data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "A plane intersects a right circular cylinder of radius $3$ forming an ellipse. If the major axis of the ellipse is $75\\%$ longer than the minor axis, the length of the major axis is:"
}
] | math | {
"style": "rule",
"ground_truth": "10.5"
} | {
"index": 2600,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A plane is expressed parametrically by\n\\[\\mathbf{v} = \\begin{pmatrix} 1 + s - t \\\\ 2 - s \\\\ 3 - 2s + 2t \\end{pmatrix}.\\]Find the equation of the plane. Enter your answer in the form\n\\[Ax + By + Cz + D = 0,\\]where $A,$ $B,$ $C,$ $D$ are integers such that $A > 0$ a... | math | {
"style": "rule",
"ground_truth": "2x + z - 5 = 0"
} | {
"index": 2601,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A plane is uniquely determined by three non-collinear points. What is the maximum possible number of planes that can be determined by 12 points in space?"
}
] | math | {
"style": "rule",
"ground_truth": "220"
} | {
"index": 2602,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A plane parallel to the base of a cone divides the height of the cone into two equal segments. What is the ratio of the lateral surface areas of the two parts of the cone?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 2603,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A plane passes through the midpoints of edges $AB$ and $CD$ of pyramid $ABCD$ and divides edge $BD$ in the ratio $1:3$. In what ratio does this plane divide edge $AC$?"
}
] | math | {
"style": "rule",
"ground_truth": "1:3"
} | {
"index": 2604,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A plastic snap-together cube has a protruding snap on one side and receptacle holes on the other five sides as shown. What is the smallest number of these cubes that can be snapped together so that only receptacle holes are showing?\n[asy] draw((0,0)--(4,0)--(4,4)--(0,4)--cycl... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 2605,
"solution": "To solve this problem, we need to determine the smallest number of cubes that can be snapped together such that all the protruding snaps are hidden and only the receptacle holes are visible.\n\n1. **Understanding the Cube Configuration**: Each cube has one protruding snap and five rece... |
DeepScaleR | [
{
"role": "user",
"content": "A play has two different male roles, two different female roles and two different roles that can be either gender. Only a man can be assigned to a male role, and only a woman can be assigned to a female role. If five men and six women audition, in how many ways can the six role... | math | {
"style": "rule",
"ground_truth": "25200"
} | {
"index": 2606,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point $ M$ is taken on the perpendicular bisector of the side $ AC$ of an acute-angled triangle $ ABC$ so that $ M$ and $ B$ are on the same side of $ AC$ . If $ \\angle BAC\\equal{}\\angle MCB$ and $ \\angle ABC\\plus{}\\angle MBC\\equal{}180^{\\circ}$ , find ... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 2607,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point $E$ on side $CD$ of a rectangle $ABCD$ is such that $\\triangle DBE$ is isosceles and $\\triangle ABE$ is right-angled. Find the ratio between the side lengths of the rectangle."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{\\sqrt{5} - 2}"
} | {
"index": 2608,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point $P$ is chosen uniformly at random in the interior of triangle $ABC$ with side lengths $AB = 5$ , $BC = 12$ , $CA = 13$ . The probability that a circle with radius $\\frac13$ centered at $P$ does not intersect the perimeter of $ABC$ can be written as $\\fr... | math | {
"style": "rule",
"ground_truth": "61"
} | {
"index": 2609,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point $P$ is given on the curve $x^4+y^4=1$ . Find the maximum distance from the point $P$ to the origin."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 2610,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point $(3\\sqrt{5},d+3)$ is $3d$ units away from the origin. What is the smallest possible value of $d$?"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 2611,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point $(x, y)$ is randomly selected from inside the rectangle with vertices $(0, 0)$, $(4, 0)$, $(4, 3)$, and $(0, 3)$. What is the probability that both $x < y$ and $x + y < 5$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{8}"
} | {
"index": 2612,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point $(x, y)$ is randomly selected such that $0 \\leq x \\leq 3$ and $0 \\leq y \\leq 3$. What is the probability that $x + 2y \\leq 6$? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 2613,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point $(x, y)$ is randomly selected such that $0 \\leq x \\leq 4$ and $0 \\leq y \\leq 5$. What is the probability that $x + y \\leq 5$? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"index": 2614,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point $(x, y)$ is selected uniformly at random from the unit square $S=\\{(x, y) \\mid 0 \\leq x \\leq 1,0 \\leq y \\leq 1\\}$. If the probability that $(3x+2y, x+4y)$ is in $S$ is $\\frac{a}{b}$, where $a, b$ are relatively prime positive integers, compute $100a+b$."
}
] | math | {
"style": "rule",
"ground_truth": "820"
} | {
"index": 2615,
"solution": "Under the transformation $(x, y) \\mapsto(3x+2y, x+4y), S$ is mapped to a parallelogram with vertices $(0,0),(3,1),(5,5)$, and $(2,4)$. Using the shoelace formula, the area of this parallelogram is 10. The intersection of the image parallelogram and $S$ is the quadrilateral with vertic... |
DeepScaleR | [
{
"role": "user",
"content": "A point $(x, y)$ is to be chosen in the coordinate plane so that it is equally distant from the x-axis, the y-axis, and the line $x+y=2$. Then $x$ is"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 2616,
"solution": "1. **Identify the Triangle and its Properties**:\n The problem involves a triangle formed by the x-axis, y-axis, and the line $x+y=2$. The vertices of this triangle are at the origin $O(0,0)$, the y-intercept $A(0,2)$, and the x-intercept $B(2,0)$.\n\n2. **Understanding the Incenter*... |
DeepScaleR | [
{
"role": "user",
"content": "A point $(x,y)$ in the plane is called a lattice point if both $x$ and $y$ are integers. The area of the largest square that contains exactly three lattice points in its interior is closest to"
}
] | math | {
"style": "rule",
"ground_truth": "5.0"
} | {
"index": 2617,
"solution": "To solve this problem, we need to find the largest square that contains exactly three lattice points in its interior. We will consider the properties of lattice points and the geometry of squares.\n\n1. **Understanding Lattice Points and Squares**:\n A lattice point is a point in the... |
DeepScaleR | [
{
"role": "user",
"content": "A point $(x,y)$ is a distance of 12 units from the $x$-axis. It is a distance of 10 units from the point $(1,6)$. It is a distance $n$ from the origin. Given that $x>1$, what is $n$?"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 2618,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point $(x,y)$ is a distance of 15 units from the $x$-axis. It is a distance of 13 units from the point $(2,7)$. It is a distance $n$ from the origin. Given that $x>2$, what is $n$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{334 + 4\\sqrt{105}}"
} | {
"index": 2619,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point $(x,y)$ is randomly and uniformly chosen inside the rectangle with vertices (0,0), (0,3), (4,3), and (4,0). What is the probability that $x + 2y < 6$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{2}{3}"
} | {
"index": 2620,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point $(x,y)$ is randomly and uniformly chosen inside the square with vertices (0,0), (0,2), (2,2), and (2,0). What is the probability that $x+y < 3$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{7}{8}"
} | {
"index": 2621,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point $(x,y)$ is randomly and uniformly chosen inside the square with vertices (0,0), (0,3), (3,3), and (3,0). What is the probability that $x+y < 5$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{17}{18}"
} | {
"index": 2622,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point $(x,y)$ is randomly picked from inside the rectangle with vertices $(0,0)$, $(3,0)$, $(3,2)$, and $(0,2)$. What is the probability that $x < y$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{3}"
} | {
"index": 2623,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point $(x,y)$ is randomly picked from inside the rectangle with vertices $(0,0)$, $(4,0)$, $(4,1)$, and $(0,1)$. What is the probability that $x < y$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"index": 2624,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point $(x,y)$ is randomly picked from inside the rectangle with vertices $(0,0)$, $(2,0)$, $(2,2)$, and $(0,2)$. What is the probability that $x^2 + y^2 < y$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{32}"
} | {
"index": 2625,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point $(x,y)$ is randomly picked from inside the rectangle with vertices $(0,0)$, $(6,0)$, $(6,2)$, and $(0,2)$. What is the probability that $x + 1 < y$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{24}"
} | {
"index": 2626,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point $(x,y)$ is randomly picked from inside the rectangle with vertices $(0,0)$, $(6,0)$, $(6,2)$, and $(0,2)$. What is the probability that $x + y < 3$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 2627,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point $(x,y)$ is randomly picked from inside the rectangle with vertices $(0,0)$, $(6,0)$, $(6,2)$, and $(0,2)$. What is the probability that $x^2 < y$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{18}"
} | {
"index": 2628,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point $(x,y)$ is randomly selected such that $0 \\le x \\le 3$ and $0 \\le y \\le 6$. What is the probability that $x+y \\le 4$? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{12}"
} | {
"index": 2629,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point $A(-2,-4)$ outside the parabola $y^{2}=2px (p > 0)$ is connected to a line $l$: $\\begin{cases} x=-2+ \\frac{\\sqrt{2}}{2}t \\\\ y=-4+ \\frac{\\sqrt{2}}{2}t \\end{cases} (t$ is a parameter, $t \\in \\mathbb{R})$ intersecting the parabola at points $M_{1}$ and $M_{2}$. T... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{10}"
} | {
"index": 2630,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point $M$ on the parabola $y=4x^{2}$ is at a distance of $1$ from the focus. The ordinate of point $M$ is __________."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{15}{16}"
} | {
"index": 2631,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point $P$ is chosen at random from the interior of a square $ABCD$. What is the probability that the triangle $ABP$ has a greater area than each of the triangles $BCP$, $CDP$, and $DAP$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 2632,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point $P$ is chosen at random in the interior of a unit square $S$. Let $d(P)$ denote the distance from $P$ to the closest side of $S$. The probability that $\\frac{1}{5}\\le d(P)\\le\\frac{1}{3}$ is equal to $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive int... | math | {
"style": "rule",
"ground_truth": "281"
} | {
"index": 2633,
"solution": "First, let's figure out $d(P) \\geq \\frac{1}{3}$ which is\\[\\left(\\frac{3}{5}\\right)^2=\\frac{9}{25}.\\]Then, $d(P) \\geq \\frac{1}{5}$ is a square inside $d(P) \\geq \\frac{1}{3}$, so\\[\\left(\\frac{1}{3}\\right)^2=\\frac{1}{9}.\\]Therefore, the probability that $\\frac{1}{5}\\le... |
DeepScaleR | [
{
"role": "user",
"content": "A point $P$ is chosen at random in the interior of equilateral triangle $ABC$. What is the probability that $\\triangle ABP$ has a greater area than each of $\\triangle ACP$ and $\\triangle BCP$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 2634,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point $P$ is chosen in the interior of $\\triangle ABC$ such that when lines are drawn through $P$ parallel to the sides of $\\triangle ABC$, the resulting smaller triangles $t_{1}$, $t_{2}$, and $t_{3}$ in the figure, have areas $4$, $9$, and $49$, respectively. Find the are... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"index": 2635,
"solution": "By the transversals that go through $P$, all four triangles are similar to each other by the $AA$ postulate. Also, note that the length of any one side of the larger triangle is equal to the sum of the sides of each of the corresponding sides on the smaller triangles. We use the identi... |
DeepScaleR | [
{
"role": "user",
"content": "A point $P$ is chosen uniformly at random inside a square of side length 2. If $P_{1}, P_{2}, P_{3}$, and $P_{4}$ are the reflections of $P$ over each of the four sides of the square, find the expected value of the area of quadrilateral $P_{1} P_{2} P_{3} P_{4}$."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 2636,
"solution": "Let $A B C D$ denote the square defined in the problem. We see that if $P_{1}$ is the reflection of $P$ over $\\overline{A B}$, then the area of $P_{1} A B$ is the same as the area of $P A B$. Furthermore, if $P_{4}$ is the reflection of $P$ over $\\overline{D A}, P_{1}, A$, and $P_{4}... |
DeepScaleR | [
{
"role": "user",
"content": "A point $P$ is outside a circle and is $13$ inches from the center. A secant from $P$ cuts the circle at $Q$ and $R$ so that the external segment of the secant $PQ$ is $9$ inches and $QR$ is $7$ inches. The radius of the circle is:"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 2637,
"solution": "1. **Identify the Relevant Theorem**: We use the Secant-Secant Power Theorem (or External Secant Segment Theorem), which states that for two secants intersecting at a point outside the circle, the product of the lengths of one secant segment and its external part equals the product of ... |
DeepScaleR | [
{
"role": "user",
"content": "A point $P$ is randomly placed in the interior of right triangle $ABC$, with coordinates $A(0,6)$, $B(9,0)$, and $C(0,0)$. What is the probability that the area of triangle $APC$ is more than a third of the area of triangle $ABC$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 2638,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point $P$ is randomly placed in the interior of the right triangle below. What is the probability that the area of triangle $PBC$ is less than half of the area of triangle $ABC$? Express your answer as a common fraction. [asy]\nsize(7cm);\ndefaultpen(linewidth(0.7));\npair ... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 2639,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point $P$ is randomly placed inside the right triangle $\\triangle XYZ$ where $X$ is at $(0,6)$, $Y$ is at $(0,0)$, and $Z$ is at $(9,0)$. What is the probability that the area of triangle $PYZ$ is less than half of the area of triangle $XYZ$?\n[asy]\nsize(7cm);\ndefaultpen(l... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 2640,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point $P$ is randomly selected from the rectangular region with vertices $(0,0), (2,0)$, $(2,1),(0,1)$. What is the probability that $P$ is closer to the origin than it is to the point $(3,1)$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 2641,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point $P$ is randomly selected from the square region with vertices at $(\\pm 2, \\pm 2)$. What is the probability that $P$ is within one unit of the origin? Express your answer as a common fraction in terms of $\\pi$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{16}"
} | {
"index": 2642,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point $P$ lies at the center of square $A B C D$. A sequence of points $\\left\\{P_{n}\\right\\}$ is determined by $P_{0}=P$, and given point $P_{i}$, point $P_{i+1}$ is obtained by reflecting $P_{i}$ over one of the four lines $A B, B C, C D, D A$, chosen uniformly at random... | math | {
"style": "rule",
"ground_truth": "\\frac{1225}{16384}"
} | {
"index": 2643,
"solution": "Solution 1. WLOG, $A B$ and $C D$ are horizontal line segments and $B C$ and $D A$ are vertical. Then observe that we can consider the reflections over vertical lines separately from those over horizontal lines, as each reflection over a vertical line moves $P_{i}$ horizontally to poin... |
DeepScaleR | [
{
"role": "user",
"content": "A point $P$ lies in the same plane as a given square of side $1$. Let the vertices of the square, taken counterclockwise, be $A, B, C$ and $D$. Also, let the distances from $P$ to $A, B$ and $C$, respectively, be $u, v$ and $w$. What is the greatest distance that $P$ can be fro... | math | {
"style": "rule",
"ground_truth": "$2 + \\sqrt{2}$"
} | {
"index": 2644,
"solution": "1. **Position the Square and Define Coordinates:**\n Place the square in the $xy$-plane such that $A$ is at the origin $(0,0)$, $B$ at $(1,0)$, $C$ at $(1,1)$, and $D$ at $(0,1)$.\n\n2. **Translate the Given Condition:**\n We are given that $u^2 + v^2 = w^2$, where $u = PA$, $v = P... |
DeepScaleR | [
{
"role": "user",
"content": "A point $P$ lies in the same plane as a given square of side $2$. Let the vertices of the square, taken counterclockwise, be $A, B, C$ and $D$. Also, let the distances from $P$ to $B, C$ and $D$, respectively, be $v, w$ and $t$. What is the greatest distance that $P$ can be fro... | math | {
"style": "rule",
"ground_truth": "\\sqrt{10}"
} | {
"index": 2645,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point $P(\\frac{\\pi}{12}, m)$ on the graph of the function $y = \\sin 2x$ can be obtained by shifting a point $Q$ on the graph of the function $y = \\cos (2x - \\frac{\\pi}{4})$ to the left by $n (n > 0)$ units. Determine the minimum value of $n \\cdot m$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5\\pi}{48}"
} | {
"index": 2646,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point $Q$ is chosen at random in the interior of equilateral triangle $DEF$. What is the probability that $\\triangle DEQ$ has a greater area than each of $\\triangle DFQ$ and $\\triangle EFQ$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 2647,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point $Q$ is chosen in the interior of $\\triangle DEF$ such that when lines are drawn through $Q$ parallel to the sides of $\\triangle DEF$, the resulting smaller triangles $t_1$, $t_2$, and $t_3$ have areas of $16$, $25$, and $36$, respectively. Find the area of $\\triangle... | math | {
"style": "rule",
"ground_truth": "225"
} | {
"index": 2648,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point $Q$ is chosen in the interior of $\\triangle DEF$ such that when lines are drawn through $Q$ parallel to the sides of $\\triangle DEF$, the resulting smaller triangles $u_{1}$, $u_{2}$, and $u_{3}$ have areas $16$, $25$, and $36$, respectively. Furthermore, a circle cen... | math | {
"style": "rule",
"ground_truth": "225"
} | {
"index": 2649,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point $Q$ is chosen in the interior of $\\triangle DEF$ such that when lines are drawn through $Q$ parallel to the sides of $\\triangle DEF$, the resulting smaller triangles $u_{1}$, $u_{2}$, and $u_{3}$ have areas $9$, $16$, and $36$, respectively. Find the area of $\\triang... | math | {
"style": "rule",
"ground_truth": "169"
} | {
"index": 2650,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point $Q$ is chosen inside $\\triangle DEF$ such that lines drawn through $Q$ parallel to $\\triangle DEF$'s sides decompose it into three smaller triangles $u_1$, $u_2$, and $u_3$, which have areas $3$, $12$, and $15$ respectively. Determine the area of $\\triangle DEF$."
... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 2651,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point $Q$ is chosen within $\\triangle DEF$ such that lines drawn through $Q$, parallel to the sides of $\\triangle DEF$, divide it into three smaller triangles with areas $9$, $16$, and $25$ respectively. Determine the area of $\\triangle DEF$."
}
] | math | {
"style": "rule",
"ground_truth": "144"
} | {
"index": 2652,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point $Q$ is randomly chosen inside an equilateral triangle $DEF$. What is the probability that the area of $\\triangle DEQ$ is greater than both $\\triangle DFQ$ and $\\triangle EFQ$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 2653,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point $Q$ is randomly placed in the interior of the right triangle $XYZ$ with $XY = 10$ units and $XZ = 6$ units. What is the probability that the area of triangle $QYZ$ is less than one-third of the area of triangle $XYZ$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 2654,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point $Q$ lies inside the triangle $\\triangle DEF$ such that lines drawn through $Q$ parallel to the sides of $\\triangle DEF$ divide it into three smaller triangles $u_1$, $u_2$, and $u_3$ with areas $16$, $25$, and $36$ respectively. Determine the area of $\\triangle DEF$.... | math | {
"style": "rule",
"ground_truth": "77"
} | {
"index": 2655,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point A is a fixed point on the circumference of a circle with a perimeter of 3. If a point B is randomly selected on the circumference, the probability that the length of the minor arc is less than 1 is ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 2656,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point M on the parabola $y=-4x^2$ is at a distance of 1 from the focus. What is the y-coordinate of point M?"
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{15}{16}"
} | {
"index": 2657,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point P is taken on the circle x²+y²=4. A vertical line segment PD is drawn from point P to the x-axis, with D being the foot of the perpendicular. As point P moves along the circle, what is the trajectory of the midpoint M of line segment PD? Also, find the focus and eccentr... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"index": 2658,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point \\( A \\) in the plane with integer coordinates is said to be visible from the origin \\( O \\) if the open segment \\( ] O A[ \\) contains no point with integer coordinates. How many such visible points are there in \\( [0,25]^{2} \\setminus \\{(0,0)\\} \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "399"
} | {
"index": 2659,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point \\( D \\) is marked on the altitude \\( BH \\) of triangle \\( ABC \\). Line \\( AD \\) intersects side \\( BC \\) at point \\( E \\), and line \\( CD \\) intersects side \\( AB \\) at point \\( F \\). It is known that \\( BH \\) divides segment \\( FE \\) in the ratio ... | math | {
"style": "rule",
"ground_truth": "1:3"
} | {
"index": 2660,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point \\( M \\) is chosen on the diameter \\( AB \\). Points \\( C \\) and \\( D \\), lying on the circumference on one side of \\( AB \\), are chosen such that \\(\\angle AMC=\\angle BMD=30^{\\circ}\\). Find the diameter of the circle given that \\( CD=12 \\)."
}
] | math | {
"style": "rule",
"ground_truth": "8\\sqrt{3}"
} | {
"index": 2661,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point \\((x, y)\\) is a distance of 14 units from the \\(x\\)-axis. It is a distance of 8 units from the point \\((1, 8)\\). Given that \\(x > 1\\), what is the distance \\(n\\) from this point to the origin?"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 2662,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point \\((x, y)\\) is randomly picked from inside the rectangle with vertices \\((0,0)\\), \\((5,0)\\), \\((5,2)\\), and \\((0,2)\\). What is the probability that \\(x < 2y\\)?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"index": 2663,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point \\((x, y)\\) is randomly selected such that \\(0 \\leq x \\leq 4\\) and \\(0 \\leq y \\leq 5\\). What is the probability that \\(x + y \\leq 5\\)? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"index": 2664,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point \\(A_{1}\\) is taken on the side \\(AC\\) of triangle \\(ABC\\), and a point \\(C_{1}\\) is taken on the extension of side \\(BC\\) beyond point \\(C\\). The length of segment \\(A_{1}C\\) is 85% of the length of side \\(AC\\), and the length of segment \\(BC_{1}\\) is ... | math | {
"style": "rule",
"ground_truth": "102"
} | {
"index": 2665,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point has rectangular coordinates $(-5,-7,4)$ and spherical coordinates $(\\rho, \\theta, \\phi).$ Find the rectangular coordinates of the point with spherical coordinates $(\\rho, \\theta, -\\phi).$"
}
] | math | {
"style": "rule",
"ground_truth": "(5,7,4)"
} | {
"index": 2666,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point has rectangular coordinates $(x,y,z)$ and spherical coordinates $\\left(2, \\frac{8 \\pi}{7}, \\frac{2 \\pi}{9} \\right).$ Find the spherical coordinates of the point with rectangular coordinates $(x,y,-z).$ Enter your answer in the form $(\\rho,\\theta,\\phi),$ where... | math | {
"style": "rule",
"ground_truth": "\\left( 2, \\frac{8 \\pi}{7}, \\frac{7 \\pi}{9} \\right)"
} | {
"index": 2667,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point in space $(x,y,z)$ is randomly selected so that $-1\\le x \\le 1$,$-1\\le y \\le 1$,$-1\\le z \\le 1$. What is the probability that $x^2+y^2+z^2\\le 1$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{6}"
} | {
"index": 2668,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point in three-space has distances $2,6,7,8,9$ from five of the vertices of a regular octahedron. What is its distance from the sixth vertex?"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{21}"
} | {
"index": 2669,
"solution": "By a simple variant of the British Flag Theorem, if $A B C D$ is a square and $P$ any point in space, $A P^{2}+C P^{2}=B P^{2}+D P^{2}$. Four of the five given vertices must form a square $A B C D$, and by experimentation we find their distances to the given point $P$ must be $A P=2, B... |
DeepScaleR | [
{
"role": "user",
"content": "A point is chosen at random from within a circular region. What is the probability that the point is closer to the center of the region than it is to the boundary of the region?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 2670,
"solution": "1. **Define the circles**: Consider a circle with radius $2$ (outer circle) and a concentric circle with radius $1$ (inner circle). The center of both circles is the same point.\n\n2. **Determine the significance of the inner circle**: The inner circle of radius $1$ consists of all poi... |
DeepScaleR | [
{
"role": "user",
"content": "A point is chosen at random on the number line between 0 and 1, and the point is colored green. Then, another point is chosen at random on the number line between 0 and 1, and this point is colored purple. What is the probability that the number of the purple point is greater t... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 2671,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point is chosen at random on the number line between 0 and 1, and the point is colored red. Then, another point is chosen at random on the number line between 0 and 1, and this point is colored blue. What is the probability that the number of the blue point is greater than th... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 2672,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point is chosen at random on the number line between 0 and 1, and this point is colored red. Another point is then chosen at random on the number line between 0 and 2, and this point is colored blue. What is the probability that the number of the blue point is greater than th... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 2673,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point is chosen at random within a rectangle in the coordinate plane whose vertices are $(0, 0), (3030, 0), (3030, 2020),$ and $(0, 2020)$. The probability that the point is within $d$ units of a lattice point is $\\tfrac{3}{4}$. Find the value of $d$ to the nearest tenth."
... | math | {
"style": "rule",
"ground_truth": "0.5"
} | {
"index": 2674,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point is chosen at random within the square in the coordinate plane whose vertices are $(0, 0), (2020, 0), (2020, 2020),$ and $(0, 2020)$. The probability that the point is within $d$ units of a lattice point is $\\frac{1}{2}$. (A point $(x, y)$ is a lattice point if $x$ and ... | math | {
"style": "rule",
"ground_truth": "0.4"
} | {
"index": 2675,
"solution": "\nWe are given a square with vertices at $(0, 0), (2020, 0), (2020, 2020),$ and $(0, 2020)$, and we need to find the radius $d$ such that the probability a randomly chosen point within the square is within $d$ units of a lattice point is $\\frac{1}{2}$. \n\n#### Step 1: Understanding t... |
DeepScaleR | [
{
"role": "user",
"content": "A point is chosen randomly from within a circular region with radius $r$. A related concentric circle with radius $\\sqrt{r}$ contains points that are closer to the center than to the boundary. Calculate the probability that a randomly chosen point lies closer to the center tha... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 2676,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point is equidistant from the coordinate axes if the vertical distance from the point to the $x$-axis is equal to the horizontal distance from the point to the $y$-axis. The point of intersection of the vertical line $x = a$ with the line with equation $3x + 8y = 24$ is equid... | math | {
"style": "rule",
"ground_truth": "-\\frac{144}{55}"
} | {
"index": 2677,
"solution": "If $a > 0$, the distance from the vertical line with equation $x = a$ to the $y$-axis is $a$. If $a < 0$, the distance from the vertical line with equation $x = a$ to the $y$-axis is $-a$. In each case, there are exactly two points on the vertical line with equation $x = a$ that are al... |
DeepScaleR | [
{
"role": "user",
"content": "A point is marked one quarter of the way along each side of a triangle. What fraction of the area of the triangle is shaded?"
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{5}{8}$"
} | {
"index": 2678,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point is randomly dropped on the interval $[5 ; 7]$ and let $k$ be the resulting value. Find the probability that the roots of the equation $\\left(k^{2}-3 k-4\\right) x^{2}+(3 k-7) x+2=0$ satisfy the condition $x_{1} \\leq 2 x_{2}$."
}
] | math | {
"style": "rule",
"ground_truth": "1/3"
} | {
"index": 2679,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point is randomly dropped onto the interval $[8, 13]$ and let $k$ be the resulting value. Find the probability that the roots of the equation $\\left(k^{2}-2k-35\\right)x^{2}+(3k-9)x+2=0$ satisfy the condition $x_{1} \\leq 2x_{2}$."
}
] | math | {
"style": "rule",
"ground_truth": "0.6"
} | {
"index": 2680,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point is randomly selected on a plane, where its Cartesian coordinates are integers with absolute values less than or equal to 4, and all such points are equally likely to be chosen. Find the probability that the selected point is at most 2 units away from the origin."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{81}"
} | {
"index": 2681,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point is randomly thrown on the segment [12, 17] and let $k$ be the resulting value. Find the probability that the roots of the equation $\\left(k^{2}+k-90\\right) x^{2}+(3 k-8) x+2=0$ satisfy the condition $x_{1} \\leq 2 x_{2}$."
}
] | math | {
"style": "rule",
"ground_truth": "2/3"
} | {
"index": 2682,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point is randomly thrown on the segment [3, 8] and let \\( k \\) be the resulting value. Find the probability that the roots of the equation \\((k^{2}-2k-3)x^{2}+(3k-5)x+2=0\\) satisfy the condition \\( x_{1} \\leq 2x_{2} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "4/15"
} | {
"index": 2683,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point is randomly thrown on the segment \\([6, 11]\\) and let \\( k \\) be the resulting value. Find the probability that the roots of the equation \\(\\left(k^{2}-2k-24\\right)x^{2}+(3k-8)x+2=0\\) satisfy the condition \\( x_{1} \\leq 2x_{2} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "2/3"
} | {
"index": 2684,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point is randomly thrown onto the interval $[6, 10]$ and let $k$ be the resulting value. Find the probability that the roots of the equation $\\left(k^{2}-3k-10\\right)x^{2}+(3k-8)x+2=0$ satisfy the condition $x_{1} \\leq 2x_{2}$."
}
] | math | {
"style": "rule",
"ground_truth": "1/3"
} | {
"index": 2685,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point is randomly thrown onto the interval \\( [11; 18] \\) and let \\( k \\) be the resulting value. Find the probability that the roots of the equation \\((k^2 + 2k - 99)x^2 + (3k - 7)x + 2 = 0\\) satisfy the condition \\( x_1 \\leq 2x_2 \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 2686,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point is randomly thrown onto the segment $[11, 18]$ and let $k$ be the resulting value. Find the probability that the roots of the equation $\\left(k^{2}+2k-99\\right)x^{2}+(3k-7)x+2=0$ satisfy the condition $x_{1} \\leq 2x_{2}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 2687,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point is randomly thrown onto the segment [3, 8] and let $k$ be the resulting value. Find the probability that the roots of the equation $\\left(k^{2}-2 k-3\\right) x^{2}+(3 k-5) x+2=0$ satisfy the condition $x_{1} \\leq 2 x_{2}$."
}
] | math | {
"style": "rule",
"ground_truth": "4/15"
} | {
"index": 2688,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point is randomly thrown onto the segment [6, 11], and let $k$ be the resulting value. Find the probability that the roots of the equation $\\left(k^{2}-2k-24\\right)x^{2}+(3k-8)x+2=0$ satisfy the condition $x_{1} \\leq 2x_{2}$."
}
] | math | {
"style": "rule",
"ground_truth": "2/3"
} | {
"index": 2689,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point is randomly thrown onto the segment [6, 11], and let \\( k \\) be the resulting value. Find the probability that the roots of the equation \\( \\left(k^{2}-2k-15\\right)x^{2}+(3k-7)x+2=0 \\) satisfy the condition \\( x_{1} \\leq 2x_{2} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "1/3"
} | {
"index": 2690,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point is selected at random from the portion of the number line shown here. What is the probability that the point is closer to 4 than to 0? Express your answer as a decimal to the nearest tenth.\n\n[asy]unitsize(0.2inch);\ndraw((0,0)--(5,0));\ndraw((0,-0.1)--(0,0.1));\ndraw(... | math | {
"style": "rule",
"ground_truth": ".6"
} | {
"index": 2691,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point moving in the positive direction of the $O x$ axis has the abscissa $x(t)=5(t+1)^{2}+\\frac{a}{(t+1)^{5}}$, where $a$ is a positive constant. Find the minimum value of $a$ such that $x(t) \\geqslant 24$ for all $t \\geqslant 0$."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{\\left( \\frac{24}{7} \\right)^7}"
} | {
"index": 2692,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point moving in the positive direction of the $OX$ axis has its horizontal coordinate given by $x(t) = 5(t + 1)^2 + \\frac{a}{(t + 1)^5}$, where $a$ is a positive constant. Find the minimum value of $a$ such that $x(t) \\geqslant 24$ for all $t \\geqslant 0$."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{\\left( \\frac{24}{7} \\right)^{7}}"
} | {
"index": 2693,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A point whose coordinates are both integers is called a lattice point. How many lattice points lie on the hyperbola $x^2 - y^2 = 2000^2$?"
}
] | math | {
"style": "rule",
"ground_truth": "98"
} | {
"index": 2694,
"solution": "\\[(x-y)(x+y)=2000^2=2^8 \\cdot 5^6\\]\n Note that $(x-y)$ and $(x+y)$ have the same parities, so both must be even. We first give a factor of $2$ to both $(x-y)$ and $(x+y)$. We have $2^6 \\cdot 5^6$ left. Since there are $7 \\cdot 7=49$ factors of $2^6 \\cdot 5^6$, and since both $x$... |
DeepScaleR | [
{
"role": "user",
"content": "A poll shows that $70\\%$ of all voters approve of the mayor's work. On three separate occasions a pollster selects a voter at random. What is the probability that on exactly one of these three occasions the voter approves of the mayor's work?"
}
] | math | {
"style": "rule",
"ground_truth": "0.189"
} | {
"index": 2695,
"solution": "1. **Define the Events:**\n Let $Y$ represent the event that a voter approves of the mayor's work, and $N$ represent the event that a voter does not approve. According to the problem, the probability that a voter approves of the mayor's work is $P(Y) = 0.7$. Consequently, the probabi... |
DeepScaleR | [
{
"role": "user",
"content": "A polygon \\(\\mathcal{P}\\) is drawn on the 2D coordinate plane. Each side of \\(\\mathcal{P}\\) is either parallel to the \\(x\\) axis or the \\(y\\) axis (the vertices of \\(\\mathcal{P}\\) do not have to be lattice points). Given that the interior of \\(\\mathcal{P}\\) incl... | math | {
"style": "rule",
"ground_truth": "8 \\sqrt{2022}"
} | {
"index": 2696,
"solution": "The minimum possible perimeter is achieved by an axis-aligned square with all four sides tangent to the circle, which has area \\(8 \\sqrt{2022}\\). To see why this is true, notice that there must be at least \\(2 \\sqrt{2022}\\) length of total perimeter facing left, \\(2 \\sqrt{2022}... |
DeepScaleR | [
{
"role": "user",
"content": "A polygon is said to be friendly if it is regular and it also has angles that, when measured in degrees, are either integers or half-integers (i.e., have a decimal part of exactly 0.5). How many different friendly polygons are there?"
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 2697,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A polyhedron has 12 faces and is such that:\n(i) all faces are isosceles triangles,\n(ii) all edges have length either \\( x \\) or \\( y \\),\n(iii) at each vertex either 3 or 6 edges meet, and\n(iv) all dihedral angles are equal.\nFind the ratio \\( x / y \\)."
}
] | math | {
"style": "rule",
"ground_truth": "3/5"
} | {
"index": 2698,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A polyhedron has faces that all either triangles or squares. No two square faces share an edge, and no two triangular faces share an edge. What is the ratio of the number of triangular faces to the number of square faces?"
}
] | math | {
"style": "rule",
"ground_truth": "4:3"
} | {
"index": 2699,
"solution": ""
} |
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