Yuting6/geoqa-r1v-8k-rotated huggingface dataset:
Problem: <image>Question: Sheila is making a regular-hexagon-shaped sign with side length $ 1$. Let $ABCDEF$ be the regular hexagon, and let $R, S,T$ and U be the midpoints of $FA$, $BC$, $CD$ and $EF$, respectively. Sheila splits the hexagon into four regions of equal width: trapezoids $ABSR$, $RSCF$ , $FCTU$, and $UTDE$. She then paints the middle two regions gold. The fraction of the total hexagon that is gold can be written in the form $m/n$ , where m and n are relatively prime positive integers. Compute $m + n$.\n<image1>
Answer: 19
Problem: <image>Question: In the star shaped figure below, if all side lengths are equal to $3$ and the three largest angles of the figure are $210$ degrees, its area can be expressed as $\frac{a \sqrt{b}}{c}$ , where $a, b$, and $c$ are positive integers such that $a$ and $c$ are relatively prime and that $b$ is square-free. Compute $a + b + c$.\n<image1>
Answer: 14
Problem: <image>Question: On the first day of school, Ashley the teacher asked some of her students what their favorite color was and used those results to construct the pie chart pictured below. During this first day, $165$ students chose yellow as their favorite color. The next day, she polled $30$ additional students and was shocked when none of them chose yellow. After making a new pie chart based on the combined results of both days, Ashley noticed that the angle measure of the sector representing the students whose favorite color was yellow had decreased. Compute the difference, in degrees, between the old and the new angle measures.\n<image1>
Answer: $\frac{90}{23}^{\circ}$
Problem: <image>Question: Consider $27$ unit-cubes assembled into one $3 \times 3  \times  3$ cube. Let $A$ and $B$ be two opposite corners  of this large cube. Remove the one unit-cube not visible from the exterior, along with all six unit-cubes in the center of each face. Compute the minimum distance an ant has to walk along the surface of the modified cube to get from $A$ to $B$.\n<image1>
Answer: $\sqrt{41}$
Problem: <image>Question: Parallelograms $ABGF$, $CDGB$ and $EFGD$ are drawn so that $ABCDEF$ is a convex hexagon, as shown. If $\angle ABG = 53^o$ and $\angle CDG = 56^o$, what is the measure of $\angle EFG$, in degrees?\n<image1>
Answer: 71
Problem: <image>Question: Let equilateral triangle $\vartriangle ABC$ be inscribed in a circle $\omega_1$ with radius $4$. Consider another circle $\omega_2$ with radius $2$ internally tangent to $\omega_1$ at $A$. Let $\omega_2$ intersect sides $AB$ and $AC$ at $D$ and $E$, respectively, as shown in the diagram. Compute the area of the shaded region.\n<image1>
Answer: $6 \sqrt{3}+4 \pi$
Problem: <image>Question: Big Chungus has been thinking of a new symbol for BMT, and the drawing below is what he came up with. If each of the $16$ small squares in the grid are unit squares, what is the area of the shaded region?\n<image1>
Answer: 6
Problem: <image>Question: Sohom constructs a square $BERK$ of side length $10$. Darlnim adds points $T$, $O$, $W$, and $N$, which are the midpoints of $\overline{BE}$, $\overline{ER}$, $\overline{RK}$, and $\overline{KB}$, respectively. Lastly, Sylvia constructs square $CALI$ whose edges contain the vertices of $BERK$, such that $\overline{CA}$ is parallel to $\overline{BO}$. Compute  the area of $CALI$.\n<image1>
Answer: 180
Problem: <image>Question: In the diagram below, all circles are tangent to each other as shown. The six outer circles are all congruent to each other, and the six inner circles are all congruent to each other. Compute the ratio of the area of one of the outer circles to the area of one of the inner circles.\n<image1>
Answer: 9
Problem: <image>Question: In the diagram below, the three circles and the three line segments are tangent as shown. Given that the radius of all of the three circles is $1$, compute the area of the triangle.\n<image1>
Answer: $6+4\sqrt{3}$
Problem: <image>Question: A $101\times 101$ square grid is given with rows and columns numbered in order from $1$ to $101$. Each square that is contained in both an even-numbered row and an even-numbered column is cut out. A small section of the grid is shown below, with the cut-out squares in black. Compute the maximum number of $L$-triominoes (pictured below) that can be placed in the grid so that each $L$-triomino lies entirely inside the grid and no two overlap. Each $L$-triomino may be placed in the orientation pictured below, or rotated by $90^\circ$, $180^\circ$, or $270^\circ$.\n<image1>
Answer: 2550
Problem: <image>Question: Consider the figure below, not drawn to scale.\nIn this figure, assume that$AB \perp BE$ and $AD  \perp DE$. Also, let $AB = \sqrt{6}$ and $\angle BED =\frac{\pi}{6}$ . Find $AC$.\n<image1>
Answer: $2\sqrt{2}$
Problem: <image>Question: Consider a $1$ by $2$ by $3$ rectangular prism. Find the length of the shortest path between opposite corners $A$ and $B$ that does not leave the surface of the prism.\n<image1>
Answer: $3\sqrt{2}$
Problem: <image>Question: In the diagram below, $A$ and $B$ trisect $DE$, $C$ and $A$ trisect $F G$, and $B$ and $C$ trisect $HI$. Given that $DI = 5$, $EF = 6$, $GH = 7$, find the area of $\vartriangle ABC$.\n<image1>
Answer: $\frac{3 \sqrt{6}}{2}$
Problem: <image>Question: Suppose we have a hexagonal grid in the shape of a hexagon of side length $4$ as shown at left. Define a “chunk” to be four tiles, two of which are adjacent to the other three, and the other two of which are adjacent to just two of the others. The three possible rotations of these are shown at right.\n\nIn how many ways can we choose a chunk from the grid?\n<image1>
Answer: 72
Problem: <image>Question: Suppose that in a group of $6$ people, if $A$ is friends with $B$, then $B$ is friends with $A$. If each of the $6$ people draws a graph of the friendships between the other $5$ people, we get these $6$ graphs, where edges represent\nfriendships and points represent people.\n\nIf Sue drew the first graph, how many friends does she have?\n<image1>
Answer: 4
Problem: <image>Question: Consider the $5\times 5$ grid $Z^2_5 = \{(a, b) : 0 \le a, b \le 4\}$.\nSay that two points $(a, b)$,$(x, y)$ are adjacent if $a - x  \equiv  -1, 0, 1$ (mod $5$) and $b - y \equiv -1, 0, 1$ (mod $5$) .\nFor example, in the diagram, all of the squares marked with $\cdot$ are adjacent to the square marked with $\times$.\n\nWhat is the largest number of $\times$ that can be placed on the grid such that no two are adjacent?\n<image1>
Answer: 5
Problem: <image>Question: Consider constructing a tower of tables of numbers as follows. <image1> <image2> The first table is a one by one array containing the single number $1$.\nThe second table is a two by two array formed underneath the first table and built as followed. For each entry, we look at the terms in the previous table that are directly up and to the left, up and to the right, and down and to the right of the entry, and we fill that entry with the sum of the numbers occurring there. If there happens to be no term at a particular location, it contributes a value of zero to the sum.\n\nThe diagram above shows how we compute the second table from the first.\nThe diagram below shows how to then compute the third table from the second.\n\nFor example, the entry in the middle row and middle column of the third table is equal the sum of the top left entry $1$, the top right entry $0$, and the bottom right entry $1$ from the second table, which is just $2$.\nSimilarly, to compute the bottom rightmost entry in the third table, we look above it to the left and see that the entry in the second table's bottom rightmost entry is $1$. There are no entries from the second table above it and to the right or below it and to the right, so we just take this entry in the third table to be $1$.\nWe continue constructing the tower by making more tables from the previous tables. Find the entry in the third (from the bottom) row of the third (from the left) column of the tenth table in this resulting tower.
Answer: 756
Problem: <image>Question: We define the $\emph{weight}$ of a path to be the sum of the numbers written on each edge of the path. Find the minimum weight among all paths in the graph below that visit each vertex precisely once. \n<image1>
Answer: 65
Problem: <image>Question: Right isosceles triangle $T$ is placed in the first quadrant of the coordinate plane.  Suppose that the projection of $T$ onto the $x$-axis has length $6$, while the projection of $T$ onto the $y$-axis has length $8$.  What is the sum of all possible areas of the triangle $T$?\n<image1>
Answer: 20
Problem: <image>Question: Ryan stands on the bottom-left square of a 2017 by 2017 grid of squares, where each square is colored either black, gray, or white according to the pattern as depicted to the right. Each second he moves either one square up, one square to the right, or both one up and to the right, selecting between these three options uniformly and independently. Noting that he begins on a black square, find the probability that Ryan is still on a black square after 2017 seconds.\n<image1>
Answer: $\frac{3^{1008}-1}{3^{1009}}$
Problem: <image>Question: The figure below depicts two congruent triangles with angle measures $40^\circ$, $50^\circ$, and $90^\circ$.  What is the measure of the obtuse angle $\alpha$ formed by the hypotenuses of these two triangles?\n<image1>
Answer: 170
Problem: <image>Question: On Misha's new phone, a passlock consists of six circles arranged in a $2\times 3$ rectangle. The lock is opened by a continuous path connecting the six circles; the path cannot pass through a circle  on the way between two others (e.g. the top left and right circles cannot be adjacent).  For example, the left path shown below is allowed but the right path is not.  (Paths are considered to be oriented, so that a path starting at $A$ and ending at $B$ is different from a path starting at $B$ and ending at $A$.  However, in the diagrams below, the paths are valid/invalid regardless of orientation.)  How many passlocks are there consisting of all six circles?\n<image1>
Answer: 336
Problem: <image>Question: Adam has a circle of radius $1$ centered at the origin.\n\n- First, he draws $6$ segments from the origin to the boundary of the circle, which splits the upper (positive $y$) semicircle into $7$ equal pieces.\n\n- Next, starting from each point where a segment hit the circle, he draws an altitude to the $x$-axis.\n\n- Finally, starting from each point where an altitude hit the $x$-axis, he draws a segment directly away from the bottommost point of the circle $(0,-1)$, stopping when he reaches the boundary of the circle.\n\nWhat is the product of the lengths of all $18$ segments Adam drew?\n<image1>
Answer: $\frac{7^3}{2^{12} 13^2}$
Problem: <image>Question: Four semicircles of radius $1$ are placed in a square, as shown below. The diameters of these semicircles lie on the sides of the square and each semicircle touches a vertex of the square. Find the absolute difference between the shaded area and the "hatched" area.\n<image1>
Answer: $4-2 \sqrt{3}$
Problem: <image>Question: A regular dodecahedron is a figure with $12$ identical pentagons for each of its faces. Let x be the number of ways to color the faces of the dodecahedron with $12$ different colors, where two colorings are identical if one can be rotated to obtain the other. Compute $\frac{x}{12!}$.\n<image1>
Answer: $\frac{1}{60}$
Problem: <image>Question: $7$ congruent squares are arranged into a 'C,' as shown below. If the perimeter and area of the 'C' are equal (ignoring units), compute the (nonzero) side length of the squares.\n<image1>
Answer: $\boxed{\frac{16}{7}}$
Problem: <image>Question: The following diagram uses $126$ sticks of length $1$ to form a “triangulated hollow hexagon” with inner side length $2$ and outer side length $4$. How many sticks would be needed for a triangulated hollow hexagon with inner side length $20$ and outer side length $23$?\n<image1>
Answer: 1290
Problem: <image>Question: Let $A, B, C$, and $D$ be equally spaced points on a circle $O$. $13$ circles of equal radius lie inside $O$ in the configuration below, where all centers lie on $\overline{AC}$ or $\overline{BD}$, adjacent circles are externally tangent, and the outer circles are internally tangent to $O$. Find the ratio of the area of the region inside $O$ but outside the smaller circles to the total area of the smaller circles.\n<image1>
Answer: rac{36}{13}
Problem: <image>Question: Triangle $T$ has side lengths $1$, $2$, and $\sqrt{7}$. It turns out that one can arrange three copies of triangle $T$ to form two equilateral triangles, one inside the other, as shown below. Compute the ratio of the area of the outer equilaterial triangle to the area of the inner equilateral triangle.\n<image1>
Answer: 7
Problem: <image>Question: Let $T$ be $7$. The diagram below features two concentric circles of radius $1$ and $T$ (not necessarily to scale). Four equally spaced points are chosen on the smaller circle, and rays are drawn from these points to the larger circle such that all of the rays are tangent to the smaller circle and no two rays intersect. If the area of the shaded region can be expressed as $k\pi$ for some integer $k$, find $k$.\n<image1>
Answer: 12
Problem: <image>Question: Let $T$ be $12$. $T^2$ congruent squares are arranged in the configuration below (shown for $T = 3$), where the squares are tilted in alternating fashion such that they form congruent rhombuses between them. If all of the rhombuses have long diagonal twice the length of their short diagonal, compute the ratio of the total area of all of the rhombuses to the total area of all of the squares. (Hint: Rather than waiting for $T$, consider small cases and try to find a general formula in terms of $T$, such a formula does exist.)\n<image1>
Answer: $\boxed{\frac{121}{180}}$
Problem: <image>Question: Rays $r_1$ and $r_2$ share a common endpoint. Three squares have sides on one of the rays and vertices on the other, as shown in the diagram. If the side lengths of the smallest two squares are $20$ and $22$, find the side length of the largest square.\n<image1>
Answer: 24.2
Problem: <image>Question: Blahaj has two rays with a common endpoint A0 that form an angle of $1^o$. They construct a sequence of points $A_0$, $. . . $, $A_n$ such that for all $1 \le i \le n$, $|A_{i-1}A_i | = 1$, and $|A_iA_0| > |A_{i-1}A_0|$. Find the largest possible value of $n$.\n<image1>
Answer: 90
Problem: <image>Question: Suppose Annie the Ant is walking on a regular icosahedron (as shown). She starts on point $A$ and will randomly create a path to go to point $Z$ which is the point directly opposite to $A$. Every move she makes never moves further from Z, and she has equal probability to go down every valid move. What is the expected number of moves she can make?\n<image1>
Answer: 6
Problem: <image>Question: Quadrilateral $ABCD$ (with $A, B, C$ not collinear and $A, D, C$ not collinear) has $AB = 4$, $BC = 7$, $CD = 10$, and $DA = 5$. Compute the number of possible integer lengths $AC$.\n<image1>
Answer: 5
Problem: <image>Question: Let $T$ be the answer from the previous part. $2T$ congruent isosceles triangles with base length $b$ and leg length $\ell$ are arranged to form a parallelogram as shown below (not necessarily the correct number of triangles). If the total length of all drawn line segments (not double counting overlapping sides) is exactly three times the perimeter of the parallelogram, find $\frac{\ell}{b}$.\n<image1>
Answer: 4
Problem: <image>Question: Let $T$ be the answer from the previous part. Rectangle $R$ has length $T$ times its width. $R$ is inscribed in a square $S$ such that the diagonals of $ S$ are parallel to the sides of $R$. What proportion of the area of $S$ is contained within $R$?\n<image1>
Answer: $\frac{8}{25}$
Problem: <image>Question: Sujay and Rishabh are taking turns marking lattice points within a square board in the Cartesian plane with opposite vertices $(1, 1)$,$(n, n)$ for some constant $n$. Sujay loses when the two-point pattern $P$ below shows up. That is, Sujay loses when there exists a pair of points $(x, y)$ and $(x + 2, y + 1)$. He and Rishabh stop marking points when the pattern $P$ appears on the board. If Rishabh goes first, let $S$ be the set of all integers $3 \le n \le 100$ such that Rishabh has a strategy to always trick Sujay into being the one who creates $P$. Find the sum of all elements of $S$.\n<image1>
Answer: 2499
Problem: <image>Question: Sujay sees a shooting star go across the night sky, and took a picture of it. The shooting star consists of a star body, which is bounded by four quarter-circle arcs, and a triangular tail. Suppose $AB = 2$, $AC = 4$. Let the area of the shooting star be $X$. If $6X = a-b\pi$ for positive integers $a, b$, find $a + b$.\n<image1>
Answer: 39
Problem: <image>Question: There are $4$ mirrors facing the inside of a $5\times 7$ rectangle as shown in the figure. A ray of light comes into the inside of a rectangle through $A$ with an angle of $45^o$. When it hits the sides of the rectangle, it bounces off at the same angle, as shown in the diagram. How many times will the ray of light bounce before it reaches any one of the corners $A$, $B$, $C$, $D$? A bounce is a time when the ray hit a mirror and reflects off it.\n<image1>
Answer: 10
Problem: <image>Question: The Olympic logo is made of $5$ circles of radius $1$, as shown in the figure. Suppose that the total area covered by these $5$ circles is $a+b\pi$ where $a, b$ are rational numbers. Find $10a + 20b$.\n<image1>
Answer: 100
Problem: <image>Question: Let $ABC$ be an equilateral triangle and $CDEF$ a square such that $E$ lies on segment $AB$ and $F$ on segment $BC$. If the perimeter of the square is equal to $4$, what is the area of triangle $ABC$?\n<image1>
Answer: $\frac{1}{2}+\frac{\sqrt{3}}{3}$
Problem: <image>Question: What is the maximum number of $T$-shaped polyominos (shown below) that we can put into a $6 \times 6$ grid without any overlaps. The blocks can be rotated.\n<image1>
Answer: 8
Problem: <image>Question: A drunkard is randomly walking through a city when he stumbles upon a $2 \times 2$ sliding tile puzzle. The puzzle consists of a $2 \times 2$ grid filled with a blank square, as well as $3$ square tiles, labeled $1$, $2$, and $3$. During each turn you may fill the empty square by sliding one of the adjacent tiles into it. The following image shows the puzzle's correct state, as well as two possible moves you can make. Assuming that the puzzle is initially in an incorrect (but solvable) state, and that the drunkard will make completely random moves to try and solve it, how many moves is he expected to make before he restores the puzzle to its correct state?\n<image1>
Answer: 7/3
Problem: <image>Question: Let $ABCD$ be a rectangle with $AB = 20$, $BC = 15$. Let $X$ and $Y$ be on the diagonal $\overline{BD}$ of $ABCD$ such that $BX > BY$ . Suppose $A$ and $X$ are two vertices of a square which has two sides on lines $\overline{AB}$ and $\overline{AD}$, and suppose that $C$ and $Y$ are vertices of a square which has sides on $\overline{CB}$ and $\overline{CD}$. Find the length $XY$.\n<image1>
Answer: \frac{25}{7}
Problem: <image>Question: In chess, a knight can move by jumping to any square whose center is $\sqrt{5}$ units away from the center of the square that it is currently on. For example, a knight on the square marked by the horse in the diagram below can move to any of the squares marked with an “X” and to no other squares. How many ways can a knight on the square marked by the horse in the diagram move to the square with a circle in exactly four moves?\n<image1>
Answer: 54
Problem: <image>Question: (See the diagram below.) $ABCD$ is a square. Points $G$, $H$, $I$, and $J$ are chosen in the interior of $ABCD$ so that:\n(i) $H$ is on $\overline{AG}$, $I$ is on $\overline{BH}$, $J$ is on $\overline{CI}$, and $G$ is on $\overline{DJ}$\n(ii) $\vartriangle ABH  \sim \vartriangle BCI  \sim  \vartriangle CDJ  \sim  \vartriangle DAG$ and \n(iii) the radii of the inscribed circles of $\vartriangle ABH$, $\vartriangle BCI$, $\vartriangle CDJ$, $\vartriangle DAK$, and $GHIJ$ are all the same.\nWhat is the ratio of $\overline{AB}$ to $\overline{GH}$?\n<image1>
Answer: 1+\sqrt{3}
Problem: <image>Question: In the diagram below, $ABCDEFGH$ is a rectangular prism, $\angle BAF = 30^o$ and $\angle DAH = 60^o$. What is the cosine of $\angle CEG$?\n<image1>
Answer: \frac{\sqrt{130}}{13}
Problem: <image>Question: Teddy works at Please Forget Meat, a contemporary vegetarian pizza chain in the city of Gridtown, as a deliveryman. Please Forget Meat (PFM) has two convenient locations, marked with “$X$” and “$Y$ ” on the street map of Gridtown shown below. Teddy, who is currently at $X$, needs to deliver an eggplant pizza to $\nabla$ en route to $Y$ , where he is urgently needed. There is currently construction taking place at $A$, $B$, and $C$, so those three intersections will be completely impassable. How many ways can Teddy get from $X$ to $Y$ while staying on the roads (Traffic tickets are expensive!), not taking paths that are longer than necessary (Gas is expensive!), and that let him pass through $\nabla$ (Losing a job is expensive!)?\n<image1>
Answer: 1144
Problem: <image>Question: Charlotte is playing the hit new web number game, Primle. In this game, the objective is to guess a two-digit positive prime integer between $10$ and $99$, called the Primle. For each guess, a digit is highlighted blue if it is in the Primle, but not in the correct place. A digit is highlighted orange if it is in the Primle and is in the correct place. Finally, a digit is left unhighlighted if it is not in the Primle. If Charlotte guesses $13$ and $47$ and is left with the following game board, what is the Primle?\n<image1>
Answer: 79
Problem: <image>Question: Suppose two circles $\Omega_1$ and $\Omega_2$ with centers $O_1$ and $O_2$ have radii $3$ and $4$, respectively. Suppose that points $A$ and $B$ lie on circles $\Omega_1$ and $\Omega_2$, respectively, such that segments $AB$ and $O_1O_2$ intersect and that $AB$ is tangent to $\Omega_1$ and $\Omega_2$. If $O_1O_2=25$, find the area of quadrilateral $O_1AO_2B$.\n<image1>
Answer: 84
Problem: <image>Question: An ant is standing at the bottom left corner of a $3$ by $3$ grid. How many ways can it get to the top right corner if it can only move up, right, and left, and it is not allowed to cross the same edge twice?\n<image1>
Answer: 9
Problem: <image>Question: In the diagram below, all seven of the small rectangles are congruent. If the perimeter of the large rectangle is $65$, what is its area?\n<image1>
Answer: 525/2
Problem: <image>Question: Will stands at a point $P$ on the edge of a circular room with perfectly reflective walls. He shines two laser pointers into the room, forming angles of $n^o$ and $(n + 1)^o$ with the tangent at $P$, where $n$ is a positive integer less than $90$. The lasers reflect off of the walls, illuminating the points they hit on the walls, until they reach $P$ again. ($P$ is also illuminated at the end.) What is the minimum possible number of illuminated points on the walls of the room?\n<image1>
Answer: 28
Problem: <image>Question: A square can be divided into four congruent figures as shown. For how many $n$ with $1 \le n \le 100$ can a unit square be divided into $n$ congruent figures?\n<image1>
Answer: 100
Problem: <image>Question: Sammy has a wooden board, shaped as a rectangle with length $2^{2014}$ and height $3^{2014}$. The board is divided into a grid of unit squares. A termite starts at either the left or bottom edge of the rectangle, and walks along the gridlines by moving either to the right or upwards, until it reaches an edge opposite the one from which the termite started. Depicted below are two possible paths of the termite. The termite's path dissects the board into two parts. Sammy is surprised to find that he can still arrange the pieces to form a new rectangle not congruent to the original rectangle. This rectangle has perimeter $P$. How many possible values of $P$ are there?\n<image1>
Answer: 4
Problem: <image>Question: How many lines pass through exactly two points in the following hexagonal grid?\n<image1>
Answer: 60