Yuting6/geoqa-r1v-8k-rotated huggingface dataset:
Problem: <image>Question: Point $P$ is inside $\triangle ABC$. Line segments $APD$, $BPE$, and $CPF$ are drawn with $D$ on $BC$, $E$ on $AC$, and $F$ on $AB$ (see the figure at right). Given that $AP=6$, $BP=9$, $PD=6$, $PE=3$, and $CF=20$, find the area of $\triangle ABC$.

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Answer: 108
Problem: <image>Question: The rectangle $ABCD$ below has dimensions $AB = 12 \sqrt{3}$ and $BC = 13 \sqrt{3}$.  Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at $P$.  If triangle $ABP$ is cut out and removed, edges $\overline{AP}$ and $\overline{BP}$ are joined, and the figure is then creased along segments $\overline{CP}$ and $\overline{DP}$, we obtain a triangular pyramid, all four of whose faces are isosceles triangles.  Find the volume of this pyramid.

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Answer: 594
Problem: <image>Question: Twelve congruent disks are placed on a circle $C$ of radius 1 in such a way that the twelve disks cover $C$, no two of the disks overlap, and so that each of the twelve disks is tangent to its two neighbors. The resulting arrangement of disks is shown in the figure below.  The sum of the areas of the twelve disks can be written in the from $\pi(a-b\sqrt{c})$, where $a,b,c$ are positive integers and $c$ is not divisible by the square of any prime. Find $a+b+c$.

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Answer: 135
Problem: <image>Question: In a game of Chomp, two players alternately take bites from a 5-by-7 grid of unit squares. To take a bite, a player chooses one of the remaining squares, then removes ("eats'') all squares in the quadrant defined by the left edge (extended upward) and the lower edge (extended rightward) of the chosen square. For example, the bite determined by the shaded square in the diagram would remove the shaded square and the four squares marked by $\times$. (The squares with two or more dotted edges have been removed form the original board in previous moves.)
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The object of the game is to make one's opponent take the last bite. The diagram shows one of the many subsets of the set of 35 unit squares that can occur during the game of Chomp. How many different subsets are there in all? Include the full board and empty board in your count.
Answer: 792
Problem: <image>Question: Two thousand points are given on a circle.  Label one of the points 1.  From this point, count 2 points in the clockwise direction and label this point 2.  From the point labeled 2, count 3 points in the clockwise direction and label this point 3.  (See figure.)  Continue this process until the labels $1, 2, 3, \dots, 1993$ are all used.  Some of the points on the circle will have more than one label and some points will not have a label.  What is the smallest integer that labels the same point as 1993?

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Answer: 118
Problem: <image>Question: A beam of light strikes $\overline{BC}$ at point $C$ with angle of incidence $\alpha=19.94^\circ$ and reflects with an equal angle of reflection as shown.  The light beam continues its path, reflecting off line segments $\overline{AB}$ and $\overline{BC}$ according to the rule: angle of incidence equals angle of reflection.  Given that $\beta=\alpha/10=1.994^\circ$ and $AB=AC,$ determine the number of times the light beam will bounce off the two line segments.  Include the first reflection at $C$ in your count.

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Answer: 71
Problem: <image>Question: Square $S_{1}$ is $1\times 1$.  For $i\ge 1,$ the lengths of the sides of square $S_{i+1}$ are half the lengths of the sides of square $S_{i},$ two adjacent sides of square $S_{i}$ are perpendicular bisectors of two adjacent sides of square $S_{i+1},$ and the other two sides of square $S_{i+1},$ are the perpendicular bisectors of two adjacent sides of square $S_{i+2}$.  The total area enclosed by at least one of $S_{1}, S_{2}, S_{3}, S_{4}, S_{5}$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers.  Find $m-n$.

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Answer: 255
Problem: <image>Question: Triangle $ABC$ is isosceles, with $AB=AC$ and altitude $AM=11$.  Suppose that there is a point $D$ on $\overline{AM}$ with $AD=10$ and $\angle BDC=3\angle BAC$.  Then the perimeter of $\triangle ABC$ may be written in the form $a+\sqrt{b},$ where $a$ and $b$ are integers.  Find $a+b$.

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Answer: 616
Problem: <image>Question: In a magic square, the sum of the three entries in any row, column, or diagonal is the same value.  The figure shows four of the entries of a magic square.  Find $x$.

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Answer: 200
Problem: <image>Question: The two squares shown share the same center $O$ and have sides of length 1. The length of $\overline{AB}$ is $43/99$ and the area of octagon $ABCDEFGH$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers.  Find $m+n$.
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Answer: 185
Problem: <image>Question: The diagram shows a rectangle that has been dissected into nine non-overlapping squares. Given that the width and the height of the rectangle are relatively prime positive integers, find the perimeter of the rectangle.

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Answer: 260
Problem: <image>Question: The diagram shows twenty congruent circles arranged in three rows and enclosed in a rectangle.  The circles are tangent to one another and to the sides of the rectangle as shown in the diagram.  The ratio of the longer dimension of the rectangle to the shorter dimension can be written as $\frac{1}{2}\left(\sqrt{p}-q\right),$ where $p$ and $q$ are positive integers.  Find $p+q$.

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Answer: 154
Problem: <image>Question: In the diagram below, angle $ABC$ is a right angle.  Point $D$ is on $\overline{BC}$, and $\overline{AD}$ bisects angle $CAB$.  Points $E$ and $F$ are on $\overline{AB}$ and $\overline{AC}$, respectively, so that $AE=3$ and $AF=10$.  Given that $EB=9$ and $FC=27$, find the integer closest to the area of quadrilateral $DCFG$.

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Answer: 148
Problem: <image>Question: Patio blocks that are hexagons $1$ unit on a side are used to outline a garden by placing the blocks edge to edge with $n$ on each side. The diagram indicates the path of blocks around the garden when $n=5$.
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If $n=202,$ then the area of the garden enclosed by the path, not including the path itself, is $m(\sqrt{3}/2)$ square units, where $m$ is a positive integer. Find the remainder when $m$ is divided by $1000$.
Answer: 803
Problem: <image>Question: $ABCD$ is a rectangular sheet of paper that has been folded so that corner $B$ is matched with point $B'$ on edge $AD$. The crease is $EF$, where $E$ is on $AB$ and $F$is on $CD$. The dimensions $AE=8$, $BE=17$, and $CF=3$ are given. The perimeter of rectangle $ABCD$ is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
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Answer: 293
Problem: <image>Question: An angle is drawn on a set of equally spaced parallel lines as shown.  The ratio of the area of shaded region $\mathcal{C}$ to the area of shaded region $\mathcal{B}$ is $11/5$.  Find the ratio of shaded region $\mathcal{D}$ to the area of shaded region $\mathcal{A}$.

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Answer: 408
Problem: <image>Question: Hexagon $ABCDEF$ is divided into four rhombuses, $\mathcal{P, Q, R, S,}$ and $\mathcal{T,}$ as shown.  Rhombuses $\mathcal{P, Q, R,}$ and $\mathcal{S}$ are congruent, and each has area $\sqrt{2006}$.  Let $K$ be the area of rhombus $\mathcal{T}$.  Given that $K$ is a positive integer, find the number of possible values for $K$.

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Answer: 89
Problem: <image>Question: Eight circles of diameter 1 are packed in the first quadrant of the coordinte plane as shown.  Let region $\mathcal{R}$ be the union of the eight circular regions.  Line $l,$ with slope 3, divides $\mathcal{R}$ into two regions of equal area.  Line $l$'s equation can be expressed in the form $ax=by+c,$ where $a, b,$ and $c$ are positive integers whose greatest common divisor is 1.  Find $a^2+b^2+c^2$.

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Answer: 65
Problem: <image>Question: In the $ 6\times4$ grid shown, $ 12$ of the $ 24$ squares are to be shaded so that there are two shaded squares in each row and three shaded squares in each column.  Let $ N$ be the number of shadings with this property.  Find the remainder when $ N$ is divided by $ 1000$.
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Answer: 860
Problem: <image>Question: Square $ABCD$ has side length $13$, and points $E$ and $F$ are exterior to the square such that $BE=DF=5$ and $AE=CF=12$. Find $EF^{2}$.
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Answer: 578
Problem: <image>Question: A triangular array of squares has one square in the first row, two in the second, and in general, $k$ squares in the $k$th row for $1 \leq k \leq 11$. With the exception of the bottom row, each square rests on two squares in the row immediately below (illustrated in given diagram). In each square of the eleventh row, a $0$ or a $1$ is placed. Numbers are then placed into the other squares, with the entry for each square being the sum of the entries in the two squares below it. For how many initial distributions of $0$'s and $1$'s in the bottom row is the number in the top square a multiple of $3$?
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Answer: 640
Problem: <image>Question: A triangular array of numbers has a first row consisting of the odd integers $ 1,3,5,\ldots,99$ in increasing order.  Each row below the first has one fewer entry than the row above it, and the bottom row has a single entry.  Each entry in any row after the top row equals the sum of the two entries diagonally above it in the row immediately above it.  How many entries in the array are multiples of $ 67$?
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Answer: 17
Problem: <image>Question: A square piece of paper has sides of length $ 100$. From each corner a wedge is cut in the following manner: at each corner, the two cuts for the wedge each start at distance $ \sqrt{17}$ from the corner, and they meet on the diagonal at an angle of $ 60^\circ$ (see the figure below). The paper is then folded up along the lines joining the vertices of adjacent cuts. When the two edges of a cut meet, they are taped together. The result is a paper tray whose sides are not at right angles to the base. The height of the tray, that is, the perpendicular distance between the plane of the base and the plane formed by the upper edges, can be written in the form $ \sqrt{n}{m}$, where $ m$ and $ n$ are positive integers, $ m < 1000$, and $ m$ is not divisible by the $ n$th power of any prime. Find $ m + n$.

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Answer: 871
Problem: <image>Question: The diagram below shows a $ 4\times4$ rectangular array of points, each of which is $ 1$ unit away from its nearest neighbors.
<image1>Define a growing path to be a sequence of distinct points of the array with the property that the distance between consecutive points of the sequence is strictly increasing. Let $ m$ be the maximum possible number of points in a growing path, and let $ r$ be the number of growing paths consisting of exactly $ m$ points. Find $ mr$.
Answer: 240
Problem: <image>Question: Equilateral triangle $ T$ is inscribed in circle $ A$, which has radius $ 10$. Circle $ B$ with radius $ 3$ is internally tangent to circle $ A$ at one vertex of $ T$. Circles $ C$ and $ D$, both with radius $ 2$, are internally tangent to circle $ A$ at the other two vertices of $ T$. Circles $ B$, $ C$, and $ D$ are all externally tangent to circle $ E$, which has radius $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m + n$.
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Answer: 32
Problem: <image>Question: In triangle $ABC$, $BC = 23$, $CA = 27$, and $AB = 30$.  Points $V$ and $W$ are on $\overline{AC}$ with $V$ on $\overline{AW}$, points $X$ and $Y$ are on $\overline{BC}$ with $X$ on $\overline{CY}$, and points $Z$ and $U$ are on $\overline{AB}$ with $Z$ on $\overline{BU}$.  In addition, the points are positioned so that $\overline{UV} \parallel \overline{BC}$, $\overline{WX} \parallel \overline{AB}$, and $\overline{YZ} \parallel \overline{CA}$.  Right angle folds are then made along $\overline{UV}$, $\overline{WX}$, and $\overline{YZ}$.  The resulting figure is placed on a level floor to make a table with triangular legs.  Let $h$ be the maximum possible height of a table constructed from triangle $ABC$ whose top is parallel to the floor.  Then $h$ can be written in the form $\frac{k \sqrt{m}}{n}$, where $k$ and $n$ are relatively prime positive integers and $m$ is a positive integer that is not divisible by the square of any prime.  Find $k + m + n$.

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Answer: 318
Problem: <image>Question: At each of the sixteen circles in the network below stands a student. A total of 3360 coins are distributed among the sixteen students. All at once, all students give away all their coins by passing an equal number of coins to each of their neighbors in the network. After the trade, all students have the same number of coins as they started with. Find the number of coins the student standing at the center circle had originally.

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Answer: 280
Problem: <image>Question: Cube $ABCDEFGH$, labeled as shown below, has edge length $1$ and is cut by a plane passing through vertex $D$ and the midpoints $M$ and $N$ of $\overline{AB}$ and $\overline{CG}$ respectively. The plane divides the cube into two solids. The volume of the larger of the two solids can be written in the form $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
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Answer: 89
Problem: <image>Question: In the accompanying figure, the outer square has side length 40. A second square S' of side length 15 is constructed inside S with the same center as S and with sides parallel to those of S. From each midpoint of a side of S, segments are drawn to the two closest vertices of S'. The result is a four-pointed starlike figure inscribed in S. The star figure is cut out and then folded to form a pyramid with base S'. Find the volume of this pyramid.
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Answer: 750
Problem: <image>Question: In the array of $13$ squares shown below, $8$ squares are colored red, and the remaining $5$ squares are colored blue. If one of all possible such colorings is chosen at random, the probability that the chosen colored array appears the same when rotated $90^{\circ}$ around the central square is $\frac{1}{n}$, where $n$ is a positive integer. Find $n$.
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Answer: 429
Problem: <image>Question: A paper equilateral triangle $ABC$ has side length $12$. The paper triangle is folded so that vertex $A$ touches a point on side $\overline{BC}$ a distance $9$ from point $B$. The length of the line segment along which the triangle is folded can be written as $\frac{m\sqrt{p}}{n}$, where $m$, $n$, and $p$ are positive integers, $m$ and $n$ are relatively prime, and $p$ is not divisible by the square of any prime. Find $m+n+p$.
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Answer: 113
Problem: <image>Question: The $8$ eyelets for the lace of a sneaker all lie on a rectangle, four equally spaced on each of the longer sides. The rectangle has a width of $50$ mm and a length of $80$ mm. There is one eyelet at each vertex of the rectangle. The lace itself must pass between the vertex eyelets along a width side of the rectangle and then crisscross between successive eyelets until it reaches the two eyelets at the other width side of the rectrangle as shown. After passing through these final eyelets, each of the ends of the lace must extend at least $200$ mm farther to allow a knot to be tied. Find the minimum length of the lace in millimeters.

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Answer: 790
Problem: <image>Question: On square $ABCD,$ points $E,F,G,$ and $H$ lie on sides $\overline{AB},\overline{BC},\overline{CD},$ and $\overline{DA},$ respectively, so that $\overline{EG} \perp \overline{FH}$ and $EG=FH = 34$. Segments $\overline{EG}$ and $\overline{FH}$ intersect at a point $P,$ and the areas of the quadrilaterals $AEPH, BFPE, CGPF,$ and $DHPG$ are in the ratio $269:275:405:411$. Find the area of square $ABCD$.


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Answer: 850
Problem: <image>Question: A rectangle has sides of length $a$ and $36$. A hinge is installed at each vertex of the rectangle and at the midpoint of each side of length $36$. The sides of length $a$ can be pressed toward each other keeping those two sides parallel so the rectangle becomes a convex hexagon as shown. When the figure is a hexagon with the sides of length $a$ parallel and separated by a distance of $24,$ the hexagon has the same area as the original rectangle. Find $a^2$.


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Answer: 720
Problem: <image>Question: Point $A,B,C,D,$ and $E$ are equally spaced on a minor arc of a circle. Points $E,F,G,H,I$ and $A$ are equally spaced on a minor arc of a second circle with center $C$ as shown in the figure below. The angle $\angle ABD$ exceeds $\angle AHG$ by $12^\circ$. Find the degree measure of $\angle BAG$.<image1>
Answer: 58
Problem: <image>Question: In the diagram below, $ABCD$ is a square. Point $E$ is the midpoint of $\overline{AD}$. Points $F$ and $G$ lie on $\overline{CE}$, and $H$ and $J$ lie on $\overline{AB}$ and $\overline{BC}$, respectively, so that $FGHJ$ is a square. Points $K$ and $L$ lie on $\overline{GH}$, and $M$ and $N$ lie on $\overline{AD}$ and $\overline{AB}$, respectively, so that $KLMN$ is a square. The area of $KLMN$ is 99. Find the area of $FGHJ$.
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Answer: 539
Problem: <image>Question: A block of wood has the shape of a right circular cylinder with radius $6$ and height $8$, and its entire surface has been painted blue. Points $A$ and $B$ are chosen on the edge on one of the circular faces of the cylinder so that $\overarc{AB}$ on that face measures $120^\circ$. The block is then sliced in half along the plane that passes through point $A$, point $B$, and the center of the cylinder, revealing a flat, unpainted face on each half. The area of one of those unpainted faces is $a\cdot\pi + b\sqrt{c}$, where $a$, $b$, and $c$ are integers and $c$ is not divisible by the square of any prime. Find $a+b+c$.

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Answer: 53
Problem: <image>Question: A cylindrical barrel with radius $4$ feet and height $10$ feet is full of water. A solid cube with side length $8$ feet is set into the barrel so that the diagonal of the cube is vertical. The volume of water thus displaced is $v$ cubic feet. Find $v^2$.

<image1>
Answer: 384
Problem: <image>Question: Circles $\mathcal{P}$ and $\mathcal{Q}$ have radii $1$ and $4$, respectively, and are externally tangent at point $A$. Point $B$ is on $\mathcal{P}$ and point $C$ is on $\mathcal{Q}$ so that line $BC$ is a common external tangent of the two circles. A line $\ell$ through $A$ intersects $\mathcal{P}$ again at $D$ and intersects $\mathcal{Q}$ again at $E$. Points $B$ and $C$ lie on the same side of $\ell$, and the areas of $\triangle DBA$ and $\triangle ACE$ are equal. This common area is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

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Answer: 129
Problem: <image>Question: A regular icosahedron is a $20$-faced solid where each face is an equilateral triangle and five triangles meet at every vertex. The regular icosahedron shown below has one vertex at the top, one vertex at the bottom, an upper pentagon of five vertices all adjacent to the top vertex and all in the same horizontal plane, and a lower pentagon of five vertices all adjacent to the bottom vertex and all in another horizontal plane. Find the number of paths from the top vertex to the bottom vertex such that each part of a path goes downward or horizontally along an edge of the icosahedron, and no vertex is repeated.<image1>
Answer: 810
Problem: <image>Question: The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and will paint each of the six sections a solid color. Find the number of ways you can choose to paint each of the six sections if no two adjacent section can be painted with the same color.

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Answer: 732
Problem: <image>Question: The area of the smallest equilateral triangle with one vertex on each of the sides of the right triangle with side lengths $2\sqrt{3}$, $5$, and $\sqrt{37}$, as shown, is $\frac{m\sqrt{p}}{n}$, where $m$, $n$, and $p$ are positive integers, $m$ and $n$ are relatively prime, and $p$ is not divisible by the square of any prime. Find $m+n+p$.
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Answer: 145
Problem: <image>Question: Circle $C_0$ has radius $1$, and the point $A_0$ is a point on the circle. Circle $C_1$ has radius $r<1$ and is internally tangent to $C_0$ at point $A_0$. Point $A_1$ lies on circle $C_1$ so that $A_1$ is located $90^{\circ}$ counterclockwise from $A_0$ on $C_1$. Circle $C_2$ has radius $r^2$ and is internally tangent to $C_1$ at point $A_1$. In this way a sequence of circles $C_1,C_2,C_3,...$ and a sequence of points on the circles $A_1,A_2,A_3,...$ are constructed, where circle $C_n$ has radius $r^n$ and is internally tangent to circle $C_{n-1}$ at point $A_{n-1}$, and point $A_n$ lies on $C_n$ $90^{\circ}$ counterclockwise from point $A_{n-1}$, as shown in the figure below. There is one point $B$ inside all of these circles. When $r=\frac{11}{60}$, the distance from the center of $C_0$ to $B$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

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Answer: 110
Problem: <image>Question: The wheel shown below consists of two circles and five spokes, with a label at each point where a spoke meets a circle. A bug walks along the wheel, starting at point \(A\). At every step of the process, the bug walks from one labeled point to an adjacent labeled point. Along the inner circle the bug only walks in a counterclockwise direction, and along the outer circle the bug only walks in a clockwise direction. For example, the bug could travel along the path \(AJABCHCHIJA\), which has \(10\) steps. Let \(n\) be the number of paths with \(15\) steps that begin and end at point \(A\). Find the remainder when \(n\) is divided by \(1000\).

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Answer: 4
Problem: <image>Question: Octagon $ABCDEFGH$ with side lengths $AB = CD = EF = GH = 10$ and $BC=  DE = FG = HA = 11$ is formed by removing four $6-8-10$ triangles from the corners of a $23\times 27$ rectangle with side $\overline{AH}$ on a short side of the rectangle, as shown. Let $J$ be the midpoint of $\overline{HA}$, and partition the octagon into $7$ triangles by drawing segments $\overline{JB}$, $\overline{JC}$, $\overline{JD}$, $\overline{JE}$, $\overline{JF}$, and $\overline{JG}$. Find the area of the convex polygon whose vertices are the centroids of these $7$ triangles.

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Answer: 184
Problem: <image>Question: In the diagram below, $ABCD$ is a rectangle with side lengths $AB=3$ and $BC=11$, and $AECF$ is a rectangle with side lengths $AF=7$ and $FC=9,$ as shown. The area of the shaded region common to the interiors of both rectangles is $\fracmn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

<image1>
Answer: 109
Problem: <image>Question: Equilateral triangle $ABC$ has side length $840$. Point $D$ lies on the same side of line $BC$ as $A$ such that $\overline{BD} \perp \overline{BC}$. The line $\ell$ through $D$ parallel to line $BC$ intersects sides $\overline{AB}$ and $\overline{AC}$ at points $E$ and $F$, respectively. Point $G$ lies on $\ell$ such that $F$ is between $E$ and $G$, $\triangle AFG$ is isosceles, and the ratio of the area of $\triangle AFG$ to the area of $\triangle BED$ is $8:9$. Find $AF$.

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Answer: 336
Problem: <image>Question: Two spheres with radii $36$ and one sphere with radius $13$ are each externally tangent to the other two spheres and to two different planes $\mathcal{P}$ and $\mathcal{Q}$. The intersection of planes $\mathcal{P}$ and $\mathcal{Q}$ is the line $\ell$. The distance from line $\ell$ to the point where the sphere with radius $13$ is tangent to plane $\mathcal{P}$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

<image1>
Answer: 335
Problem: <image>Question: Let $\triangle ABC$ be an acute triangle with circumcenter $O$ and centroid $G$. Let $X$ be the intersection of the line tangent to the circumcircle of $\triangle ABC$ at $A$ and the line perpendicular to $GO$ at $G$. Let $Y$ be the intersection of lines $XG$ and $BC$. Given that the measures of $\angle ABC, \angle BCA, $ and $\angle XOY$ are in the ratio $13 : 2 : 17, $ the degree measure of $\angle BAC$ can be written as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
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Answer: 592