data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Two concentric squares share the same center $O$, each with sides of length 2. The length of the line segment $\\overline{AB}$ is $\\frac{1}{3}$, and the area of the octagon $ABCDEFGH$ formed by the intersection of lines drawn from the vertices of the inner square to the midpoi... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 36300,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two cones share a common base and the vertices of both cones and the circumference of the base are all on the same sphere. If the area of the base of the cone is $\\frac{3}{16}$ of the area of the sphere, calculate the ratio of the volumes of the two cones."
}
] | math | {
"style": "rule",
"ground_truth": "1:3"
} | {
"index": 36301,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two congruent 30-60-90 triangles are placed such that one triangle is translated 2 units vertically upwards, while their hypotenuses originally coincide when not translated. The hypotenuse of each triangle is 10. Calculate the area common to both triangles when one is translate... | math | {
"style": "rule",
"ground_truth": "25\\sqrt{3} - 10"
} | {
"index": 36302,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two congruent circles centered at points $A$ and $B$ each pass through the other circle's center. The line containing both $A$ and $B$ is extended to intersect the circles at points $C$ and $D$. The circles intersect at two points, one of which is $E$. What is the degree measur... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 36303,
"solution": "\n1. **Identify the Shape of $\\triangle EAB$**:\n Since the circles are congruent and each circle's center lies on the other circle, the distance between $A$ and $B$ equals the radius of the circles. Thus, $EA = EB = AB$, making $\\triangle EAB$ equilateral.\n\n2. **Calculate $\\an... |
DeepScaleR | [
{
"role": "user",
"content": "Two congruent cones with radius 12 cm and height 12 cm are enclosed within a cylinder. The base of each cone is a base of the cylinder, and the height of the cylinder is 24 cm. What is the number of cubic centimeters in the volume of the cylinder not occupied by the cones? Expr... | math | {
"style": "rule",
"ground_truth": "2304\\pi"
} | {
"index": 36304,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two congruent cones, each with a radius of 15 cm and a height of 10 cm, are enclosed within a cylinder. The bases of the cones are the bases of the cylinder, and the height of the cylinder is 30 cm. Determine the volume in cubic centimeters of the space inside the cylinder that... | math | {
"style": "rule",
"ground_truth": "5250\\pi"
} | {
"index": 36305,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two congruent cylinders each have radius 8 inches and height 3 inches. The radius of one cylinder and the height of the other are both increased by the same nonzero number of inches. The resulting volumes are equal. How many inches is the increase? Express your answer as a comm... | math | {
"style": "rule",
"ground_truth": "\\frac{16}{3}"
} | {
"index": 36306,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two congruent right circular cones each with base radius $3$ and height $8$ have the axes of symmetry that intersect at right angles at a point in the interior of the cones a distance $3$ from the base of each cone. A sphere with radius $r$ lies within both cones. The maximum p... | math | {
"style": "rule",
"ground_truth": "298"
} | {
"index": 36307,
"solution": "Consider the cross section of the cones and sphere by a plane that contains the two axes of symmetry of the cones as shown below. The sphere with maximum radius will be tangent to the sides of each of the cones. The center of that sphere must be on the axis of symmetry of each of the ... |
DeepScaleR | [
{
"role": "user",
"content": "Two congruent right circular cones each with base radius $3$ and height $8$ have the axes of symmetry that intersect at right angles at a point in the interior of the cones a distance $3$ from the base of each cone. A sphere with radius $r$ lies withing both cones. The maximum ... | math | {
"style": "rule",
"ground_truth": "298"
} | {
"index": 36308,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two congruent right circular cones each with base radius $5$ and height $12$ have the axes of symmetry that intersect at right angles at a point in the interior of the cones a distance $4$ from the base of each cone. Determine the maximum possible value of the radius $r$ of a s... | math | {
"style": "rule",
"ground_truth": "\\frac{40}{13}"
} | {
"index": 36309,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two congruent squares share the same center $O$ and have sides of length 1. The length of $\\overline{AB}$ is $\\frac{17}{40}$ and the area of octagon $ABCDEFGH$ is $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Determine $p+q$."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 36310,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two congruent squares, $ABCD$ and $EFGH$, each with a side length of $20$ units, overlap to form a $20$ by $35$ rectangle $AEGD$. Calculate the percentage of the area of rectangle $AEGD$ that is shaded."
}
] | math | {
"style": "rule",
"ground_truth": "14\\%"
} | {
"index": 36311,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two congruent squares, $ABCD$ and $JKLM$, each have side lengths of 12 units. Square $JKLM$ is placed such that its center coincides with vertex $C$ of square $ABCD$. Determine the area of the region covered by these two squares in the plane."
}
] | math | {
"style": "rule",
"ground_truth": "216"
} | {
"index": 36312,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two congruent squares, $ABCD$ and $PQRS$, have side length $15$. They overlap to form the $15$ by $25$ rectangle $AQRD$ shown. What percent of the area of rectangle $AQRD$ is shaded?"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 36313,
"solution": "1. **Identify the dimensions of the rectangle and squares**: \n - Each square $ABCD$ and $PQRS$ has a side length of $15$.\n - The rectangle $AQRD$ formed by the overlap of these squares has dimensions $15 \\times 25$.\n\n2. **Calculate the area of rectangle $AQRD$**:\n - The ar... |
DeepScaleR | [
{
"role": "user",
"content": "Two connected rooms have different sizes and temperatures. One room has a length of $5 \\,\\mathrm{m}$, a width of $3 \\,\\mathrm{m}$, a height of $4 \\,\\mathrm{m}$, and a temperature of $22 \\,^{\\circ}\\mathrm{C}$; the other room has a length of $6 \\,\\mathrm{m}$, a width o... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 36314,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two counterfeit coins of equal weight are mixed with $8$ identical genuine coins. The weight of each of the counterfeit coins is different from the weight of each of the genuine coins. A pair of coins is selected at random without replacement from the $10$ coins. A second pair ... | math | {
"style": "rule",
"ground_truth": "\\frac{15}{19}"
} | {
"index": 36315,
"solution": "To solve this problem, we need to calculate the probability that all four selected coins are genuine, given that the combined weight of the first pair is equal to the combined weight of the second pair.\n\n#### Step 1: Define the events\nLet $A$ be the event that all four selected coi... |
DeepScaleR | [
{
"role": "user",
"content": "Two couples each bring one child to visit the zoo. After purchasing tickets, they line up to enter the zoo one by one. For safety reasons, the two fathers must be positioned at the beginning and the end of the line. Moreover, the two children must be positioned together. Determ... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 36316,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two couriers start from two locations $A$ and $B$; the first heading towards $B$, and the second towards $A$. When they meet, the first courier has traveled 12 miles more than the second. If they continue their journeys at their original speeds, the first courier will reach the... | math | {
"style": "rule",
"ground_truth": "84"
} | {
"index": 36317,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two cross sections of a right hexagonal pyramid are obtained by cutting the pyramid with planes parallel to the hexagonal base. The areas of the cross sections are $216\\sqrt{3}$ square feet and $486\\sqrt{3}$ square feet. The two planes are $8$ feet apart. How far from the ape... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 36318,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two cross sections of a right octagonal pyramid are obtained by cutting the pyramid with planes parallel to the octagonal base. The areas of the cross sections are $300\\sqrt{2}$ square feet and $675\\sqrt{2}$ square feet. The two planes are $10$ feet apart. How far from the ap... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 36319,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two cubes with the faces numbered 1 through 6 are tossed and the numbers shown on the top faces are added. What is the probability that the sum is even? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac12"
} | {
"index": 36320,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two cubic dice are thrown in succession, where \\\\(x\\\\) represents the number shown by the first die, and \\\\(y\\\\) represents the number shown by the second die. \n\\\\((1)\\\\) Find the probability that point \\\\(P(x,y)\\\\) lies on the line \\\\(y=x-1\\\\); \n\\\\((2... | math | {
"style": "rule",
"ground_truth": "\\dfrac{17}{36}"
} | {
"index": 36321,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two cubical dice each have removable numbers $1$ through $6$. The twelve numbers on the two dice are removed, put into a bag, then drawn one at a time and randomly reattached to the faces of the cubes, one number to each face. The dice are then rolled and the numbers on the two... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"index": 36322,
"solution": "To solve this problem, we need to consider the probability of rolling a sum of 7 with two dice, where the numbers on the dice have been randomly reassigned.\n\n#### Step 1: Understand the possible sums\nThe possible combinations of two numbers summing to 7 are:\n- $(1,6)$\n- $(2,5)$\n... |
DeepScaleR | [
{
"role": "user",
"content": "Two cyclists started a trip at the same time from the same location. They traveled the same route and returned together. Both rested along the way. The first cyclist rode twice as long as the second cyclist rested. The second cyclist rode four times as long as the first cyclist... | math | {
"style": "rule",
"ground_truth": "1.5"
} | {
"index": 36323,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two cyclists, $k$ miles apart, and starting at the same time, would be together in $r$ hours if they traveled in the same direction, but would pass each other in $t$ hours if they traveled in opposite directions. The ratio of the speed of the faster cyclist to that of the slowe... | math | {
"style": "rule",
"ground_truth": "\\frac {r + t}{r - t}"
} | {
"index": 36324,
"solution": "Let's denote the speeds of the faster and slower cyclists as $v_1$ and $v_2$ respectively, where $v_1 > v_2$. The distance between the cyclists is $k$ miles.\n\n1. **Traveling in the same direction**: When the cyclists travel in the same direction, the relative speed between them is $... |
DeepScaleR | [
{
"role": "user",
"content": "Two cylindrical cans have the same volume. The height of one can is triple the height of the other. If the radius of the narrower can is 12 units, how many units are in the length of the radius of the wider can? Express your answer in simplest radical form."
}
] | math | {
"style": "rule",
"ground_truth": "12\\sqrt{3}"
} | {
"index": 36325,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two cylindrical poles, with diameters of $10$ inches and $30$ inches respectively, are placed side by side and bound together with a wire. Calculate the length of the shortest wire that will go around both poles.\n**A)** $20\\sqrt{3} + 24\\pi$\n**B)** $20\\sqrt{3} + \\frac{70\\... | math | {
"style": "rule",
"ground_truth": "20\\sqrt{3} + \\frac{70\\pi}{3}"
} | {
"index": 36326,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two decimals are multiplied, and the resulting product is rounded to 27.6. It is known that both decimals have one decimal place and their units digits are both 5. What is the exact product of these two decimals?"
}
] | math | {
"style": "rule",
"ground_truth": "27.55"
} | {
"index": 36327,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two decks, each containing 36 cards, were placed on a table. The first deck was shuffled and placed on top of the second deck. For each card in the first deck, the number of cards between it and the same card in the second deck was counted. What is the sum of these 36 numbers?"... | math | {
"style": "rule",
"ground_truth": "1260"
} | {
"index": 36328,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two diagonals of a regular decagon (a 10-sided polygon) are chosen. What is the probability that their intersection lies inside the decagon and forms a convex quadrilateral?"
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{42}{119}"
} | {
"index": 36329,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two diameters and one radius are drawn in a circle of radius 1, dividing the circle into 5 sectors. The largest possible area of the smallest sector can be expressed as $\\frac{a}{b} \\pi$, where $a, b$ are relatively prime positive integers. Compute $100a+b$."
}
] | math | {
"style": "rule",
"ground_truth": "106"
} | {
"index": 36330,
"solution": "Let the two diameters split the circle into four sectors of areas $A, B$, $A$, and $B$, where $A+B=\\frac{\\pi}{2}$. Without loss of generality, let $A \\leq B$. If our radius cuts into a sector of area $A$, the area of the smallest sector will be of the form $\\min (x, A-x)$. Note th... |
DeepScaleR | [
{
"role": "user",
"content": "Two dice appear to be normal dice with their faces numbered from $1$ to $6$, but each die is weighted so that the probability of rolling the number $k$ is directly proportional to $k$. The probability of rolling a $7$ with this pair of dice is $\\frac{m}{n}$, where $m$ and $n$ ... | math | {
"style": "rule",
"ground_truth": "71"
} | {
"index": 36331,
"solution": "It is easier to think of the dice as $21$ sided dice with $6$ sixes, $5$ fives, etc. Then there are $21^2=441$ possible rolls. There are $2\\cdot(1\\cdot 6+2\\cdot 5+3\\cdot 4)=56$ rolls that will result in a seven. The odds are therefore $\\frac{56}{441}=\\frac{8}{63}$. The answer is... |
DeepScaleR | [
{
"role": "user",
"content": "Two dice are differently designed. The first die has faces numbered $1$, $1$, $2$, $2$, $3$, and $5$. The second die has faces numbered $2$, $3$, $4$, $5$, $6$, and $7$. What is the probability that the sum of the numbers on the top faces of the two dice is $3$, $7$, or $8$?\nA... | math | {
"style": "rule",
"ground_truth": "\\frac{13}{36}"
} | {
"index": 36332,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two dice are made so that the chances of getting an even sum are twice that of getting an odd sum. What is the probability of getting an odd sum in a single roll of these two dice?\n(a) \\(\\frac{1}{9}\\)\n(b) \\(\\frac{2}{9}\\)\n(c) \\(\\frac{4}{9}\\)\n(d) \\(\\frac{5}{9}\\)"
... | math | {
"style": "rule",
"ground_truth": "$\\frac{4}{9}"
} | {
"index": 36333,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two dice are rolled consecutively, and the numbers obtained are denoted as $a$ and $b$. \n(Ⅰ) Find the probability that the point $(a, b)$ lies on the graph of the function $y=2^x$. \n(Ⅱ) Using the values of $a$, $b$, and $4$ as the lengths of three line segments, find the pr... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{18}"
} | {
"index": 36334,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two dice are thrown one after the other, and the numbers obtained are denoted as $a$ and $b$.\n\n(Ⅰ) Find the probability that $a^2 + b^2 = 25$;\n\n(Ⅱ) Given that the lengths of three line segments are $a$, $b$, and $5$, find the probability that these three line segments can f... | math | {
"style": "rule",
"ground_truth": "\\dfrac{7}{18}"
} | {
"index": 36335,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two dice are tossed. What is the probability that the sum is greater than three?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11}{12}"
} | {
"index": 36336,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two different numbers are selected simultaneously and at random from the set $\\{1, 2, 3, 4, 5, 6, 7\\}$. What is the probability that the positive difference between the two numbers is $2$ or greater? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{7}"
} | {
"index": 36337,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two different cubes of the same size are to be painted, with the color of each face being chosen independently and at random to be either black or white. What is the probability that after they are painted, the cubes can be rotated to be identical in appearance?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{147}{1024}"
} | {
"index": 36338,
"solution": "To solve this problem, we need to consider the different ways two cubes can be painted such that they are identical in appearance after possible rotations. We will calculate the number of ways to paint each cube and then find the probability that they match.\n\n#### Step 1: Count the ... |
DeepScaleR | [
{
"role": "user",
"content": "Two different digits from 1 to 9 are chosen. One digit is placed in each box to complete the two 2-digit numbers shown. The result of subtracting the bottom number from the top number is calculated. How many of the possible results are positive?"
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 36339,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two different integers are randomly chosen from the set $$\\{ -5, -8, 7, 4, -2 \\}.$$ What is the probability that their product is negative? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"index": 36340,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two different integers from 1 through 20 inclusive are chosen at random. What is the probability that both numbers are prime? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{14}{95}"
} | {
"index": 36341,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two different natural numbers are chosen from the set $\\{1, 2, 3, 4, 5, 6, 7, 8\\}$. What is the probability that their greatest common factor is one? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 36342,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two different natural numbers are selected from the set $\\ \\allowbreak \\{1, 2, 3, \\ldots, 6\\}$. What is the probability that the greatest common factor of these two numbers is one? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11}{15}"
} | {
"index": 36343,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two different natural numbers are selected from the set $\\{1, 2, 3, 4, 5, 6, 7, 8\\}$. What is the probability that the greatest common factor of these two numbers is one? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 36344,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two different natural numbers are selected from the set $\\{1, 2, 3, \\ldots, 7\\}$. What is the probability that the greatest common divisor of these two numbers is one? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{17}{21}"
} | {
"index": 36345,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two different natural numbers are selected from the set $\\{1, 2, 3, \\ldots, 8\\}$. What is the probability that the greatest common factor (GCF) of these two numbers is one? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 36346,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two different natural numbers are selected from the set $\\{1, 2, 3, \\ldots, 8\\}$. What is the probability that the greatest common factor of these two numbers is one? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11}{14}"
} | {
"index": 36347,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two different natural numbers end with 7 zeros and have exactly 72 divisors. Find their sum."
}
] | math | {
"style": "rule",
"ground_truth": "70000000"
} | {
"index": 36348,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two different numbers are randomly selected from the set $S = \\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\\}$. The probability that their sum is 12 would be greater if the number $n$ had first been removed from set $S$. What is the value of $n$?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 36349,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two different numbers are randomly selected from the set $\\{ - 2, -1, 0, 3, 4, 5\\}$ and multiplied together. What is the probability that the product is $0$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 36350,
"solution": "\n#### Step-by-step Analysis:\n\n1. **Identify the Total Number of Ways to Choose Two Numbers:**\n The set given is $\\{-2, -1, 0, 3, 4, 5\\}$, which contains 6 elements. The number of ways to choose 2 different numbers from this set is given by the combination formula $\\binom{n}{k... |
DeepScaleR | [
{
"role": "user",
"content": "Two different numbers are randomly selected from the set $\\{-3, -1, 0, 2, 4\\}$ and then multiplied together. What is the probability that the product of the two numbers chosen is 0?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"index": 36351,
"solution": "There are ten possible pairs of numbers that can be chosen: -3 and $-1$; -3 and $0$; -3 and 2; -3 and $4$; -1 and $0$; -1 and $2$; -1 and $4$; 0 and $2$; 0 and $4$; 2 and 4. Each pair is equally likely to be chosen. Pairs that include 0 (4 pairs) have a product of 0; pairs that do not... |
DeepScaleR | [
{
"role": "user",
"content": "Two different numbers are randomly selected from the set $\\{-3, -2, 0, 0, 5, 6, 7\\}$. What is the probability that the product of these two numbers is $0$?\n**A)** $\\frac{1}{4}$\n**B)** $\\frac{1}{5}$\n**C)** $\\frac{5}{21}$\n**D)** $\\frac{1}{3}$\n**E)** $\\frac{1}{2}$"
}... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{21}"
} | {
"index": 36352,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two different numbers are selected from 1, 2, 3, 4, 5 to form a point (x, y). Find the probability that this point lies above the line x+y-5=0."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"index": 36353,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two different points, $C$ and $D$, lie on the same side of line $AB$ so that $\\triangle ABC$ and $\\triangle BAD$ are congruent with $AB = 9$, $BC=AD=10$, and $CA=DB=17$. The intersection of these two triangular regions has area $\\tfrac mn$, where $m$ and $n$ are relatively p... | math | {
"style": "rule",
"ground_truth": "59"
} | {
"index": 36354,
"solution": "Let $a = \\angle{CAB}$. By Law of Cosines, \\[\\cos a = \\frac{17^2+9^2-10^2}{2*9*17} = \\frac{15}{17}\\] \\[\\sin a = \\sqrt{1-\\cos^2 a} = \\frac{8}{17}\\] \\[\\tan a = \\frac{8}{15}\\] \\[A = \\frac{1}{2}* 9*\\frac{9}{2}\\tan a = \\frac{54}{5}\\] And $54+5=\\boxed{059}.$\n - by Mat... |
DeepScaleR | [
{
"role": "user",
"content": "Two different prime numbers between $4$ and $18$ are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained? $$\n\\text A. \\ \\ 21 \\qquad \\text B. \\ \\ 60 \\qquad \\text C. \\ \\ 119 \\qquad \\text D. \\ \\ 180 \\qquad \\te... | math | {
"style": "rule",
"ground_truth": "119"
} | {
"index": 36355,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two differentiable real functions \\( f(x) \\) and \\( g(x) \\) satisfy\n\\[ \\frac{f^{\\prime}(x)}{g^{\\prime}(x)} = e^{f(x) - g(x)} \\]\nfor all \\( x \\), and \\( f(0) = g(2003) = 1 \\). Find the largest constant \\( c \\) such that \\( f(2003) > c \\) for all such functions... | math | {
"style": "rule",
"ground_truth": "1 - \\ln 2"
} | {
"index": 36356,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two digits of a number were swapped, and as a result, it increased by more than 3 times. The resulting number is 8453719. Find the original number."
}
] | math | {
"style": "rule",
"ground_truth": "1453789"
} | {
"index": 36357,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two distinct integers $x$ and $y$ are factors of 48. One of these integers must be even. If $x\\cdot y$ is not a factor of 48, what is the smallest possible value of $x\\cdot y$?"
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 36358,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two distinct integers, $x$ and $y$, are randomly chosen from the set $\\{1,2,3,4,5,6,7,8,9,10,11,12\\}$. What is the probability that $(x-1)(y-1)$ is odd?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{22}"
} | {
"index": 36359,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two distinct integers, $x$ and $y$, are randomly chosen from the set $\\{1,2,3,4,5,6,7,8,9,10\\}$. What is the probability that $xy-x-y$ is even?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{9}"
} | {
"index": 36360,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two distinct lines pass through the center of three concentric circles of radii 3, 2, and 1. The area of the shaded region in the diagram is $\\frac{8}{13}$ of the area of the unshaded region. What is the radian measure of the acute angle formed by the two lines? (Note: $\\pi$ ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{7}"
} | {
"index": 36361,
"solution": "1. **Identify the areas of the circles**: \n - The largest circle has radius 3, so its area is $9\\pi$.\n - The middle circle has radius 2, so its area is $4\\pi$.\n - The smallest circle has radius 1, so its area is $\\pi$.\n\n2. **Set up the equations for shaded and unshaded r... |
DeepScaleR | [
{
"role": "user",
"content": "Two distinct natural numbers end in 5 zeros and have exactly 42 divisors. Find their sum."
}
] | math | {
"style": "rule",
"ground_truth": "700000"
} | {
"index": 36362,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two distinct natural numbers end with 7 zeros and have exactly 72 divisors. Find their sum."
}
] | math | {
"style": "rule",
"ground_truth": "70000000"
} | {
"index": 36363,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two distinct natural numbers end with 8 zeros and have exactly 90 divisors. Find their sum."
}
] | math | {
"style": "rule",
"ground_truth": "700000000"
} | {
"index": 36364,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two distinct non-consecutive positive integers $x$ and $y$ are factors of 48. If $x\\cdot y$ is not a factor of 48, what is the smallest possible value of $x\\cdot y$?"
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 36365,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two distinct numbers $a$ and $b$ are chosen randomly from the set $\\{2, 2^2, 2^3, ..., 2^{25}\\}$. What is the probability that $\\log_a b$ is an integer?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{31}{300}"
} | {
"index": 36366,
"solution": "1. **Define the problem in terms of logarithms and powers of 2:**\n Let $a = 2^x$ and $b = 2^y$ where $x$ and $y$ are integers such that $1 \\leq x, y \\leq 25$. We are given that $\\log_a b$ is an integer, say $z$. Therefore, we have:\n \\[\n \\log_a b = z \\implies a^z = b \\i... |
DeepScaleR | [
{
"role": "user",
"content": "Two distinct numbers a and b are chosen randomly from the set $\\{3, 3^2, 3^3, ..., 3^{15}\\}$. What is the probability that $\\mathrm{log}_a b$ is an integer?\nA) $\\frac{1}{10}$\nB) $\\frac{2}{7}$\nC) $\\frac{1}{7}$\nD) $\\frac{1}{5}$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{7}"
} | {
"index": 36367,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two distinct numbers a and b are chosen randomly from the set $\\{3, 3^2, 3^3, ..., 3^{20}\\}$. What is the probability that $\\mathrm{log}_a b$ is an integer?\nA) $\\frac{12}{19}$\nB) $\\frac{1}{4}$\nC) $\\frac{24}{95}$\nD) $\\frac{1}{5}$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{24}{95}"
} | {
"index": 36368,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two distinct numbers are selected from the set $\\{1,2,3,4,\\dots,36,37\\}$ so that the sum of the remaining $35$ numbers is the product of these two numbers. What is the difference of these two numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 36369,
"solution": "1. **Calculate the sum of all numbers in the set**: The sum of the first $n$ integers is given by the formula $\\frac{n(n+1)}{2}$. For the set $\\{1, 2, 3, \\ldots, 37\\}$, we have $n = 37$. Thus, the sum is:\n \\[\n \\frac{37 \\times (37 + 1)}{2} = \\frac{37 \\times 38}{2} = 703.... |
DeepScaleR | [
{
"role": "user",
"content": "Two distinct numbers are selected from the set $\\{1,2,3,4,\\dots,38\\}$ so that the sum of the remaining $36$ numbers equals the product of these two selected numbers plus one. Find the difference of these two numbers."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 36370,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two distinct numbers are selected simultaneously and at random from the set $\\{1, 2, 3, 4, 5\\}$. What is the probability that the smaller one divides the larger one? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 36371,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two distinct numbers are selected simultaneously and at random from the set $\\{1, 2, 3, 4, 5\\}$. What is the probability that their product is an even number? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{10}"
} | {
"index": 36372,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two distinct numbers are selected simultaneously and at random from the set $\\{1, 2, 3, 6, 9\\}$. What is the probability that the smaller one divides the larger one? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"index": 36373,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two distinct positive integers $a$ and $b$ are factors of 48. If $a\\cdot b$ is not a factor of 48, what is the smallest possible value of $a\\cdot b$?"
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 36374,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two distinct positive integers $x$ and $y$ are factors of 36. If $x\\cdot y$ is not a factor of 36, what is the smallest possible value of $x\\cdot y$?"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 36375,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two distinct positive integers \\( x \\) and \\( y \\) are factors of 48. If \\( x \\cdot y \\) is not a factor of 48, what is the smallest possible value of \\( x \\cdot y \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 36376,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two distinct positive integers from 1 to 50 inclusive are chosen. Let the sum of the integers equal $S$ and the product equal $P$. What is the probability that $P+S$ is one less than a multiple of 5?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{89}{245}"
} | {
"index": 36377,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two distinct positive integers from 1 to 60 inclusive are chosen. Let the sum of the integers equal $S$ and the product equal $P$. What is the probability that $P+S$ is one less than a multiple of 7?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{148}{590}"
} | {
"index": 36378,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two distinct primes, each greater than 20, are multiplied. What is the least possible product of these two primes?"
}
] | math | {
"style": "rule",
"ground_truth": "667"
} | {
"index": 36379,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two distinct similar rhombi share a diagonal. The smaller rhombus has area 1, and the larger rhombus has area 9. Compute the side length of the larger rhombus."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{15}"
} | {
"index": 36380,
"solution": "Let $d$ be the length of the smaller diagonal of the smaller rhombus. Since the ratio of the areas is $9: 1$, the ratio of the lengths is $3: 1$. This means that the smaller diagonal of the larger rhombus (which is also the longer diagonal of the smaller rhombus) has length $3 d$. The... |
DeepScaleR | [
{
"role": "user",
"content": "Two distinct squares on a $4 \\times 4$ chessboard are chosen, with each pair of squares equally likely to be chosen. A knight is placed on one of the squares. The expected value of the minimum number of moves it takes for the knight to reach the other squarecan be written as $... | math | {
"style": "rule",
"ground_truth": "1205"
} | {
"index": 36381,
"solution": "We can do casework based on the position of the knight: corner, edge, or center. In each case, we can quickly compute all 15 distances by writing a 1 down in all squares reachable from the original square, then writing a 2 down in all blank squares reachable from a square with a 1 , w... |
DeepScaleR | [
{
"role": "user",
"content": "Two distinct, real, infinite geometric series each have a sum of $1$ and have the same second term. The third term of one of the series is $1/8$, and the second term of both series can be written in the form $\\frac{\\sqrt{m}-n}p$, where $m$, $n$, and $p$ are positive integers ... | math | {
"style": "rule",
"ground_truth": "518"
} | {
"index": 36382,
"solution": "Let the second term of each series be $x$. Then, the common ratio is $\\frac{1}{8x}$, and the first term is $8x^2$.\n So, the sum is $\\frac{8x^2}{1-\\frac{1}{8x}}=1$. Thus, $64x^3-8x+1 = (4x-1)(16x^2+4x-1) = 0 \\Rightarrow x = \\frac{1}{4}, \\frac{-1 \\pm \\sqrt{5}}{8}$.\n The only s... |
DeepScaleR | [
{
"role": "user",
"content": "Two divisions of an oil extraction company compete with each other for the amount of oil extracted. Depending on the availability of free tanks, each of them can extract the amount of tons defined by the following payoff matrix:\n\n$$\n\\text { Extraction }=\\left(\\begin{array... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 36383,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two equal circles in the same plane cannot have the following number of common tangents."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 36384,
"solution": "To solve this problem, we need to consider the possible configurations of two equal circles in the same plane and determine the number of common tangents in each case. The configurations depend on the relative positions of the circles:\n\n1. **Circles are separate (do not intersect)**... |
DeepScaleR | [
{
"role": "user",
"content": "Two equal parallel chords are drawn $8$ inches apart in a circle of radius $8$ inches. The area of that part of the circle that lies between the chords is:"
}
] | math | {
"style": "rule",
"ground_truth": "$32\\sqrt{3}+21\\frac{1}{3}\\pi$"
} | {
"index": 36385,
"solution": "1. **Identify the Geometry and Setup**: We are given a circle with radius $8$ inches and two equal parallel chords that are $8$ inches apart. Let's denote the center of the circle as $O$ and the chords as $BD$ and $CE$. Draw the diameter $A$ perpendicular to the chords $BD$ and $CE$. ... |
DeepScaleR | [
{
"role": "user",
"content": "Two equilateral triangles are contained in a square whose side length is $2\\sqrt{3}$. The bases of these triangles are the opposite sides of the square, and their intersection is a rhombus. What is the area of the rhombus?"
}
] | math | {
"style": "rule",
"ground_truth": "8\\sqrt{3} - 12"
} | {
"index": 36386,
"solution": "1. **Identify the configuration**: We have two equilateral triangles within a square of side length $2\\sqrt{3}$. The bases of these triangles are opposite sides of the square, and their intersection forms a rhombus.\n\n2. **Calculate the side length of the triangles**: Since the tria... |
DeepScaleR | [
{
"role": "user",
"content": "Two equilateral triangles with perimeters of 12 and 15 are positioned such that their sides are respectively parallel. Find the perimeter of the resulting hexagon."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 36387,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two externally tangent circles $\\omega_1$ and $\\omega_2$ have centers $O_1$ and $O_2$, respectively. A third circle $\\Omega$ passing through $O_1$ and $O_2$ intersects $\\omega_1$ at $B$ and $C$ and $\\omega_2$ at $A$ and $D$, as shown. Suppose that $AB = 2$, $O_1O_2 = 15$, ... | math | {
"style": "rule",
"ground_truth": "140"
} | {
"index": 36388,
"solution": "Let points $A'$ and $B'$ be the reflections of $A$ and $B,$ respectively, about the perpendicular bisector of $O_1 O_2.$ \\[B'O_2 = BO_1 = O_1 P = O_1 C,\\] \\[A'O_1 = AO_2 = O_2 P = O_2 D.\\] We establish the equality of the arcs and conclude that the corresponding chords are equal \... |
DeepScaleR | [
{
"role": "user",
"content": "Two faces of a tetrahedron are equilateral triangles with side lengths of one unit, and the other two faces are isosceles right triangles. What is the volume of the tetrahedron?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{12}"
} | {
"index": 36389,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two fair 6-sided dice are rolled. What is the probability at least one of the dice shows a 1?"
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{11}{36}"
} | {
"index": 36390,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two fair 8-sided dice, with sides numbered from 1 to 8, are rolled once. The sum of the numbers rolled determines the radius of a circle. Calculate the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference.... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{64}"
} | {
"index": 36391,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two fair coins are simultaneously flipped. This is done repeatedly until at least one of the coins comes up heads, at which point the process stops. What is the probability that the other coin also came up heads on this last flip?"
}
] | math | {
"style": "rule",
"ground_truth": "1/3"
} | {
"index": 36392,
"solution": "$1 / 3$. Let the desired probability be $p$. There is a $1 / 4$ chance that both coins will come up heads on the first toss. Otherwise, both can come up heads simultaneously only if both are tails on the first toss, and then the process restarts as if from the beginning; thus this sit... |
DeepScaleR | [
{
"role": "user",
"content": "Two fair coins are to be tossed once. For each head that results, one fair die is to be rolled. What is the probability that the sum of the die rolls is odd?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{8}"
} | {
"index": 36393,
"solution": "We start by considering the different cases based on the number of heads that appear when two fair coins are tossed. For each head, a fair die is rolled. We need to find the probability that the sum of the die rolls is odd.\n\n#### Case Analysis:\n1. **Case 1: 0 Heads (2 Tails)**\n ... |
DeepScaleR | [
{
"role": "user",
"content": "Two fair dice, each with at least $6$ faces are rolled. On each face of each die is printed a distinct integer from $1$ to the number of faces on that die, inclusive. The probability of rolling a sum of $7$ is $\\frac34$ of the probability of rolling a sum of $10,$ and the prob... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 36394,
"solution": "1. **Define Variables:**\n Let the number of faces on the two dice be $a$ and $b$, respectively, with $a \\geq b$. Assume each die has distinct integers from $1$ to the number of faces on that die.\n\n2. **Analyze the Probability of Rolling a Sum of $7$:**\n Since each die has at ... |
DeepScaleR | [
{
"role": "user",
"content": "Two fair octahedral dice, each with the numbers 1 through 8 on their faces, are rolled. Let $N$ be the remainder when the product of the numbers showing on the two dice is divided by 8. Find the expected value of $N$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11}{4}"
} | {
"index": 36395,
"solution": "If the first die is odd, which has $\\frac{1}{2}$ probability, then $N$ can be any of $0,1,2,3,4,5,6,7$ with equal probability, because multiplying each element of $\\{0, \\ldots, 7\\}$ with an odd number and taking modulo 8 results in the same numbers, as all odd numbers are relative... |
DeepScaleR | [
{
"role": "user",
"content": "Two fair six-sided dice are rolled. What is the probability that their sum is at least 10?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"index": 36396,
"solution": "There are $3,2,1$ outcomes with sum $10,11,12$, so the probability is $\\frac{3+2+1}{6^{2}}=\\frac{1}{6}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Two fair, eight-sided dice are rolled. What is the probability that the sum of the two numbers showing is less than 12?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{49}{64}"
} | {
"index": 36397,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two fair, six-sided dice are rolled. What is the probability that the sum of the two numbers showing is less than 11?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11}{12}"
} | {
"index": 36398,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two fair, six-sided dice are rolled. What is the probability that the sum of the two numbers showing is less than or equal to 10 and at least one die shows a number greater than 3?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 36399,
"solution": ""
} |
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