original_problem string | original_answer string | original_dag string | variation_1_node_deletion_problem string | variation_1_node_deletion_dag string | variation_2_edge_deletion_problem string | variation_2_edge_deletion_dag string |
|---|---|---|---|---|---|---|
Every morning Aya goes for a $9$-kilometer-long walk and stops at a coffee shop afterwards. When she walks at a constant speed of $s$ kilometers per hour, the walk takes her 4 hours, including $t$ minutes spent in the coffee shop. When she walks $s+2$ kilometers per hour, the walk takes her 2 hours and 24 minutes, incl... | 204 | {"nodes": [{"id": "n1", "type": "given", "value": 9, "label": "walk_distance", "description": "The distance Aya walks each morning is 9 kilometers"}, {"id": "n2", "type": "given", "value": 4, "label": "total_time_slow", "description": "Total time at speed s is 4 hours (including coffee shop time)"}, {"id": "n3", "type"... | Every morning Aya goes for a $9$-kilometer-long walk and stops at a coffee shop afterwards. When she walks at a constant speed of $s$ kilometers per hour, the walk takes her some amount of time, including $t$ minutes spent in the coffee shop. When she walks $s+2$ kilometers per hour, the walk takes her less time, but s... | {"nodes": [{"id": "n1", "type": "given", "value": 9, "label": "walk_distance", "description": "The distance Aya walks each morning is 9 kilometers"}, {"id": "n4", "type": "given", "value": 2, "label": "speed_increase_fast", "description": "The speed increase for the faster walk is 2 km/h"}, {"id": "n5", "type": "given"... | Every morning Aya goes for a $9$-kilometer-long walk and stops at a coffee shop afterwards. When she walks at a constant speed of $s$ kilometers per hour, the walk takes her a certain amount of time, including $t$ minutes spent in the coffee shop. When she walks $s+2$ kilometers per hour, the walk takes her a different... | {"nodes": [{"id": "n1", "type": "given", "value": 9, "label": "walk_distance", "description": "The distance Aya walks each morning is 9 kilometers"}, {"id": "n2", "type": "unknown", "value": null, "label": "total_time_slow", "description": "Total time at speed s is unknown (including coffee shop time)"}, {"id": "n3", "... |
Let $ABC$ be a triangle inscribed in circle $\omega$. Let the tangents to $\omega$ at $B$ and $C$ intersect at point $D$, and let $\overline{AD}$ intersect $\omega$ at $P$. If $AB=5$, $BC=9$, and $AC=10$, $AP$ can be written as the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m + n$. | 113 | {"nodes": [{"id": "n1", "type": "given", "value": "5", "label": "AB", "description": "Length of side AB of triangle ABC"}, {"id": "n2", "type": "given", "value": "9", "label": "BC", "description": "Length of side BC of triangle ABC"}, {"id": "n3", "type": "given", "value": "10", "label": "AC", "description": "Length of... | Let $ABC$ be a triangle inscribed in circle $\omega$. Let the tangents to $\omega$ at $B$ and $C$ intersect at point $D$, and let $\overline{AD}$ intersect $\omega$ at $P$. If $AP$ can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$. | {"nodes": [{"id": "n4", "type": "given", "value": null, "label": "circle \u03c9", "description": "Circumscribed circle of triangle ABC"}, {"id": "n5", "type": "constraint", "value": null, "label": "tangent_B", "description": "Tangent to circle \u03c9 at point B"}, {"id": "n6", "type": "constraint", "value": null, "labe... | Let $ABC$ be a triangle inscribed in circle $\omega$. Let the tangents to $\omega$ at $B$ and $C$ intersect at point $D$, and let $\overline{AD}$ intersect $\omega$ at $P$. Find $AP$. | {"nodes": [{"id": "n1", "type": "given", "value": "unknown", "label": "AB", "description": "Length of side AB of triangle ABC (not specified)"}, {"id": "n2", "type": "given", "value": "unknown", "label": "BC", "description": "Length of side BC of triangle ABC (not specified)"}, {"id": "n3", "type": "given", "value": "u... |
Rectangles $ABCD$ and $EFGH$ are drawn such that $D,E,C,F$ are collinear. Also, $A,D,H,G$ all lie on a circle. If $BC=16$,$AB=107$,$FG=17$, and $EF=184$, what is the length of $CE$? | 104 | {"nodes": [{"id": "n1", "type": "given", "value": "ABCD", "label": "Rectangle ABCD", "description": "Rectangle ABCD is drawn"}, {"id": "n2", "type": "given", "value": "EFGH", "label": "Rectangle EFGH", "description": "Rectangle EFGH is drawn"}, {"id": "n3", "type": "given", "value": 16, "label": "BC = 16", "description... | Rectangles $ABCD$ and $EFGH$ are drawn such that $D,E,C,F$ are collinear. Also, $A,D,H,G$ all lie on a circle. If $BC=16$, $FG=17$, and $EF=184$, what is the length of $CE$? | {"nodes": [{"id": "n1", "type": "given", "value": "ABCD", "label": "Rectangle ABCD", "description": "Rectangle ABCD is drawn"}, {"id": "n2", "type": "given", "value": "EFGH", "label": "Rectangle EFGH", "description": "Rectangle EFGH is drawn"}, {"id": "n3", "type": "given", "value": 16, "label": "BC = 16", "description... | Rectangles $ABCD$ and $EFGH$ are drawn such that $D,E,C,F$ are collinear. If $BC=16$, $AB=107$, $FG=17$, and $EF=184$, what is the length of $CE$? | {"nodes": [{"id": "n1", "type": "given", "value": "ABCD", "label": "Rectangle ABCD", "description": "Rectangle ABCD is drawn"}, {"id": "n2", "type": "given", "value": "EFGH", "label": "Rectangle EFGH", "description": "Rectangle EFGH is drawn"}, {"id": "n3", "type": "given", "value": 16, "label": "BC = 16", "description... |
Consider the paths of length $16$ that follow the lines from the lower left corner to the upper right corner on an $8\times 8$ grid. Find the number of such paths that change direction exactly four times, as in the examples shown below. | 294 | {"nodes": [{"id": "n1", "type": "given", "value": "8x8", "label": "Grid Size", "description": "The grid is 8 by 8"}, {"id": "n2", "type": "given", "value": 16, "label": "Path Length", "description": "Paths have length 16 (8 right + 8 up moves)"}, {"id": "n3", "type": "given", "value": "(0,0)", "label": "Start Point", "... | Consider the paths that follow the lines from the lower left corner to the upper right corner on a square grid. Find the number of such paths that change direction a certain number of times, as in the examples shown below. | {"nodes": [{"id": "n1", "type": "unknown", "value": null, "label": "Grid Size", "description": "The grid dimensions are not specified"}, {"id": "n2", "type": "unknown", "value": null, "label": "Path Length", "description": "Path length is not specified"}, {"id": "n3", "type": "given", "value": "lower left", "label": "S... | Consider the paths of length $16$ that follow the lines from the lower left corner to the upper right corner on an $8\times 8$ grid. Find the number of such paths that change direction some number of times. | {"nodes": [{"id": "n1", "type": "given", "value": "8x8", "label": "Grid size", "description": "The grid is 8x8"}, {"id": "n2", "type": "given", "value": 16, "label": "Path length", "description": "Paths have length 16"}, {"id": "n3", "type": "given", "value": "(0,0)", "label": "Start point", "description": "Paths start... |
Find the largest possible real part of \[(75+117i)z+\frac{96+144i}{z}\]where $z$ is a complex number with $|z|=4$. | 540 | {"nodes": [{"id": "n1", "type": "given", "value": "75+117i", "label": "First coefficient", "description": "The complex coefficient (75+117i) in the expression"}, {"id": "n2", "type": "given", "value": "96+144i", "label": "Second coefficient", "description": "The complex coefficient (96+144i) in the expression"}, {"id":... | Find the largest possible real part of \[(75+117i)z+\frac{96+144i}{z}\]where $z$ is a complex number. | {"nodes": [{"id": "n1", "type": "given", "value": "75+117i", "label": "First coefficient", "description": "The complex coefficient (75+117i) in the expression"}, {"id": "n2", "type": "given", "value": "96+144i", "label": "Second coefficient", "description": "The complex coefficient (96+144i) in the expression"}, {"id":... | Find the largest possible real part of \[\alpha z+\frac{\beta}{z}\]where $z$ is a complex number with $|z|=4$, and $\alpha$ and $\beta$ are complex numbers. | {"nodes": [{"id": "n1", "type": "unknown", "value": null, "label": "First coefficient", "description": "An unspecified complex number \u03b1 in the expression"}, {"id": "n2", "type": "unknown", "value": null, "label": "Second coefficient", "description": "An unspecified complex number \u03b2 in the expression"}, {"id":... |
Eight circles of radius $34$ are sequentially tangent, and two of the circles are tangent to $AB$ and $BC$ of triangle $ABC$, respectively. $2024$ circles of radius $1$ can be arranged in the same manner. The inradius of triangle $ABC$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive i... | 197 | {"nodes": [{"id": "n1", "type": "given", "value": "8", "label": "num_circles_1", "description": "Eight circles in the first configuration"}, {"id": "n2", "type": "given", "value": "34", "label": "radius_1", "description": "Radius of each of the eight circles"}, {"id": "n3", "type": "given", "value": "2024", "label": "n... | Eight circles are sequentially tangent, and two of the circles are tangent to $AB$ and $BC$ of triangle $ABC$, respectively. $2024$ circles of radius $1$ can be arranged in the same manner. The inradius of triangle $ABC$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $... | {"nodes": [{"id": "n1", "type": "given", "value": "8", "label": "num_circles_1", "description": "Eight circles in the first configuration"}, {"id": "n3", "type": "given", "value": "2024", "label": "num_circles_2", "description": "2024 circles in the second configuration"}, {"id": "n4", "type": "given", "value": "1", "l... | Several circles of radius $34$ are sequentially tangent, and two of the circles are tangent to $AB$ and $BC$ of triangle $ABC$, respectively. Many circles of radius $1$ can be arranged in the same manner. The inradius of triangle $ABC$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive i... | {"nodes": [{"id": "n1", "type": "given", "value": "unknown", "label": "num_circles_1", "description": "Several circles in the first configuration (count not specified)"}, {"id": "n2", "type": "given", "value": "34", "label": "radius_1", "description": "Radius of each circle in the first configuration"}, {"id": "n3", "t... |
Let $A$, $B$, $C$, and $D$ be point on the hyperbola $\frac{x^2}{20}- \frac{y^2}{24} = 1$ such that $ABCD$ is a rhombus whose diagonals intersect at the origin. Find the greatest real number that is less than $BD^2$ for all such rhombi. | 480 | {"nodes": [{"id": "n1", "type": "given", "value": "x^2/20 - y^2/24 = 1", "label": "hyperbola_equation", "description": "The hyperbola on which points A, B, C, D lie"}, {"id": "n2", "type": "given", "value": "20", "label": "a_squared", "description": "The denominator under x^2 in the hyperbola equation"}, {"id": "n3", "... | Let $A$, $B$, $C$, and $D$ be points on a hyperbola $\frac{x^2}{a^2}- \frac{y^2}{b^2} = 1$ such that $ABCD$ is a rhombus whose diagonals intersect at the origin. Find the greatest real number that is less than $BD^2$ for all such rhombi. | {"nodes": [{"id": "n1", "type": "given", "value": "x^2/a^2 - y^2/b^2 = 1", "label": "hyperbola_equation", "description": "A hyperbola on which points A, B, C, D lie (parameters unspecified)"}, {"id": "n4", "type": "given", "value": "ABCD", "label": "quadrilateral", "description": "Four points forming a quadrilateral"},... | Let $A$, $B$, $C$, and $D$ be points on the hyperbola $\frac{x^2}{p}- \frac{y^2}{q} = 1$, where $p$ and $q$ are positive constants, such that $ABCD$ is a rhombus whose diagonals intersect at the origin. Find the greatest real number that is less than $BD^2$ for all such rhombi. | {"nodes": [{"id": "n1", "type": "given", "value": "x^2/p - y^2/q = 1", "label": "hyperbola_equation", "description": "The hyperbola on which points A, B, C, D lie, with unspecified parameters"}, {"id": "n2", "type": "given", "value": "unknown", "label": "p_parameter", "description": "The denominator under x^2 in the hy... |
Among the 900 residents of Aimeville, there are 195 who own a diamond ring, 367 who own a set of golf clubs, and 562 who own a garden spade. In addition, each of the 900 residents owns a bag of candy hearts. There are 437 residents who own exactly two of these things, and 234 residents who own exactly three of these th... | 73 | {"nodes": [{"id": "n1", "type": "given", "value": 900, "label": "total_residents", "description": "Total number of residents in Aimeville"}, {"id": "n2", "type": "given", "value": 195, "label": "diamond_ring_owners", "description": "Number of residents who own a diamond ring"}, {"id": "n3", "type": "given", "value": 36... | Among the 900 residents of Aimeville, there are 195 who own a diamond ring, 367 who own a set of golf clubs, and 562 who own a garden spade. In addition, each of the 900 residents owns a bag of candy hearts. There are 234 residents who own exactly three of these things. Find the number of residents of Aimeville who own... | {"nodes": [{"id": "n1", "type": "given", "value": 900, "label": "total_residents", "description": "Total number of residents in Aimeville"}, {"id": "n2", "type": "given", "value": 195, "label": "diamond_ring_owners", "description": "Number of residents who own a diamond ring"}, {"id": "n3", "type": "given", "value": 36... | Among the 900 residents of Aimeville, there are 195 who own a diamond ring, 367 who own a set of golf clubs, and 562 who own a garden spade. In addition, each of the 900 residents owns a bag of candy hearts. Some residents own exactly two of these things, and some residents own exactly three of these things. Find the n... | {"nodes": [{"id": "n1", "type": "given", "value": 900, "label": "total_residents", "description": "Total number of residents in Aimeville"}, {"id": "n2", "type": "given", "value": 195, "label": "diamond_ring_owners", "description": "Number of residents who own a diamond ring"}, {"id": "n3", "type": "given", "value": 36... |
Let $\triangle ABC$ have circumcenter $O$ and incenter $I$ with $\overline{IA}\perp\overline{OI}$, circumradius $13$, and inradius $6$. Find $AB\cdot AC$. | 468 | {"nodes": [{"id": "n1", "type": "given", "value": "13", "label": "Circumradius R", "description": "The circumradius of triangle ABC is 13"}, {"id": "n2", "type": "given", "value": "6", "label": "Inradius r", "description": "The inradius of triangle ABC is 6"}, {"id": "n3", "type": "given", "value": null, "label": "Tria... | Let $\triangle ABC$ have circumcenter $O$ and incenter $I$ with $\overline{IA}\perp\overline{OI}$ and inradius $6$. Find $AB\cdot AC$. | {"nodes": [{"id": "n2", "type": "given", "value": "6", "label": "Inradius r", "description": "The inradius of triangle ABC is 6"}, {"id": "n3", "type": "given", "value": null, "label": "Triangle ABC", "description": "A triangle ABC is given"}, {"id": "n4", "type": "given", "value": null, "label": "Circumcenter O", "des... | Let $\triangle ABC$ have circumcenter $O$ and incenter $I$ with circumradius $13$ and inradius $6$. Find $AB\cdot AC$. | {"nodes": [{"id": "n1", "type": "given", "value": "13", "label": "Circumradius R", "description": "The circumradius of triangle ABC is 13"}, {"id": "n2", "type": "given", "value": "6", "label": "Inradius r", "description": "The inradius of triangle ABC is 6"}, {"id": "n3", "type": "given", "value": null, "label": "Tria... |
Find the number of triples of nonnegative integers \((a,b,c)\) satisfying \(a + b + c = 300\) and
\begin{equation*}
a^2b + a^2c + b^2a + b^2c + c^2a + c^2b = 6,000,000.
\end{equation*} | 601 | {"nodes": [{"id": "n1", "type": "given", "value": "nonnegative integers", "label": "Domain of (a,b,c)", "description": "a, b, c are nonnegative integers"}, {"id": "n2", "type": "given", "value": 300, "label": "Sum constraint value", "description": "The target sum value is 300"}, {"id": "n3", "type": "given", "value": 6... | Find the number of triples of nonnegative integers \((a,b,c)\) satisfying \(a + b + c = 300\) and
\begin{equation*}
a^2b + a^2c + b^2a + b^2c + c^2a + c^2b
\end{equation*}
is equal to some value. | {"nodes": [{"id": "n1", "type": "given", "value": "nonnegative integers", "label": "Domain of (a,b,c)", "description": "a, b, c are nonnegative integers"}, {"id": "n2", "type": "given", "value": 300, "label": "Sum constraint value", "description": "The target sum value is 300"}, {"id": "n4", "type": "constraint", "valu... | Find the number of triples of nonnegative integers $(a,b,c)$ satisfying $a + b + c = 300$ and
\begin{equation*}
a^2b + a^2c + b^2a + b^2c + c^2a + c^2b = k
\end{equation*}
where $k$ is some positive integer. | {"nodes": [{"id": "n1", "type": "given", "value": "nonnegative integers", "label": "Domain of (a,b,c)", "description": "a, b, c are nonnegative integers"}, {"id": "n2", "type": "given", "value": 300, "label": "Sum constraint value", "description": "The target sum value is 300"}, {"id": "n3", "type": "given", "value": "... |
Let \(O=(0,0)\), \(A=\left(\tfrac{1}{2},0\right)\), and \(B=\left(0,\tfrac{\sqrt{3}}{2}\right)\) be points in the coordinate plane. Let \(\mathcal{F}\) be the family of segments \(\overline{PQ}\) of unit length lying in the first quadrant with \(P\) on the \(x\)-axis and \(Q\) on the \(y\)-axis. There is a unique point... | 23 | {"nodes": [{"id": "n1", "type": "given", "value": "(0,0)", "label": "Point O", "description": "Origin point O at coordinates (0,0)"}, {"id": "n2", "type": "given", "value": "(1/2, 0)", "label": "Point A", "description": "Point A at (1/2, 0) on the x-axis"}, {"id": "n3", "type": "given", "value": "(0, sqrt(3)/2)", "labe... | Let \(O=(0,0)\) be a point in the coordinate plane. Let \(A\) be a point on the \(x\)-axis and \(B\) be a point on the \(y\)-axis, both in the first quadrant, such that \(\overline{AB}\) has unit length. Let \(\mathcal{F}\) be the family of segments \(\overline{PQ}\) of unit length lying in the first quadrant with \(P\... | {"nodes": [{"id": "n1", "type": "given", "value": "(0,0)", "label": "Point O", "description": "Origin point O at coordinates (0,0)"}, {"id": "n2", "type": "given", "value": "unknown", "label": "Point A", "description": "Point A on the x-axis (specific coordinates not given)"}, {"id": "n3", "type": "given", "value": "un... | Let \(O=(0,0)\), \(A\), and \(B\) be points in the coordinate plane, where \(A\) lies on the positive \(x\)-axis and \(B\) lies on the positive \(y\)-axis. Let \(\mathcal{F}\) be the family of segments \(\overline{PQ}\) lying in the first quadrant with \(P\) on the \(x\)-axis and \(Q\) on the \(y\)-axis. There is a uni... | {"nodes": [{"id": "n1", "type": "given", "value": "(0,0)", "label": "Point O", "description": "Origin point O at coordinates (0,0)"}, {"id": "n2", "type": "given", "value": "unknown", "label": "Point A", "description": "Point A at unspecified coordinates on the x-axis"}, {"id": "n3", "type": "given", "value": "unknown"... |
Let $\omega\neq 1$ be a 13th root of unity. Find the remainder when
\[\prod_{k=0}^{12}(2-2\omega^k+\omega^{2k})\]
is divided by 1000. | 321 | {"nodes": [{"id": "n1", "type": "given", "value": "13", "label": "root_order", "description": "\u03c9 is a 13th root of unity"}, {"id": "n2", "type": "given", "value": "\u03c9 \u2260 1", "label": "omega_constraint", "description": "\u03c9 is not equal to 1 (primitive root)"}, {"id": "n3", "type": "given", "value": "\u2... | Let $\omega\neq 1$ be a 13th root of unity. Find the remainder when
\[\prod_{k=0}^{12}(2-\omega^k+\omega^{2k})\]
is divided by 1000. | {"nodes": [{"id": "n1", "type": "given", "value": "omega^13 = 1, omega != 1", "label": "13th root of unity", "description": "omega is a primitive 13th root of unity (not equal to 1)"}, {"id": "n2", "type": "given", "value": "k = 0, 1, 2, ..., 12", "label": "Product index range", "description": "The product runs from k=... | Let $\omega\neq 1$ be a 13th root of unity. Find the remainder when
\[\prod_{k=0}^{12}(a-b\omega^k+c\omega^{2k})\]
is divided by 1000, where $a$, $b$, and $c$ are certain constants. | {"nodes": [{"id": "n1", "type": "given", "value": "13", "label": "root_order", "description": "\u03c9 is a 13th root of unity"}, {"id": "n2", "type": "given", "value": "\u03c9 \u2260 1", "label": "omega_constraint", "description": "\u03c9 is not equal to 1 (primitive root)"}, {"id": "n3", "type": "given", "value": "\u2... |
Each vertex of a regular octagon is independently colored either red or blue with equal probability. The probability that the octagon can then be rotated so that all of the blue vertices end up at positions where there were originally red vertices is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integ... | 371 | {"nodes": [{"id": "n1", "type": "given", "value": "8", "label": "Number of vertices", "description": "A regular octagon has 8 vertices"}, {"id": "n2", "type": "given", "value": "red or blue", "label": "Coloring options", "description": "Each vertex is colored either red or blue"}, {"id": "n3", "type": "given", "value":... | Each vertex of a regular octagon is independently colored either red or blue with equal probability. The probability that the octagon can then be rotated so that a certain condition is satisfied is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? | {"nodes": [{"id": "n1", "type": "given", "value": "8", "label": "Number of vertices", "description": "A regular octagon has 8 vertices"}, {"id": "n2", "type": "given", "value": "red or blue", "label": "Coloring options", "description": "Each vertex is colored either red or blue"}, {"id": "n3", "type": "given", "value":... | Each vertex of a regular octagon is independently colored either red or blue with equal probability. The probability that the octagon can then be rotated so that some specific arrangement condition is satisfied is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? | {"nodes": [{"id": "n1", "type": "given", "value": "8", "label": "Number of vertices", "description": "A regular octagon has 8 vertices"}, {"id": "n2", "type": "given", "value": "red or blue", "label": "Coloring options", "description": "Each vertex is colored either red or blue"}, {"id": "n3", "type": "given", "value":... |
Let \(b\ge 2\) be an integer. Call a positive integer \(n\) \(b\text-\textit{eautiful}\) if it has exactly two digits when expressed in base \(b\) and these two digits sum to \(\sqrt n\). For example, \(81\) is \(13\text-\textit{eautiful}\) because \(81 = \underline{6} \ \underline{3}_{13} \) and \(6 + 3 = \sqrt{81}... | 211 | {"nodes": [{"id": "n1", "type": "given", "value": "b >= 2", "label": "base_constraint", "description": "b is an integer with b >= 2"}, {"id": "n2", "type": "given", "value": "exactly two digits", "label": "digit_count", "description": "n has exactly two digits when expressed in base b"}, {"id": "n3", "type": "given", "... | Let \(b\ge 2\) be an integer. Call a positive integer \(n\) \(b\text-\textit{eautiful}\) if it has exactly two digits when expressed in base \(b\) and these two digits satisfy a certain special property. Find the least integer \(b\ge 2\) for which there are more than ten \(b\text-\textit{eautiful}\) integers. | {"nodes": [{"id": "n1", "type": "given", "value": "b >= 2", "label": "base_constraint", "description": "b is an integer with b >= 2"}, {"id": "n2", "type": "given", "value": "exactly two digits", "label": "digit_count", "description": "n has exactly two digits when expressed in base b"}, {"id": "n4", "type": "constrain... | Let \(b\ge 2\) be an integer. Call a positive integer \(n\) \(b\text-\textit{eautiful}\) if it has exactly two digits when expressed in base \(b\) and these two digits satisfy a special property related to \(n\). Find the least integer \(b\ge 2\) for which there are more than ten \(b\text-\textit{eautiful}\) integers. | {"nodes": [{"id": "n1", "type": "given", "value": "b >= 2", "label": "base_constraint", "description": "b is an integer with b >= 2"}, {"id": "n2", "type": "given", "value": "exactly two digits", "label": "digit_count", "description": "n has exactly two digits when expressed in base b"}, {"id": "n3", "type": "given", "... |
Find the number of rectangles that can be formed inside a fixed regular dodecagon ($12$-gon) where each side of the rectangle lies on either a side or a diagonal of the dodecagon. The diagram below shows three of those rectangles.
[asy] unitsize(0.6 inch); for(int i=0; i<360; i+=30) { dot(dir(i), 4+black); draw(dir(i)-... | 315 | {"nodes": [{"id": "n1", "type": "given", "value": "12", "label": "Regular dodecagon", "description": "A fixed regular 12-sided polygon"}, {"id": "n2", "type": "constraint", "value": null, "label": "Rectangle side constraint", "description": "Each side of the rectangle must lie on either a side or a diagonal of the dode... | Find the number of rectangles that can be formed inside a fixed regular polygon where each side of the rectangle lies on either a side or a diagonal of the polygon. | {"nodes": [{"id": "n1", "type": "given", "value": "unknown", "label": "Regular polygon", "description": "A fixed regular polygon with unspecified number of sides"}, {"id": "n2", "type": "constraint", "value": null, "label": "Rectangle side constraint", "description": "Each side of the rectangle must lie on either a sid... | Find the number of quadrilaterals that can be formed inside a fixed regular dodecagon ($12$-gon) where each side of the quadrilateral lies on either a side or a diagonal of the dodecagon. | {"nodes": [{"id": "n1", "type": "given", "value": 12, "label": "Regular 12-gon", "description": "A fixed regular dodecagon (12-sided polygon)"}, {"id": "n2", "type": "constraint", "value": null, "label": "Quadrilateral sides constraint", "description": "Each side of the quadrilateral must lie on either a side or a diag... |
A list of positive integers has the following properties:
$\bullet$ The sum of the items in the list is $30$.
$\bullet$ The unique mode of the list is $9$.
$\bullet$ The median of the list is a positive integer that does not appear in the list itself.
Find the sum of the squares of all the items in the list. | 236 | {"nodes": [{"id": "n1", "type": "given", "value": 30, "label": "sum_of_list", "description": "The sum of all items in the list equals 30"}, {"id": "n2", "type": "given", "value": 9, "label": "unique_mode", "description": "The unique mode of the list is 9"}, {"id": "n3", "type": "constraint", "value": null, "label": "me... | A list of positive integers has the following properties:
$\bullet$ The unique mode of the list is $9$.
$\bullet$ The median of the list is a positive integer that does not appear in the list itself.
Find the sum of the squares of all the items in the list. | {"nodes": [{"id": "n2", "type": "given", "value": 9, "label": "unique_mode", "description": "The unique mode of the list is 9"}, {"id": "n3", "type": "constraint", "value": null, "label": "median_constraint", "description": "The median is a positive integer that does not appear in the list"}, {"id": "n4", "type": "cons... | A list of positive integers has the following properties:
$\bullet$ The sum of the items in the list is a certain value S.
$\bullet$ The unique mode of the list is a certain value M.
$\bullet$ The median of the list is a positive integer that does not appear in the list itself.
Find the sum of the squares of all the it... | {"nodes": [{"id": "n1", "type": "unknown", "value": null, "label": "sum_of_list", "description": "The sum of all items in the list equals some unknown value S"}, {"id": "n2", "type": "unknown", "value": null, "label": "unique_mode", "description": "The unique mode of the list is some unknown value M"}, {"id": "n3", "ty... |
Find the number of ways to place a digit in each cell of a 2x3 grid so that the sum of the two numbers formed by reading left to right is $999$, and the sum of the three numbers formed by reading top to bottom is $99$. The grid below is an example of such an arrangement because $8+991=999$ and $9+9+81=99$.
\[\begin{arr... | 45 | {"nodes": [{"id": "n1", "type": "given", "value": "2x3", "label": "Grid dimensions", "description": "A 2x3 grid where each cell contains a single digit"}, {"id": "n2", "type": "given", "value": "[0-9]", "label": "Digit range", "description": "Each cell contains a digit from 0 to 9"}, {"id": "n3", "type": "given", "valu... | Find the number of ways to place a digit in each cell of a 2x3 grid so that the sum of the two numbers formed by reading left to right equals some specific value, and the sum of the three numbers formed by reading top to bottom equals some specific value. | {"nodes": [{"id": "n1", "type": "given", "value": "2x3", "label": "Grid dimensions", "description": "A 2x3 grid where each cell contains a single digit"}, {"id": "n2", "type": "given", "value": "[0-9]", "label": "Digit range", "description": "Each cell contains a digit from 0 to 9"}, {"id": "n5", "type": "constraint", ... | Find the number of ways to place a digit in each cell of a 2x3 grid so that the sum of the two numbers formed by reading left to right is $R$, and the sum of the three numbers formed by reading top to bottom is $C$.
\[\begin{array}{|c|c|c|} \hline & & \\ \hline & & \\ \hline \end{array}\] | {"nodes": [{"id": "n1", "type": "given", "value": "2x3", "label": "Grid dimensions", "description": "A 2x3 grid where each cell contains a single digit"}, {"id": "n2", "type": "given", "value": "[0-9]", "label": "Digit range", "description": "Each cell contains a digit from 0 to 9"}, {"id": "n3", "type": "given", "valu... |
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]\[\log_2\left({y \over xz}\right) = {1 \over 3}\]\[\log_2\left({z \over xy}\right) = {1 \over 4}\]
Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is $\tfrac{m}{n}$ where $m$ an... | 33 | {"nodes": [{"id": "n1", "type": "given", "value": "1/2", "label": "log_2(x/yz)", "description": "First equation: log_2(x/yz) = 1/2"}, {"id": "n2", "type": "given", "value": "1/3", "label": "log_2(y/xz)", "description": "Second equation: log_2(y/xz) = 1/3"}, {"id": "n3", "type": "given", "value": "1/4", "label": "log_2(... | Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = a\]\[\log_2\left({y \over xz}\right) = b\]\[\log_2\left({z \over xy}\right) = c\]
where $a$, $b$, and $c$ are real constants. Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is $\tfrac{m}{n}... | {"nodes": [{"id": "n1", "type": "given", "value": "unknown", "label": "log_2(x/yz) = ?", "description": "First equation: log_2(x/yz) = some unknown value"}, {"id": "n2", "type": "given", "value": "unknown", "label": "log_2(y/xz) = ?", "description": "Second equation: log_2(y/xz) = some unknown value"}, {"id": "n3", "ty... | Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = p\]\[\log_2\left({y \over xz}\right) = q\]\[\log_2\left({z \over xy}\right) = r\]
where $p$, $q$, and $r$ are certain values.
Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is $\tfrac{m}{n}... | {"nodes": [{"id": "n1", "type": "given", "value": "unknown_p", "label": "log_2(x/yz)", "description": "First equation: log_2(x/yz) = p (value unknown)"}, {"id": "n2", "type": "given", "value": "unknown_q", "label": "log_2(y/xz)", "description": "Second equation: log_2(y/xz) = q (value unknown)"}, {"id": "n3", "type": "... |
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