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 walk is 9 kilometers long"}, {"id": "n2", "type": "given", "value": 4, "label": "total_time_slow", "description": "Total time at speed s is 4 hours (including coffee shop)"}, {"id": "n3", "type": "given", "value": 2.4, "l... | Every morning Aya goes for a walk of a certain distance D kilometers and stops at a coffee shop afterwards. When she walks at a constant speed of s kilometers per hour, the walk takes her H1 hours, including t minutes spent in the coffee shop. When she walks s+2 kilometers per hour, the walk takes her H2 hours, includi... | {"nodes": [{"id": "n1", "type": "unknown", "value": null, "label": "walk_distance", "description": "The walk is D kilometers long (D is unspecified)"}, {"id": "n2", "type": "unknown", "value": null, "label": "total_time_slow", "description": "Total time at speed s is H1 hours (H1 is unspecified)"}, {"id": "n3", "type":... | Every morning Aya goes for a 9-kilometer-long walk and stops at a coffee shop afterwards. When she walks at her usual constant speed, the walk takes her 4 hours, including some minutes spent in the coffee shop. When she walks at a faster constant speed, the walk takes her 2 hours and 24 minutes, including the same amou... | {"nodes": [{"id": "n1", "type": "given", "value": 9, "label": "walk_distance", "description": "The walk is 9 kilometers long"}, {"id": "n2", "type": "given", "value": 4, "label": "total_time_slow", "description": "Total time at her usual speed is 4 hours (including coffee shop)"}, {"id": "n3", "type": "given", "value":... |
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 $AB=p$, $BC=9$, and $AC=q$, $AP$ can be written as the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m + n$. | {"nodes": [{"id": "n1", "type": "given", "value": "p (undefined)", "label": "AB", "description": "Length of side AB of triangle ABC, where p is an unspecified positive real number"}, {"id": "n2", "type": "given", "value": "9", "label": "BC", "description": "Length of side BC of triangle ABC"}, {"id": "n3", "type": "giv... | Let $ABC$ be a triangle inscribed in circle $\omega$. Let $D$ be a point in the plane, and let line $\overline{AD}$ intersect $\omega$ at $P$. If $AB=5$, $BC=9$, and $AC=10$, $AP$ can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m + n$. | {"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... |
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 with vertices A, B, C, D"}, {"id": "n2", "type": "given", "value": "EFGH", "label": "Rectangle EFGH", "description": "Rectangle with vertices E, F, G, H"}, {"id": "n3", "type": "given", "value": "collinear", "... | Rectangles $ABCD$ and $EFGH$ are drawn such that $D,E,C,F$ are collinear. The rectangles are positioned under certain geometric conditions. If $AB=107$, $FG=17$, and $EF=184$, what is the length of $CE$? | {"nodes": [{"id": "n1", "type": "given", "value": "ABCD", "label": "Rectangle ABCD", "description": "Rectangle with vertices A, B, C, D"}, {"id": "n2", "type": "given", "value": "EFGH", "label": "Rectangle EFGH", "description": "Rectangle with vertices E, F, G, H"}, {"id": "n3", "type": "given", "value": "collinear", "... | 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 with vertices A, B, C, D"}, {"id": "n2", "type": "given", "value": "EFGH", "label": "Rectangle EFGH", "description": "Rectangle with vertices E, F, G, H"}, {"id": "n3", "type": "given", "value": "collinear", "... |
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 8x8"}, {"id": "n2", "type": "given", "value": 16, "label": "path_length", "description": "Total path length is 16 steps"}, {"id": "n3", "type": "given", "value": "(0,0)", "label": "start_point", "description": "Pa... | Consider lattice paths that follow grid lines from the lower left corner to the upper right corner on a rectangular grid. The paths must change direction some fixed number of times. Find the number of such valid paths. | {"nodes": [{"id": "n1", "type": "given", "value": "p\u00d7q", "label": "grid_size", "description": "The grid is p\u00d7q where p and q are unspecified positive integers"}, {"id": "n2", "type": "given", "value": "L", "label": "path_length", "description": "Total path length is L steps (unspecified)"}, {"id": "n3", "type... | Consider the paths of length 16 that follow the lines from the lower left corner to the upper right corner on an 8×8 grid. Find the number of such paths that change direction multiple 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": "Total path length is 16 steps"}, {"id": "n3", "type": "given", "value": "(0,0)", "label": "start_point", "description": "Pa... |
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": "coefficient_1", "description": "First complex coefficient in the expression"}, {"id": "n2", "type": "given", "value": "96+144i", "label": "coefficient_2", "description": "Second complex coefficient in the expression"}, {"id": "n3", "type": "given", ... | Find the largest possible real part of \[(75+117i)z+\frac{w}{z}\]where $z$ is a complex number with $|z|=r$, $w$ is a complex constant, and $r$ is a positive real number. | {"nodes": [{"id": "n1", "type": "given", "value": "75+117i", "label": "coefficient_1", "description": "First complex coefficient in the expression"}, {"id": "n2", "type": "unknown", "value": "w", "label": "coefficient_2", "description": "Second complex coefficient in the expression (undefined)"}, {"id": "n3", "type": "... | 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": "coefficient_1", "description": "First complex coefficient \u03b1 (unspecified)"}, {"id": "n2", "type": "unknown", "value": null, "label": "coefficient_2", "description": "Second complex coefficient \u03b2 (unspecified)"}, {"id": "n3", "type": "given", ... |
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": "First configuration has 8 circles"}, {"id": "n2", "type": "given", "value": "34", "label": "radius_1", "description": "Radius of circles in first configuration"}, {"id": "n3", "type": "given", "value": "2024", "label": "num... | Some circles of radius β are sequentially tangent, and two of the circles are tangent to AB and BC of triangle ABC, respectively. There are α such circles in this configuration. Another configuration of γ circles of radius δ can be arranged in the same manner within the same triangle. The inradius of triangle ABC can b... | {"nodes": [{"id": "n1", "type": "given", "value": "\u03b1 (undefined)", "label": "num_circles_1", "description": "First configuration has \u03b1 circles (\u03b1 is unspecified)"}, {"id": "n2", "type": "given", "value": "\u03b2 (undefined)", "label": "radius_1", "description": "Radius of circles in first configuration i... | 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 within the same triangle. The inradius of triangle $ABC$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are re... | {"nodes": [{"id": "n1", "type": "unknown", "value": null, "label": "num_circles_1", "description": "First configuration has an unspecified number of circles"}, {"id": "n2", "type": "given", "value": "34", "label": "radius_1", "description": "Radius of circles in first configuration"}, {"id": "n3", "type": "unknown", "v... |
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 value a^2 in the hyperbola equation"}, {"id": "n3", "type": "given"... | 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 undefined parameters p and q"}, {"id": "n2", "type": "unknown", "value": "p", "label": "a_squared", "description": "The value a^2 in the hyperbola equa... | Let $A$, $B$, $C$, and $D$ be points on the hyperbola $\frac{x^2}{20}- \frac{y^2}{24} = 1$ such that $ABCD$ is a quadrilateral whose diagonals intersect at the origin. Find the greatest real number that is less than $BD^2$ for all such quadrilaterals. | {"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 value a^2 in the hyperbola equation"}, {"id": "n3", "type": "given"... |
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 P residents who own exactly two of these things, and Q residents who own exactly three of these things... | {"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, residents may own multiple items from different categories. Some items are owned by everyone in town, while other items are owned by only some residents. There are 437 residents who own exactly two items in total, and 234 residents who own exactly three items in total. Find the num... | {"nodes": [{"id": "n1", "type": "given", "value": 900, "label": "total_residents", "description": "Total number of residents in Aimeville"}, {"id": "n2", "type": "unknown", "value": null, "label": "item_category_counts", "description": "Number of items in each category (unspecified)"}, {"id": "n6", "type": "given", "va... |
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": "circ... | Let $\triangle ABC$ have circumcenter $O$ and incenter $I$ with $\overline{IA}\perp\overline{OI}$, circumradius $P$, and inradius $Q$. Find $AB\cdot AC$. | {"nodes": [{"id": "n1", "type": "given", "value": "P (undefined)", "label": "circumradius P", "description": "The circumradius of triangle ABC is some value P (not specified)"}, {"id": "n2", "type": "given", "value": "Q (undefined)", "label": "inradius Q", "description": "The inradius of triangle ABC is some value Q (n... | Let △ABC have circumcenter O and incenter I with IA⊥OI. The triangle has some circumradius and some inradius, both unspecified positive values. Find AB·AC. | {"nodes": [{"id": "n1", "type": "given", "value": "unspecified", "label": "circumradius R", "description": "The circumradius of triangle ABC exists but its value is not given"}, {"id": "n2", "type": "given", "value": "unspecified", "label": "inradius r", "description": "The inradius of triangle ABC exists but its value... |
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 constant 300 in the sum equation"}, {"id": "n3", "type": "given",... | Find the number of triples of nonnegative integers $(a,b,c)$ satisfying $a + b + c = S$ and
\begin{equation*}
a^2b + a^2c + b^2a + b^2c + c^2a + c^2b = T
\end{equation*}
where $S$ and $T$ are positive constants. | {"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": "S (undefined)", "label": "Sum constraint value", "description": "An undefined positive constant S in the sum equation"}, {"i... | Find the number of triples of nonnegative integers (a,b,c) satisfying a + b + c = 300 and such that the expression a²b + a²c + b²a + b²c + c²a + c²b satisfies a certain condition. | {"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 constant 300 in the sum equation"}, {"id": "n3", "type": "given",... |
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)", "la... | Let \(O=(0,0)\), \(A=(\alpha,0)\), and \(B=(0,\beta)\) be points in the coordinate plane, where \(\alpha\) and \(\beta\) are positive real numbers. 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 i... | {"nodes": [{"id": "n1", "type": "given", "value": "(0, 0)", "label": "Point O", "description": "Origin point O at coordinates (0, 0)"}, {"id": "n2", "type": "given", "value": "(\u03b1, 0)", "label": "Point A", "description": "Point A at (\u03b1, 0) on the x-axis where \u03b1 is unspecified"}, {"id": "n3", "type": "give... | Let \(O=(0,0)\) be the origin in the coordinate plane. Let \(A\) be a point on the positive \(x\)-axis and \(B\) be a point on the positive \(y\)-axis. Let \(\mathcal{F}\) be the family of segments \(\overline{PQ}\) of some fixed length lying in the first quadrant with \(P\) on the \(x\)-axis and \(Q\) on the \(y\)-axi... | {"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 on x-axis", "label": "Point A", "description": "Point A somewhere on the positive x-axis (coordinates not specified)"}, {"id": "n3", "ty... |
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": "omega is a 13th root of unity"}, {"id": "n2", "type": "given", "value": "omega != 1", "label": "omega_constraint", "description": "omega is not equal to 1"}, {"id": "n3", "type": "given", "value": "omega^13 = 1", "label": "ro... | Let $\omega\neq 1$ be an Nth root of unity. Find the remainder when
\[\prod_{k=0}^{N-1}(2-2\omega^k+\omega^{2k})\]
is divided by 1000. | {"nodes": [{"id": "n1", "type": "given", "value": "N (undefined)", "label": "root_order", "description": "omega is an Nth root of unity, where N is unspecified"}, {"id": "n2", "type": "given", "value": "omega != 1", "label": "omega_constraint", "description": "omega is not equal to 1"}, {"id": "n3", "type": "given", "v... | Let $\omega\neq 1$ be a 13th root of unity. Find the remainder when
\[\prod_{k=0}^{12} f(\omega^k)\]
is divided by 1000, where $f$ is an expression involving $\omega^k$. | {"nodes": [{"id": "n1", "type": "given", "value": "13", "label": "root_order", "description": "omega is a 13th root of unity"}, {"id": "n2", "type": "given", "value": "omega != 1", "label": "omega_constraint", "description": "omega is not equal to 1"}, {"id": "n3", "type": "given", "value": "omega^13 = 1", "label": "ro... |
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": "color_options", "description": "Each vertex is colored either red or blue"}, {"id": "n3", "type": "given", "value": "0... | Each vertex of a regular convex polygon is independently colored either red or blue. The probability of coloring a vertex red is p, and the probability of coloring it blue is q. The probability that the polygon can then be rotated so that all of the blue vertices end up at positions where there were originally red vert... | {"nodes": [{"id": "n1", "type": "unknown", "value": "V", "label": "number_of_vertices", "description": "A regular polygon has V vertices, where V is unspecified"}, {"id": "n2", "type": "given", "value": "red or blue", "label": "color_options", "description": "Each vertex is colored either red or blue"}, {"id": "n3", "t... | Each vertex of a regular octagon is independently colored either red or blue with equal probability. The probability that the resulting coloring satisfies a certain geometric condition 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": "color_options", "description": "Each vertex is colored either red or blue"}, {"id": "n3", "type": "given", "value": "0... |
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 greater than or equal to 2"}, {"id": "n2", "type": "given", "value": "n is a positive integer", "label": "n_positive", "description": "n must be a positive integer"}, {"id": "n3", "type": "constraint"... | Let b be an integer with b ≥ Q for some value Q. Call a positive integer n b-eautiful if it has exactly two digits when expressed in base b and these two digits sum to f(n) for some function f. For example, if f and Q were specified appropriately, certain integers would qualify as b-eautiful. Find the least integer b ≥... | {"nodes": [{"id": "n1", "type": "given", "value": "b >= Q (Q undefined)", "label": "base_constraint", "description": "b is an integer greater than or equal to some unspecified value Q"}, {"id": "n2", "type": "given", "value": "n is a positive integer", "label": "n_positive", "description": "n must be a positive integer... | Let b ≥ 2 be an integer. Call a positive integer n b-special if it has exactly two digits when expressed in base b and these two digits satisfy a particular condition involving n. Find the least integer b ≥ 2 for which there are sufficiently many b-special integers. | {"nodes": [{"id": "n1", "type": "given", "value": "b >= 2", "label": "base_constraint", "description": "b is an integer greater than or equal to 2"}, {"id": "n2", "type": "given", "value": "n is a positive integer", "label": "n_positive", "description": "n must be a positive integer"}, {"id": "n3", "type": "constraint"... |
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 (dodecagon)"}, {"id": "n2", "type": "given", "value": "sides_and_diagonals", "label": "allowed_lines", "description": "Rectangle sides must lie on sides or diagonals of the dodecagon"}, {... | Find the number of quadrilaterals of type Q that can be formed inside a fixed regular convex polygon with p sides, where each side of the quadrilateral lies on either a side or a diagonal of the polygon, and the quadrilateral is entirely contained within the polygon. | {"nodes": [{"id": "n1", "type": "given", "value": "p", "label": "regular_polygon", "description": "A fixed regular p-sided polygon where p is unspecified"}, {"id": "n2", "type": "given", "value": "sides_and_diagonals", "label": "allowed_lines", "description": "Quadrilateral sides must lie on sides or diagonals of the p... | 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_dodecagon", "description": "A fixed regular 12-sided polygon (dodecagon)"}, {"id": "n2", "type": "given", "value": "sides_and_diagonals", "label": "allowed_lines", "description": "Quadrilateral sides must lie on sides or diagonals of the dodecagon"... |
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_constraint", "description": "The sum of the items in the list is 30"}, {"id": "n2", "type": "given", "value": 9, "label": "mode_value", "description": "The unique mode of the list is 9"}, {"id": "n3", "type": "constraint", "value": null, "label": "posi... | A list of positive integers has the following properties:
• The sum of the items in the list is S.
• The unique mode of the list is M.
• 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": "n1", "type": "unknown", "value": null, "label": "sum_constraint", "description": "The sum of the items in the list is S (undefined)"}, {"id": "n2", "type": "unknown", "value": null, "label": "mode_value", "description": "The unique mode of the list is M (undefined)"}, {"id": "n3", "type": "constraint... | A list of positive integers has the following properties:
• The sum of the items in the list is a certain positive integer.
• The list has a unique mode.
• 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": "n1", "type": "given", "value": "unknown_S", "label": "sum_constraint", "description": "The sum of the items in the list is some positive integer S (value not specified)"}, {"id": "n2", "type": "given", "value": "unknown_M", "label": "mode_value", "description": "The unique mode of the list is some po... |
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 (0-9)"}, {"id": "n2", "type": "given", "value": "999", "label": "Row sum target", "description": "The sum of the two 3-digit numbers formed by reading each row left to ... | Find the number of ways to place a digit in each cell of a rectangular grid so that the sum of the numbers formed by reading left to right equals R, and the sum of the numbers formed by reading top to bottom equals C, where each cell contains a digit from 0 to 9. | {"nodes": [{"id": "n1", "type": "unknown", "value": null, "label": "Grid dimensions", "description": "A grid of unknown dimensions where each cell contains a single digit"}, {"id": "n2", "type": "unknown", "value": null, "label": "Row sum target", "description": "The sum of numbers formed by reading rows must equal an ... | Find the number of ways to place a digit in each cell of a 2x3 grid so that certain numbers can be formed from the rows and combined in a specific way to satisfy a particular condition, and certain numbers can be formed from the columns and combined in a specific way to satisfy another particular condition. | {"nodes": [{"id": "n1", "type": "given", "value": "2x3", "label": "Grid dimensions", "description": "A 2x3 grid where each cell contains a single digit (0-9)"}, {"id": "n2", "type": "given", "value": "unknown", "label": "Row constraint target", "description": "Some unspecified target value for row-based numbers combine... |
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