $-1-2i$. We can interpret these as points in the Cartesian plane: $A = (1, 2), B = (-2, 1), C = (-1, -2)$.\n\n2. **Determine the slopes of lines AB and BC**:\n - The slope of line ... |
DeepScaleR | [
{
"role": "user",
"content": "Four concentric circles are drawn with radii of 1, 3, 5 and 7. The inner circle is painted black, the ring around it is white, the next ring is black and the outer ring is white. What is the ratio of the black area to the white area? Express your answer as a common fraction."
... | math | {
"style": "rule",
"ground_truth": "\\frac{17}{32}"
} | {
"index": 12150,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four congruent rectangles and a square are assembled without overlapping to form a large square. Each of the rectangles has a perimeter of 40 cm. What is the total area of the large square?"
}
] | math | {
"style": "rule",
"ground_truth": "400 \\mathrm{~cm}^{2}"
} | {
"index": 12151,
"solution": "Suppose that each of the smaller rectangles has a longer side of length $x \\mathrm{~cm}$ and a shorter side of length $y \\mathrm{~cm}$. Since the perimeter of each of the rectangles is 40 cm, then $2x+2y=40$ or $x+y=20$. But the side length of the large square is $x+y \\mathrm{~cm}$... |
DeepScaleR | [
{
"role": "user",
"content": "Four congruent rectangles are placed as shown. The area of the outer square is $4$ times that of the inner square. What is the ratio of the length of the longer side of each rectangle to the length of its shorter side?\n\n[asy]\nunitsize(6mm);\ndefaultpen(linewidth(.8pt));\npat... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 12152,
"solution": "1. **Identify the dimensions of the squares and rectangles**: \n Let the side length of the inner square be $s$. Assume the shorter side of each rectangle is $y$ and the longer side is $x$. The rectangles are congruent and placed around the inner square such that their longer sides ... |
DeepScaleR | [
{
"role": "user",
"content": "Four congruent rectangles are placed as shown. The area of the outer square is 4 times that of the inner square. What is the ratio of the length of the longer side of each rectangle to the length of its shorter side?\n\n[asy] unitsize(6mm); defaultpen(linewidth(.8pt)); path p=... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 12153,
"solution": "1. **Identify the dimensions of the squares and rectangles**: \n Let the side length of the inner square be $s$. Assume the shorter side of each rectangle is $y$ and the longer side is $x$. The rectangles are congruent and placed around the inner square such that their longer sides ... |
DeepScaleR | [
{
"role": "user",
"content": "Four consecutive even integers have a product of 6720. What is the largest of these four integers?"
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 12154,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four consecutive positive integers have a product of 840. What is the largest of the four integers?"
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 12155,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four consecutive positive prime numbers have a sum that is divisible by three. What is the smallest possible value of this sum?"
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 12156,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four contestants \\\\(A\\\\), \\\\(B\\\\), \\\\(C\\\\), and \\\\(D\\\\) participate in three competitions: shooting, ball throwing, and walking on a balance beam. Each contestant has an equal chance of passing or failing each competition. At the end of the competitions, the jud... | math | {
"style": "rule",
"ground_truth": "\\dfrac {2}{3}"
} | {
"index": 12157,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four cubes with edge lengths $1$, $2$, $3$, and $4$ are stacked as shown. What is the length of the portion of $\\overline{XY}$ contained in the cube with edge length $3$?"
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"index": 12158,
"solution": "1. **Identify the coordinates of points $X$ and $Y$:**\n - Assume the cubes are stacked such that the bottom face of each cube lies on the $xy$-plane and one vertex of each cube is at the origin $(0,0,0)$.\n - The cube with edge length $1$ is entirely within the first unit cube.\n... |
DeepScaleR | [
{
"role": "user",
"content": "Four cubes with edge lengths $2$, $3$, $4$, and $5$ are stacked with their bottom faces on the $xy$-plane, and one vertex at the origin $(0,0,0)$. The stack sequence follows the increasing order of cube sizes from the bottom. If point $X$ is at $(0,0,0)$ and point $Y$ is at the... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{3}"
} | {
"index": 12159,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four cyclists. Four identical circles represent four tracks. The four cyclists start from the center at noon. Each moves along their track at speeds: the first at 6 km/h, the second at 9 km/h, the third at 12 km/h, and the fourth at 15 km/h. They agreed to ride until they all m... | math | {
"style": "rule",
"ground_truth": "12:26:40"
} | {
"index": 12160,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four different balls are to be placed into five boxes numbered 1, 2, 3, 4, and 5. (Write the result as a number) \n(1) How many ways can there be with at least one ball in box number 1? \n(2) How many ways are there to have exactly two empty boxes? \n(3) How many ways are th... | math | {
"style": "rule",
"ground_truth": "375"
} | {
"index": 12161,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four different natural numbers, of which one is 1, have the following properties: the sum of any two of them is a multiple of 2, the sum of any three of them is a multiple of 3, and the sum of all four numbers is a multiple of 4. What is the minimum possible sum of these four n... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 12162,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four distinct integers $a, b, c$, and $d$ are chosen from the set $\\{1,2,3,4,5,6,7,8,9,10\\}$. What is the greatest possible value of $ac+bd-ad-bc$?"
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"index": 12163,
"solution": "We note that $ac+bd-ad-bc=a(c-d)-b(c-d)=(a-b)(c-d)$. Since each of $a, b, c, d$ is taken from the set $\\{1,2,3,4,5,6,7,8,9,10\\}$, then $a-b \\leq 9$ since the greatest possible difference between two numbers in the set is 9 . Similarly, $c-d \\leq 9$. Now, if $a-b=9$, we must have $... |
DeepScaleR | [
{
"role": "user",
"content": "Four distinct natural numbers, one of which is an even prime number, have the following properties:\n- The sum of any two numbers is a multiple of 2.\n- The sum of any three numbers is a multiple of 3.\n- The sum of all four numbers is a multiple of 4.\n\nFind the smallest poss... | math | {
"style": "rule",
"ground_truth": "44"
} | {
"index": 12164,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four distinct points are arranged on a plane so that the segments connecting them have lengths $a$, $a$, $a$, $a$, $2a$, and $b$. What is the ratio of $b$ to $a$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 12165,
"solution": "1. **Identify the possible configurations of the points**: Given that there are four segments of length $a$, one of length $2a$, and one of length $b$, we need to determine how these points can be arranged. \n\n2. **Check the triangle inequality**: The triangle inequality states that ... |
DeepScaleR | [
{
"role": "user",
"content": "Four distinct points are arranged on a plane such that they have segments connecting them with lengths $a$, $a$, $a$, $b$, $b$, and $2a$. Determine the ratio $\\frac{b}{a}$ assuming the formation of a non-degenerate triangle with one of the side lengths being $2a$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 12166,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four distinct points, $A$, $B$, $C$, and $D$, are to be selected from $1996$ points evenly spaced around a circle. All quadruples are equally likely to be chosen. What is the probability that the chord $\\overline{AB}$ intersects the chord $\\overline{CD}$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 12167,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four elevators in a skyscraper, differing in color (red, blue, green, and yellow), move in different directions at different but constant speeds. An observer timed the events as follows: At the 36th second, the red elevator caught up with the blue one (moving in the same direct... | math | {
"style": "rule",
"ground_truth": "46"
} | {
"index": 12168,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four equal circles with diameter $6$ are arranged such that three circles are tangent to one side of a rectangle and the fourth circle is tangent to the opposite side. All circles are tangent to at least one other circle with their centers forming a straight line that is parall... | math | {
"style": "rule",
"ground_truth": "648"
} | {
"index": 12169,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four fair coins are to be flipped. What is the probability that all four will be heads or all four will be tails? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"index": 12170,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four fair coins are tossed once. For every head that appears, two six-sided dice are rolled. What is the probability that the sum of all dice rolled is exactly ten?\nA) $\\frac{1} {48}$\nB) $\\frac{1} {20}$\nC) $\\frac{1} {16}$\nD) $\\frac{1} {30}$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1} {20}"
} | {
"index": 12171,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four fair dice are tossed at random. What is the probability that the four numbers on the dice can be arranged to form an arithmetic progression with a common difference of one?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{432}"
} | {
"index": 12172,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four fair eight-sided dice (with faces showing 1 to 8) are rolled. What is the probability that the sum of the numbers on the top faces equals 32?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4096}"
} | {
"index": 12173,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four fair six-sided dice are rolled. What is the probability that at least three of the four dice show the same value?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{72}"
} | {
"index": 12174,
"solution": "To solve this problem, we need to calculate the probability of two scenarios: \n1. Exactly three dice show the same number.\n2. All four dice show the same number.\n\n#### Case 1: All four dice show the same number\n- The probability that the first die shows any number is 1 (as it can... |
DeepScaleR | [
{
"role": "user",
"content": "Four families visit a tourist spot that has four different routes available for exploration. Calculate the number of scenarios in which exactly one route is not visited by any of the four families."
}
] | math | {
"style": "rule",
"ground_truth": "144"
} | {
"index": 12175,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four football teams participate in a round-robin tournament, where each team plays a match against every other team. In each match, the winning team earns 3 points, the losing team earns 0 points, and in the case of a draw, both teams earn 1 point each. After all matches are co... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 12176,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four foreign guests visit a school and need to be accompanied by two security personnel. Six people enter the school gate in sequence. For safety reasons, the two security personnel must be at the beginning and the end. If the guests A and B must be together, calculate the tota... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 12177,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four friends came back from fishing. Each pair of them counted the sum of their catches. They obtained six numbers: $7, 9, 14, 14, 19, 21$. How many fish were caught in total?"
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 12178,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four friends do yardwork for their neighbors over the weekend, earning $15, $20, $25, and $40, respectively. They decide to split their earnings equally among themselves. In total, how much will the friend who earned $40 give to the others?"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 12179,
"solution": "1. **Calculate the total earnings**: The four friends earn different amounts: $15, $20, $25, and $40. To find the total earnings, sum these amounts:\n \\[\n 15 + 20 + 25 + 40 = 100\n \\]\n Therefore, the total earnings are $\\$100$.\n\n2. **Determine the equal share**: Since t... |
DeepScaleR | [
{
"role": "user",
"content": "Four friends have a total of 8 identical pencils, and each friend has at least one pencil. Additionally, the first friend always has at least two pencils. In how many ways can this happen?"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 12180,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four friends initially plan a road trip and decide to split the fuel cost equally. However, 3 more friends decide to join at the last minute. Due to the increase in the number of people sharing the cost, the amount each of the original four has to pay decreases by $\\$$8. What ... | math | {
"style": "rule",
"ground_truth": "74.67"
} | {
"index": 12181,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four friends make cookies from the same amount of dough with the same thickness. Art's cookies are circles with a radius of 2 inches, and Trisha's cookies are squares with a side length of 4 inches. If Art can make 18 cookies in his batch, determine the number of cookies Trisha... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 12182,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four friends rent a cottage for a total of £300 for the weekend. The first friend pays half of the sum of the amounts paid by the other three friends. The second friend pays one third of the sum of the amounts paid by the other three friends. The third friend pays one quarter o... | math | {
"style": "rule",
"ground_truth": "65"
} | {
"index": 12183,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four friends went fishing one day and caught a total of 11 fish. Each person caught at least one fish. Which statement must be true: (A) At least one person caught exactly one fish. (B) At least one person caught exactly three fish. (C) At least one person caught more than thre... | math | {
"style": "rule",
"ground_truth": "D"
} | {
"index": 12184,
"solution": "Choice (A) is not necessarily true, since the four friends could have caught 2,3, 3, and 3 fish. Choice (B) is not necessarily true, since the four friends could have caught 1, 1, 1, and 8 fish. Choice (C) is not necessarily true, since the four friends could have caught 2, 3, 3, and ... |
DeepScaleR | [
{
"role": "user",
"content": "Four friends — Alex, Betty, Clara, and Dave — participated in a relay race by running in pairs, with one pair sitting out each race. Dave ran in 8 races, which was more than any other friend, and Betty ran in 3 races, which was fewer than any other friend. Determine the total n... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 12185,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four girls and eight boys came for a class photograph. Children approach the photographer in pairs and take a joint photo. Among how many minimum photos must there necessarily be either a photo of two boys, a photo of two girls, or two photos with the same children?"
}
] | math | {
"style": "rule",
"ground_truth": "33"
} | {
"index": 12186,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four girls — Mary, Alina, Tina, and Hanna — sang songs in a concert as trios, with one girl sitting out each time. Hanna sang $7$ songs, which was more than any other girl, and Mary sang $4$ songs, which was fewer than any other girl. How many songs did these trios sing?"
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 12187,
"solution": "1. **Identify the number of songs each girl sang**: \n - Hanna sang 7 songs.\n - Mary sang 4 songs.\n - Alina and Tina sang more than Mary but less than Hanna, so they sang either 5 or 6 songs each.\n\n2. **Set up the equation for total songs**:\n - Let $a$ be the number of so... |
DeepScaleR | [
{
"role": "user",
"content": "Four horizontal lines and four vertical lines are drawn in a plane. In how many ways can four lines be chosen such that a rectangular region is enclosed?"
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 12188,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four identical isosceles triangles $A W B, B X C, C Y D$, and $D Z E$ are arranged with points $A, B, C, D$, and $E$ lying on the same straight line. A new triangle is formed with sides the same lengths as $A X, A Y,$ and $A Z$. If $A Z = A E$, find the largest integer value of... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 12189,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four identical isosceles triangles $A W B, B X C, C Y D$, and $D Z E$ are arranged with points $A, B, C, D$, and $E$ lying on the same straight line. A new triangle is formed with sides the same lengths as $A X, A Y,$ and $A Z$. If $A Z = A E$, what is the largest integer value... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 12190,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four identical point charges are initially placed at the corners of a square, storing a total energy of 20 Joules. Determine the total amount of energy stored if one of these charges is moved to the center of the square."
}
] | math | {
"style": "rule",
"ground_truth": "10\\sqrt{2} + 10"
} | {
"index": 12191,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four identical regular tetrahedrons are thrown simultaneously on a table. Calculate the probability that the product of the four numbers on the faces touching the table is divisible by 4."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{16}"
} | {
"index": 12192,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four identical squares and one rectangle are placed together to form one large square as shown. The length of the rectangle is how many times as large as its width?\n\n[asy]\nunitsize(8mm); defaultpen(linewidth(.8pt));\ndraw((0,0)--(4,0)--(4,4)--(0,4)--cycle);\ndraw((0,3)--(0,4... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 12193,
"solution": "1. **Identify the dimensions of the small squares and the large square**: Let the side length of each small square be $x$. Since there are four small squares placed side by side along one side of the large square, the side length of the large square is $4x$.\n\n2. **Analyze the placem... |
DeepScaleR | [
{
"role": "user",
"content": "Four lighthouses are located at points $A$, $B$, $C$, and $D$. The lighthouse at $A$ is $5$ kilometers from the lighthouse at $B$, the lighthouse at $B$ is $12$ kilometers from the lighthouse at $C$, and the lighthouse at $A$ is $13$ kilometers from the lighthouse at $C$. To an... | math | {
"style": "rule",
"ground_truth": "96"
} | {
"index": 12194,
"solution": "Let $O$ be the intersection of $BC$ and $AD$. By the Angle Bisector Theorem, $\\frac {5}{BO}$ = $\\frac {13}{CO}$, so $BO$ = $5x$ and $CO$ = $13x$, and $BO$ + $OC$ = $BC$ = $12$, so $x$ = $\\frac {2}{3}$, and $OC$ = $\\frac {26}{3}$. Let $P$ be the foot of the altitude from $D$ to $OC... |
DeepScaleR | [
{
"role": "user",
"content": "Four male students and five female students are lined up in a row. Calculate the number of different arrangements with alternating male and female students."
}
] | math | {
"style": "rule",
"ground_truth": "2880"
} | {
"index": 12195,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four math teams in a region consist of 6, 8, 9, and 11 students respectively. Each team has three co-captains. If you randomly select a team, and then randomly select two members from that team to gift a copy of *Introduction to Geometry*, what is the probability that both reci... | math | {
"style": "rule",
"ground_truth": "\\frac{1115}{18480}"
} | {
"index": 12196,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four mathletes and two coaches sit at a circular table. How many distinct arrangements are there of these six people if the two coaches sit opposite each other?"
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 12197,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four members of Barnett family, including two grandparents, one adult parent, and one child, visit a zoo. The grandparents, being senior citizens, get a 20% discount. The child receives a 60% discount due to being under the age of 12, while the adult pays the full ticket price.... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 12198,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four mice: White, Gray, Fat, and Thin were dividing a cheese wheel. They cut it into 4 apparently identical slices. Some slices had more holes, so Thin's slice weighed 20 grams less than Fat's slice, and White's slice weighed 8 grams less than Gray's slice. However, White wasn'... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 12199,
"solution": ""
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.