data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "A triangle $ABC$ with $AC=20$ is inscribed in a circle $\\omega$ . A tangent $t$ to $\\omega$ is drawn through $B$ . The distance $t$ from $A$ is $25$ and that from $C$ is $16$ .If $S$ denotes the area of the triangle $ABC$ , find the largest integer not e... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 4000,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle $ABC$ with orthocenter $H$ is inscribed in a circle with center $K$ and radius $1$ , where the angles at $B$ and $C$ are non-obtuse. If the lines $HK$ and $BC$ meet at point $S$ such that $SK(SK -SH) = 1$ , compute the area of the concave quadrilate... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"index": 4001,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle $\\triangle ABC$ satisfies $AB = 13$ , $BC = 14$ , and $AC = 15$ . Inside $\\triangle ABC$ are three points $X$ , $Y$ , and $Z$ such that:\n\n\n\n- $Y$ is the centroid of $\\triangle ABX$\n- $Z$ is the centroid of $\\triangle BCY$\n- $X$ is the centro... | math | {
"style": "rule",
"ground_truth": "84/13"
} | {
"index": 4002,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle $DEF$ has an inradius of $2$ and a circumradius of $9$. If $2\\cos{E} = \\cos{D} + \\cos{F}$, then what is the area of triangle $DEF$?"
}
] | math | {
"style": "rule",
"ground_truth": "54"
} | {
"index": 4003,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle $H$ is inscribed in a regular hexagon $S$ such that one side of $H$ is parallel to one side of $S$. What is the maximum possible ratio of the area of $H$ to the area of $S$?"
}
] | math | {
"style": "rule",
"ground_truth": "3/8"
} | {
"index": 4004,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle $X Y Z$ and a circle $\\omega$ of radius 2 are given in a plane, such that $\\omega$ intersects segment $\\overline{X Y}$ at the points $A, B$, segment $\\overline{Y Z}$ at the points $C, D$, and segment $\\overline{Z X}$ at the points $E, F$. Suppose that $X B>X A, ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{10}-1"
} | {
"index": 4005,
"solution": "Let $d=A B$ and $x=d / 2$ for ease of notation. Let the center of $(A B C D E F)$ be $I$. Because $A B=C D=E F$, the distance from $I$ to $A B, C D$, and $E F$ are the same, so $I$ is the incenter of $\\triangle X Y Z$. Let $\\triangle X Y Z$ have inradius $r$. By symmetry, we have $X ... |
DeepScaleR | [
{
"role": "user",
"content": "A triangle $\\bigtriangleup ABC$ has vertices lying on the parabola defined by $y = x^2 + 4$. Vertices $B$ and $C$ are symmetric about the $y$-axis and the line $\\overline{BC}$ is parallel to the $x$-axis. The area of $\\bigtriangleup ABC$ is $100$. $A$ is the point $(2,8)$. D... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 4006,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle ABC has vertices at points $A = (0,2)$, $B = (0,0)$, and $C = (10,0)$. A vertical line $x = a$ divides the triangle into two regions. Find the value of $a$ such that the area to the left of the line is one-third of the total area of triangle ABC.\nA) $\\frac{10}{3}$\... | math | {
"style": "rule",
"ground_truth": "\\frac{10}{3}"
} | {
"index": 4007,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle \\( ABC \\) is given. It is known that \\( AB=4 \\), \\( AC=2 \\), and \\( BC=3 \\). The angle bisector of \\( \\angle BAC \\) intersects side \\( BC \\) at point \\( K \\). A line passing through point \\( B \\) parallel to \\( AC \\) intersects the extension of the... | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{6}"
} | {
"index": 4008,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle \\(A B C\\) is considered. Point \\(F\\) is the midpoint of side \\(A B\\). Point \\(S\\) lies on the ray \\(A C\\) such that \\(C S = 2 A C\\). In what ratio does the line \\(S F\\) divide side \\(B C\\)?"
}
] | math | {
"style": "rule",
"ground_truth": "2:3"
} | {
"index": 4009,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle \\(ABC\\) is to be constructed so that \\(A\\) is at \\((3,2)\\), \\(B\\) is on the line \\(y=x\\), and \\(C\\) is on the \\(x\\)-axis. Find the minimum possible perimeter of \\(\\triangle ABC\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{26}"
} | {
"index": 4010,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle and a trapezoid are equal in area. They also have the same altitude. If the base of the triangle is 18 inches, the median of the trapezoid is:"
}
] | math | {
"style": "rule",
"ground_truth": "9 \\text{ inches}"
} | {
"index": 4011,
"solution": "1. **Identify the formula for the area of the triangle and trapezoid:**\n - The area of a triangle is given by the formula:\n \\[\n \\text{Area}_{\\text{triangle}} = \\frac{1}{2} \\times \\text{base} \\times \\text{height} = \\frac{1}{2}bh\n \\]\n - The area of a trapez... |
DeepScaleR | [
{
"role": "user",
"content": "A triangle can be formed having side lengths 4, 5, and 8. It is impossible, however, to construct a triangle with side lengths 4, 5, and 9. Ron has eight sticks, each having an integer length. He observes that he cannot form a triangle using any three of these sticks as side le... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 4012,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle has a base of 20 inches. Two lines are drawn parallel to the base, intersecting the other two sides and dividing the triangle into four regions of equal area. Determine the length of the parallel line closer to the base."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 4013,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle has a side of length 6 cm, a side of length 8 cm and a right angle. What is the shortest possible length of the remaining side of the triangle? Express your answer in centimeters as a decimal to the nearest hundredth."
}
] | math | {
"style": "rule",
"ground_truth": "5.29"
} | {
"index": 4014,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle has altitudes of lengths 15, 21, and 35. Find its area."
}
] | math | {
"style": "rule",
"ground_truth": "210"
} | {
"index": 4015,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle has area $30$, one side of length $10$, and the median to that side of length $9$. Let $\\theta$ be the acute angle formed by that side and the median. What is $\\sin{\\theta}$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 4016,
"solution": "1. **Identify the given values and the formula for the area of a triangle:**\n - Given: Area of the triangle $= 30$, one side (let's call it $a$) $= 10$, and the median to that side $= 9$.\n - The formula for the area of a triangle using a side and the median to that side is not di... |
DeepScaleR | [
{
"role": "user",
"content": "A triangle has one side of length $13$, and the angle opposite this side is $60^{\\circ}$. The ratio of the other two sides is $4:3$. Calculate the area of this triangle."
}
] | math | {
"style": "rule",
"ground_truth": "39 \\sqrt{3}"
} | {
"index": 4017,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle has one side of length 5 cm, another side of length 12 cm, and includes a right angle. What is the shortest possible length of the third side of the triangle? Express your answer in centimeters as a decimal to the nearest hundredth."
}
] | math | {
"style": "rule",
"ground_truth": "10.91"
} | {
"index": 4018,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle has side lengths 7, 8, and 9. There are exactly two lines that simultaneously bisect the perimeter and area of the triangle. Let $\\theta$ be the acute angle between these two lines. Find $\\tan \\theta.$\n\n[asy]\nunitsize(0.5 cm);\n\npair A, B, C, P, Q, R, S, X;... | math | {
"style": "rule",
"ground_truth": "3 \\sqrt{5} + 2 \\sqrt{10}"
} | {
"index": 4019,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle has side lengths of $x,75,100$ where $x<75$ and altitudes of lengths $y,28,60$ where $y<28$ . What is the value of $x+y$ ?\n\n*2019 CCA Math Bonanza Team Round #2*"
}
] | math | {
"style": "rule",
"ground_truth": "56"
} | {
"index": 4020,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle has side lengths of 8, 15 and 17 units. What is the area of the triangle, in square units?"
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 4021,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle has sides of length $48$ , $55$ , and $73$ . A square is inscribed in the triangle such that one side of the square lies on the longest side of the triangle, and the two vertices not on that side of the square touch the other two sides of the triangle. If $c$ ... | math | {
"style": "rule",
"ground_truth": "200689"
} | {
"index": 4022,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle has sides of length $7$ and $23$. What is the smallest whole number greater than the perimeter of any triangle with these side lengths?"
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 4023,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle has sides of length 888, 925, and $x>0$. Find the value of $x$ that minimizes the area of the circle circumscribed about the triangle."
}
] | math | {
"style": "rule",
"ground_truth": "259"
} | {
"index": 4024,
"solution": "259."
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle has sides of lengths 20 and 19. If the triangle is not acute, how many possible integer lengths can the third side have?"
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 4025,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle has sides of lengths 30, 70, and 80. When an altitude is drawn to the side of length 80, the longer segment of this side that is intercepted by the altitude is:"
}
] | math | {
"style": "rule",
"ground_truth": "65"
} | {
"index": 4026,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle has sides of lengths 40 units, 50 units, and 70 units. An altitude is dropped from the vertex opposite the side of length 70 units. Calculate the length of this altitude.\nA) $\\frac{40\\sqrt{7}}{7}$ units\nB) $\\frac{80\\sqrt{7}}{7}$ units\nC) $\\frac{120\\sqrt{7}}{... | math | {
"style": "rule",
"ground_truth": "\\frac{80\\sqrt{7}}{7}"
} | {
"index": 4027,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle has sides of lengths 6 cm and 8 cm that create a 45-degree angle between them. Calculate the length of the third side."
}
] | math | {
"style": "rule",
"ground_truth": "5.67"
} | {
"index": 4028,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle has sides with lengths of $18$, $24$, and $30$. Calculate the length of the shortest altitude."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 4029,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle has three different integer side lengths and a perimeter of 20 units. What is the maximum length of any one side?"
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 4030,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle has three different integer side lengths and a perimeter of 30 units. What is the maximum length of any one side?"
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 4031,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle has three sides of the following side lengths: $7$, $10$, and $x^2$. What are all of the positive integer values of $x$ such that the triangle exists? Separate your answers using commas and express them in increasing order."
}
] | math | {
"style": "rule",
"ground_truth": "2, 3, \\text{ and } 4"
} | {
"index": 4032,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle has three sides that are three consecutive natural numbers, and the largest angle is twice the smallest angle. The perimeter of this triangle is __________."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 4033,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle has two medians of lengths 9 and 12. Find the largest possible area of the triangle. (Note: A median is a line segment joining a vertex of the triangle to the midpoint of the opposite side.)"
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 4034,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle has vertices $(0,0)$, $(1,1)$, and $(6m,0)$. The line $y = mx$ divides the triangle into two triangles of equal area. What is the sum of all possible values of $m$?"
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac {1}{6}"
} | {
"index": 4035,
"solution": "1. **Identify the vertices and the line dividing the triangle**: \n Let's label the vertices of the triangle as $A = (0,0)$, $B = (1,1)$, and $C = (6m,0)$. The line $y = mx$ is supposed to divide the triangle into two triangles of equal area.\n\n2. **Determine the intersection point*... |
DeepScaleR | [
{
"role": "user",
"content": "A triangle has vertices $A(0,0)$, $B(12,0)$, and $C(8,10)$. The probability that a randomly chosen point inside the triangle is closer to vertex $B$ than to either vertex $A$ or vertex $C$ can be written as $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integer... | math | {
"style": "rule",
"ground_truth": "409"
} | {
"index": 4036,
"solution": "Calculate the area of the triangle using the Shoelace Theorem on $(0,0), (12,0), (8,10)$ \\[\\frac{1}{2}|(0+120+0)-(0+0+0)|=60\\]\n Get the four points $(6,0), (6,\\frac{17}{5}), (10,5)$, and $(12,0)$ by any method from the above solutions. Then use the Shoelace Theorem to find the are... |
DeepScaleR | [
{
"role": "user",
"content": "A triangle has vertices $A=(4,3)$, $B=(-4,-1)$, and $C=(9,-7)$. Calculate the equation of the bisector of $\\angle A$ in the form $3x - by + c = 0$. Determine the value of $b+c$."
}
] | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"index": 4037,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle has vertices $P=(-8,5)$, $Q=(-15,-19)$, and $R=(1,-7)$. The equation of the bisector of $\\angle P$ can be written in the form $ax+2y+c=0$. Find $a+c$.\n[asy] import graph; pointpen=black;pathpen=black+linewidth(0.7);pen f = fontsize(10); pair P=(-8,5),Q=(-15,-19),R=... | math | {
"style": "rule",
"ground_truth": "89"
} | {
"index": 4038,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle has vertices $P_{}^{}=(-8,5)$, $Q_{}^{}=(-15,-19)$, and $R_{}^{}=(1,-7)$. The equation of the bisector of $\\angle P$ can be written in the form $ax+2y+c=0_{}^{}$. Find $a+c_{}^{}$.\n[asy] import graph; pointpen=black;pathpen=black+linewidth(0.7);pen f = fontsize(10)... | math | {
"style": "rule",
"ground_truth": "89"
} | {
"index": 4039,
"solution": "[asy] import graph; pointpen=black;pathpen=black+linewidth(0.7);pen f = fontsize(10); pair P=(-8,5),Q=(-15,-19),R=(1,-7),S=(7,-15),T=(-4,-17),U=IP(P--T,Q--R); MP(\"P\",P,N,f);MP(\"Q\",Q,W,f);MP(\"R\",R,E,f);MP(\"P'\",U,SE,f); D(P--Q--R--cycle);D(U);D(P--U); D((-17,0)--(4,0),Arrows(2mm)... |
DeepScaleR | [
{
"role": "user",
"content": "A triangle has vertices \\( P=(-7,4) \\), \\( Q=(-14,-20) \\), and \\( R=(2,-8) \\). The equation of the bisector of \\( \\angle P \\) can be written in the form \\( ax+2y+c=0 \\). Find \\( a+c \\)."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 4040,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle has vertices at $(-2,3),(7,-3),(4,6)$. Calculate the area of this triangle in square units and express your answer as a decimal to the nearest tenth."
}
] | math | {
"style": "rule",
"ground_truth": "31.5"
} | {
"index": 4041,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle has vertices at $(-3,2),(6,-2),(3,5)$. How many square units are in the area of the triangle? Express your answer as a decimal to the nearest tenth."
}
] | math | {
"style": "rule",
"ground_truth": "25.5"
} | {
"index": 4042,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle has vertices at coordinates (2,2), (5,6) and (6,2). What is the number of units in the length of the longest side of the triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 4043,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle in a Cartesian coordinate plane has vertices (5, -2), (10, 5) and (5, 5). How many square units are in the area of the triangle? Express your answer as a decimal to the nearest tenth."
}
] | math | {
"style": "rule",
"ground_truth": "17.5"
} | {
"index": 4044,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle in the $x y$-plane is such that when projected onto the $x$-axis, $y$-axis, and the line $y=x$, the results are line segments whose endpoints are $(1,0)$ and $(5,0),(0,8)$ and $(0,13)$, and $(5,5)$ and $(7.5,7.5)$, respectively. What is the triangle's area?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{17}{2}"
} | {
"index": 4045,
"solution": "Sketch the lines $x=1, x=5, y=8, y=13, y=10-x$, and $y=15-x$. The triangle has to be contained in the hexagonal region contained in all these lines. If all the projections are correct, every other vertex of the hexagon must be a vertex of the triangle, which gives us two possibilities ... |
DeepScaleR | [
{
"role": "user",
"content": "A triangle is divided into 1000 smaller triangles. What is the minimum number of distinct points that can be the vertices of these triangles?"
}
] | math | {
"style": "rule",
"ground_truth": "503"
} | {
"index": 4046,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle is divided into 1000 triangles. What is the maximum number of distinct points that can be vertices of these triangles?"
}
] | math | {
"style": "rule",
"ground_truth": "1002"
} | {
"index": 4047,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle is formed with one vertex at the vertex of the parabola $y=x^2-1$ and the other two vertices at the intersections of the line $y=r$ and the parabola. If the area of the triangle is between $8$ and $64$ inclusive, find all possible values of $r$. Express your answer i... | math | {
"style": "rule",
"ground_truth": "[3,15]"
} | {
"index": 4048,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle is inscribed in a circle. The vertices of the triangle divide the circle into three arcs of lengths $3$ , $4$ , and $5$ . What is the area of the triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{9}{\\pi^2}\\left(\\sqrt{3}+3\\right)$"
} | {
"index": 4049,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle is inscribed in a circle. The vertices of the triangle divide the circle into three arcs of lengths 3, 4, and 5. What is the area of the triangle?\n$\\mathrm{(A) \\ 6 } \\qquad \\mathrm{(B) \\frac{18}{\\pi^2} } \\qquad \\mathrm{(C) \\frac{9}{\\pi^2}(\\sqrt{3}-1) } \\... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{\\pi^2}(\\sqrt{3}+3)"
} | {
"index": 4050,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle is inscribed in a circle. The vertices of the triangle divide the circle into three arcs of lengths 5, 6, and 7. What is the area of the triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{202.2192}{\\pi^2}"
} | {
"index": 4051,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle is inscribed in a circle. The vertices of the triangle divide the circle into three arcs of lengths 5, 7, and 8. What is the area of the triangle and the radius of the circle?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{10}{\\pi}"
} | {
"index": 4052,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle is inscribed in a circle. The vertices of the triangle divide the circle into three arcs of lengths 5, 7, and 8. What is the area of the triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{119.84}{\\pi^2}"
} | {
"index": 4053,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle is made of wood sticks of lengths 8, 15 and 17 inches joined end-to-end. Pieces of the same integral length are cut from each of the sticks so that the three remaining pieces can no longer form a triangle. How many inches are in the length of the smallest piece that ... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 4054,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle is partitioned into three triangles and a quadrilateral by drawing two lines from vertices to their opposite sides. The areas of the three triangles are 3, 7, and 7, as shown. What is the area of the shaded quadrilateral?"
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 4055,
"solution": "1. **Identify the triangles and their areas**: We are given that the areas of triangles $EFA$, $FAB$, and $FBD$ are $3$, $7$, and $7$ respectively. We need to find the area of the shaded quadrilateral $CEDF$.\n\n2. **Analyze triangles $EFA$ and $FAB$**: These triangles share an altitud... |
DeepScaleR | [
{
"role": "user",
"content": "A triangle with angles \\( A, B, C \\) satisfies the following conditions:\n\\[\n\\frac{\\sin A + \\sin B + \\sin C}{\\cos A + \\cos B + \\cos C} = \\frac{12}{7},\n\\]\nand\n\\[\n\\sin A \\sin B \\sin C = \\frac{12}{25}.\n\\]\n\nGiven that \\( \\sin C \\) takes on three possibl... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 4056,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle with integral sides has perimeter $8$. The area of the triangle is"
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"index": 4057,
"solution": "1. **Identify Possible Side Lengths:**\n Given a triangle with integral sides and a perimeter of $8$, we denote the sides as $a$, $b$, and $c$. Thus, $a + b + c = 8$.\n\n2. **Check for Valid Triangles:**\n To form a valid triangle, the sum of the lengths of any two sides must be gr... |
DeepScaleR | [
{
"role": "user",
"content": "A triangle with interior angles $60^{\\circ}, 45^{\\circ}$ and $75^{\\circ}$ is inscribed in a circle of radius 2. What is the area of the triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "3 + \\sqrt{3}"
} | {
"index": 4058,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle with side lengths $5,7,8$ is inscribed in a circle $C$. The diameters of $C$ parallel to the sides of lengths 5 and 8 divide $C$ into four sectors. What is the area of either of the two smaller ones?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{49}{18} \\pi"
} | {
"index": 4059,
"solution": "Let $\\triangle P Q R$ have sides $p=7, q=5, r=8$. Of the four sectors determined by the diameters of $C$ that are parallel to $P Q$ and $P R$, two have angles equal to $P$ and the other two have angles equal to $\\pi-P$. We first find $P$ using the law of cosines: $49=25+64-2(5)(8) \\... |
DeepScaleR | [
{
"role": "user",
"content": "A triangle with side lengths 8, 13, and 17 has an incircle. The side length of 8 is divided by the point of tangency into segments \\( r \\) and \\( s \\), with \\( r < s \\). Find the ratio \\( r : s \\)."
}
] | math | {
"style": "rule",
"ground_truth": "1: 3"
} | {
"index": 4060,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle with side lengths in the ratio 2:3:4 is inscribed in a circle of radius 4. What is the area of the triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{15}"
} | {
"index": 4061,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle with side lengths in the ratio 3:4:5 is inscribed in a circle of radius 3. What is the area of the triangle? Provide your answer as a decimal rounded to the nearest hundredth."
}
] | math | {
"style": "rule",
"ground_truth": "8.64"
} | {
"index": 4062,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle with sides of 5, 12, and 13 has both an inscribed and a circumscribed circle. What is the distance between the centers of those circles?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{65}}{2}"
} | {
"index": 4063,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle with vertices $(6, 5)$, $(8, -3)$, and $(9, 1)$ is reflected about the line $x=8$ to create a second triangle. What is the area of the union of the two triangles?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{32}{3}"
} | {
"index": 4064,
"solution": "1. **Identify the vertices and their reflections**: \n - Let $A = (6, 5)$, $B = (8, -3)$, and $C = (9, 1)$. \n - Reflecting over the line $x=8$, we find:\n - $A' = (10, 5)$ since reflecting $(6, 5)$ over $x=8$ moves it 2 units to the left of $x=8$, so it moves 2 units to the ri... |
DeepScaleR | [
{
"role": "user",
"content": "A triangle with vertices \\(A = (4, 3)\\), \\(B = (6, -2)\\), and \\(C = (7, 1)\\) is reflected about the line \\(x = 6\\) to create a second triangle. Determine the area of the union of the two triangles."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 4065,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangle with vertices as $A=(1,3)$, $B=(5,1)$, and $C=(4,4)$ is plotted on a $6\\times5$ grid. What fraction of the grid is covered by the triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"index": 4066,
"solution": "\nTo find the fraction of the grid covered by the triangle with vertices $A=(1,3)$, $B=(5,1)$, and $C=(4,4)$, we first need to calculate the area of the triangle and then compare it to the area of the grid.\n\n#### Step 1: Calculate the area of the triangle using the Shoelace Theorem\n... |
DeepScaleR | [
{
"role": "user",
"content": "A triangle with vertices at \\((1003,0), (1004,3),\\) and \\((1005,1)\\) in the \\(xy\\)-plane is revolved all the way around the \\(y\\)-axis. Find the volume of the solid thus obtained."
}
] | math | {
"style": "rule",
"ground_truth": "5020 \\pi"
} | {
"index": 4067,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangular array of $2016$ coins has $1$ coin in the first row, $2$ coins in the second row, $3$ coins in the third row, and so on up to $N$ coins in the $N$th row. What is the sum of the digits of $N$?"
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 4068,
"solution": "1. **Identify the formula for the sum of the first $N$ natural numbers**: The sum of the first $N$ natural numbers is given by the formula:\n \\[\n S = 1 + 2 + 3 + \\cdots + N = \\frac{N(N+1)}{2}\n \\]\n We need to find $N$ such that this sum equals $2016$.\n\n2. **Set up the e... |
DeepScaleR | [
{
"role": "user",
"content": "A triangular array of numbers has a first row consisting of the odd integers $1,3,5,\\ldots,99$ in increasing order. Each row below the first has one fewer entry than the row above it, and the bottom row has a single entry. Each entry in any row after the top row equals the sum... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 4069,
"solution": "The result above is fairly intuitive if we write out several rows and then divide all numbers in row $r$ by $2^{r-1}$ (we can do this because dividing by a power of 2 doesn't affect divisibility by $67$). The second row will be $2, 4, 6, \\cdots , 98$, the third row will be $3, 5, \\cd... |
DeepScaleR | [
{
"role": "user",
"content": "A triangular array of squares has one square in the first row, two in the second, and in general, $k$ squares in the $k$th row for $1 \\leq k \\leq 11.$ With the exception of the bottom row, each square rests on two squares in the row immediately below (illustrated in the given... | math | {
"style": "rule",
"ground_truth": "640"
} | {
"index": 4070,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangular array of squares has one square in the first row, two in the second, and in general, $k$ squares in the $k$th row for $1 \\leq k \\leq 11.$ With the exception of the bottom row, each square rests on two squares in the row immediately below (illustrated in the given... | math | {
"style": "rule",
"ground_truth": "640"
} | {
"index": 4071,
"solution": "Label each of the bottom squares as $x_0, x_1 \\ldots x_9, x_{10}$.\n Through induction, we can find that the top square is equal to ${10\\choose0}x_0 + {10\\choose1}x_1 + {10\\choose2}x_2 + \\ldots {10\\choose10}x_{10}$. (This also makes sense based on a combinatorial argument: the nu... |
DeepScaleR | [
{
"role": "user",
"content": "A triangular array of squares has one square in the first row, two in the second, and in general, $k$ squares in the $k$th row for $1 \\leq k \\leq 16$. With the exception of the bottom row, each square rests on two squares in the row immediately below. In each square of the si... | math | {
"style": "rule",
"ground_truth": "16384"
} | {
"index": 4072,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangular box is to be cut from an equilateral triangle of length 30 cm. Find the largest possible volume of the box (in cm³)."
}
] | math | {
"style": "rule",
"ground_truth": "500"
} | {
"index": 4073,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangular corner with side lengths $DB=EB=1$ is cut from equilateral triangle ABC of side length $3$. The perimeter of the remaining quadrilateral is"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 4074,
"solution": "1. **Identify the Similar Triangles**: \n - $\\triangle DBE$ is cut from $\\triangle ABC$. Since $\\triangle ABC$ is equilateral, all its angles are $60^\\circ$. \n - $\\triangle DBE$ is also equilateral because $DB = EB$ and each angle in $\\triangle DBE$ is $60^\\circ$.\n\n2. **D... |
DeepScaleR | [
{
"role": "user",
"content": "A triangular lattice is formed with seven points arranged as follows: six points form a regular hexagon and one point is at the center. Each point is one unit away from its nearest neighbor. Determine how many equilateral triangles can be formed where all vertices are on this l... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 4075,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangular pyramid \\( S-ABC \\) has a base in the shape of an equilateral triangle with side lengths of 4. It is known that \\( AS = BS = \\sqrt{19} \\) and \\( CS = 3 \\). Find the surface area of the circumscribed sphere of the triangular pyramid \\( S-ABC \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{268\\pi}{11}"
} | {
"index": 4076,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangular pyramid \\( S-ABC \\) has a base that is an equilateral triangle with a side length of 4. It is given that \\( AS = BS = \\sqrt{19} \\) and \\( CS = 3 \\). Find the surface area of the circumscribed sphere of the triangular pyramid \\( S-ABC \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{268\\pi}{11}"
} | {
"index": 4077,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangular pyramid stack of apples is formed with a base consisting of 6 apples on each side. Each higher layer is formed by 3 apples per row, with the top layer consisting of a single apple. Calculate the total number of apples used in the entire stack."
}
] | math | {
"style": "rule",
"ground_truth": "56"
} | {
"index": 4078,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangular region is bounded by the two coordinate axes and the line given by the equation $2x + y = 6$. What is the area of the region, in square units?"
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 4079,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangular region is bounded by the two coordinate axes and the line given by the equation $3x + y = 9$. Check if the point (1,1) lies inside this triangular region and find the region's area in square units."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{27}{2}"
} | {
"index": 4080,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangular region is enclosed by the lines with equations $y = \\frac{1}{2} x + 3$, $y = -2x + 6$ and $y = 1$. What is the area of the triangular region? Express your answer as a decimal to the nearest hundredth."
}
] | math | {
"style": "rule",
"ground_truth": "8.45"
} | {
"index": 4081,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triangular wire frame with side lengths of $13, 14, 15$ is fitted over a sphere with a radius of 10. Find the distance between the plane containing the triangle and the center of the sphere."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{21}"
} | {
"index": 4082,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triline is a line with the property that three times its slope is equal to the sum of its \\(x\\)-intercept and its \\(y\\)-intercept. For how many integers \\(q\\) with \\(1 \\leq q \\leq 10000\\) is there at least one positive integer \\(p\\) so that there is exactly one tr... | math | {
"style": "rule",
"ground_truth": "57"
} | {
"index": 4083,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A triple of integers \\((a, b, c)\\) satisfies \\(a+b c=2017\\) and \\(b+c a=8\\). Find all possible values of \\(c\\)."
}
] | math | {
"style": "rule",
"ground_truth": "-6,0,2,8"
} | {
"index": 4084,
"solution": "Add and subtract the two equations to find \\((b+a)(c+1)=8+2017\\) and \\((b-a)(c-1)=2017-8\\). We see that \\(c\\) is even and then that every integer \\(c\\) with \\(c+1|2025, c-1| 2009\\) works. We factor and solve. The full solutions are \\((2017,8,0),(-667,1342,2),(-59,-346,-6),(-... |
DeepScaleR | [
{
"role": "user",
"content": "A triple of positive integers $(a, b, c)$ is tasty if $\\operatorname{lcm}(a, b, c) \\mid a+b+c-1$ and $a<b<c$. Find the sum of $a+b+c$ across all tasty triples."
}
] | math | {
"style": "rule",
"ground_truth": "44"
} | {
"index": 4085,
"solution": "The condition implies $c \\mid b+a-1$. WLOG assume $c>b>a$; since $b+a-1<2 c$ we must have $b+a-1=c$. Substituting into $b \\mid a+c-1$ and $a \\mid c+b-1$ gives $$\\begin{aligned} & b \\mid 2 a-2 \\\\ & a \\mid 2 b-2 \\end{aligned}$$ Since $2 a-2<2 b$ we must either have $a=1$ (implyi... |
DeepScaleR | [
{
"role": "user",
"content": "A tripod has three legs each of length $5$ feet. When the tripod is set up, the angle between any pair of legs is equal to the angle between any other pair, and the top of the tripod is $4$ feet from the ground. In setting up the tripod, the lower 1 foot of one leg breaks off. ... | math | {
"style": "rule",
"ground_truth": "183"
} | {
"index": 4086,
"solution": "[asy] size(200); import three; pointpen=black;pathpen=black+linewidth(0.65);pen ddash = dashed+linewidth(0.65); currentprojection = perspective(1,-10,3.3); triple O=(0,0,0),T=(0,0,5),C=(0,3,0),A=(-3*3^.5/2,-3/2,0),B=(3*3^.5/2,-3/2,0); triple M=(B+C)/2,S=(4*A+T)/5; draw(T--S--B--T--C--B... |
DeepScaleR | [
{
"role": "user",
"content": "A trirectangular tetrahedron $M-ABC$ has three pairs of adjacent edges that are perpendicular, and a point $N$ inside the base triangle $ABC$ is at distances of $2\\sqrt{2}$, $4$, and $5$ from the three faces respectively. Find the surface area of the smallest sphere that passe... | math | {
"style": "rule",
"ground_truth": "49\\pi"
} | {
"index": 4087,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A truck delivered 4 bags of cement. They are stacked in the truck. A worker can carry one bag at a time either from the truck to the gate or from the gate to the shed. The worker can carry the bags in any order, each time taking the top bag, carrying it to the respective destin... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"index": 4088,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A truck has new tires fitted on all four wheels. A tire is considered completely worn out if it has traveled $15000 \\mathrm{~km}$ on the rear wheel or $25000 \\mathrm{~km}$ on the front wheel. How far can the truck travel before all four tires are completely worn out if the fr... | math | {
"style": "rule",
"ground_truth": "18750"
} | {
"index": 4089,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A truck travels $\\frac{b}{6}$ feet every $t$ seconds. There are $3$ feet in a yard. How many yards does the truck travel in $3$ minutes?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{10b}{t}"
} | {
"index": 4090,
"solution": "1. **Identify the rate of travel**: The truck travels $\\frac{b}{6}$ feet every $t$ seconds.\n\n2. **Convert time from minutes to seconds**: Since there are $60$ seconds in a minute, $3$ minutes is $3 \\times 60 = 180$ seconds.\n\n3. **Set up a proportion to find the distance in feet**... |
DeepScaleR | [
{
"role": "user",
"content": "A truck travels due west at $\\frac{3}{4}$ mile per minute on a straight road. At the same time, a circular storm, whose radius is $60$ miles, moves southwest at $\\frac{1}{2}\\sqrt{2}$ mile per minute. At time $t=0$, the center of the storm is $130$ miles due north of the truc... | math | {
"style": "rule",
"ground_truth": "208"
} | {
"index": 4091,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A truncated cone has horizontal bases with radii 18 and 2. A sphere is tangent to the top, bottom, and lateral surface of the truncated cone. What is the radius of the sphere?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 4092,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A truncated pyramid has a square base with a side length of 4 units, and every lateral edge is also 4 units. The side length of the top face is 2 units. What is the greatest possible distance between any two vertices of the truncated pyramid?"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{32}"
} | {
"index": 4093,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A truncated right circular cone has a large base radius of 10 cm and a small base radius of 5 cm. The height of the truncated cone is 10 cm. Calculate the volume of this solid."
}
] | math | {
"style": "rule",
"ground_truth": "583.33\\pi"
} | {
"index": 4094,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A two-digit integer $AB$ equals $\\frac{1}{9}$ of the three-digit integer $AAB$, where $A$ and $B$ represent distinct digits from 1 to 9. What is the smallest possible value of the three-digit integer $AAB$?"
}
] | math | {
"style": "rule",
"ground_truth": "225"
} | {
"index": 4095,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A two-digit integer $AB$ equals $\\frac{1}{9}$ of the three-digit integer $CCB$, where $C$ and $B$ represent distinct digits from 1 to 9. What is the smallest possible value of the three-digit integer $CCB$?"
}
] | math | {
"style": "rule",
"ground_truth": "225"
} | {
"index": 4096,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A two-digit integer between 10 and 99, inclusive, is chosen at random. Each possible integer is equally likely to be chosen. What is the probability that its tens digit is a multiple of its units (ones) digit?"
}
] | math | {
"style": "rule",
"ground_truth": "23/90"
} | {
"index": 4097,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A two-digit number has its unit digit greater than the tens digit by 2. If this two-digit number is less than 30, find the number."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 4098,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A two-digit number is divided by the sum of its digits. The result is a number between 2.6 and 2.7. Find all of the possible values of the original two-digit number."
}
] | math | {
"style": "rule",
"ground_truth": "29"
} | {
"index": 4099,
"solution": ""
} |
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