id stringdate 2231-01-01 00:00:00 3093-01-01 00:00:00 | media images listlengths 1 4 | messages stringlengths 219 3.56k | answer stringlengths 1 100 | question_type stringclasses 2
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2231 | [{"role": "user", "question": "Turbo the snail sits on a point on a circle with circumference 1. Given an infinite sequence of positive real numbers $c_{1}, c_{2}, c_{3}, \\ldots$. Turbo successively crawls distances $c_{1}, c_{2}, c_{3}, \\ldots$ around the circle, each time choosing to crawl either clockwise or count... | $\frac{1}{2}$ | open | Geometry | Competition | Numerical | |
2237 | [{"role": "user", "question": "In the diagram, $\\angle A B F=41^{\\circ}, \\angle C B F=59^{\\circ}, D E$ is parallel to $B F$, and $E F=25$. If $A E=E C$, determine the length of $A E$, to 2 decimal places.\n\n<image_1>", "hint": "", "options": {}, "choices": [], "n_images": 1, "context": "", "subject": "Math", "lang... | 79.67 | numeric | Geometry | Competition | Numerical | |
2240 | [{"role": "user", "question": "In triangle $A B C, A B=B C=25$ and $A C=30$. The circle with diameter $B C$ intersects $A B$ at $X$ and $A C$ at $Y$. Determine the length of $X Y$.\n\n<image_1>", "hint": "", "options": {}, "choices": [], "n_images": 1, "context": "", "subject": "Math", "language": "English"}] | 15 | numeric | Geometry | Competition | Numerical | |
2245 | [{"role": "user", "question": "Points $P$ and $Q$ are located inside the square $A B C D$ such that $D P$ is parallel to $Q B$ and $D P=Q B=P Q$. Determine the minimum possible value of $\\angle A D P$.\n\n<image_1>", "hint": "", "options": {}, "choices": [], "n_images": 1, "context": "", "subject": "Math", "language":... | $15$ | open | Geometry | Competition | Numerical | |
2246 | [{"role": "user", "question": "In the diagram, $\\angle E A D=90^{\\circ}, \\angle A C D=90^{\\circ}$, and $\\angle A B C=90^{\\circ}$. Also, $E D=13, E A=12$, $D C=4$, and $C B=2$. Determine the length of $A B$.\n\n<image_1>", "hint": "", "options": {}, "choices": [], "n_images": 1, "context": "", "subject": "Math", "... | $\sqrt{5}$ | open | Geometry | Competition | Numerical | |
2250 | [{"role": "user", "question": "In the diagram, $A B C D$ is a quadrilateral with $A B=B C=C D=6, \\angle A B C=90^{\\circ}$, and $\\angle B C D=60^{\\circ}$. Determine the length of $A D$.\n\n<image_1>", "hint": "", "options": {}, "choices": [], "n_images": 1, "context": "", "subject": "Math", "language": "English"}] | $6\sqrt{2-\sqrt{3}}$ | open | Geometry | Competition | Numerical | |
2252 | [{"role": "user", "question": "A triangle has vertices $A(0,3), B(4,0)$, $C(k, 5)$, where $0<k<4$. If the area of the triangle is 8 , determine the value of $k$.\n\n<image_1>", "hint": "", "options": {}, "choices": [], "n_images": 1, "context": "", "subject": "Math", "language": "English"}] | $\frac{8}{3}$ | open | Geometry | Competition | Numerical | |
2264 | [{"role": "user", "question": "A helicopter hovers at point $H$, directly above point $P$ on level ground. Lloyd sits on the ground at a point $L$ where $\\angle H L P=60^{\\circ}$. A ball is droppped from the helicopter. When the ball is at point $B, 400 \\mathrm{~m}$ directly below the helicopter, $\\angle B L P=30^{... | $200 \sqrt{3}$ | open | Geometry | Competition | Numerical | |
2267 | [{"role": "user", "question": "In the diagram, $A B C D$ is a quadrilateral in which $\\angle A+\\angle C=180^{\\circ}$. What is the length of $C D$ ?\n\n<image_1>", "hint": "", "options": {}, "choices": [], "n_images": 1, "context": "", "subject": "Math", "language": "English"}] | 5 | numeric | Geometry | Competition | Numerical | |
2269 | [{"role": "user", "question": "In the diagram, the parabola\n\n$$\ny=-\\frac{1}{4}(x-r)(x-s)\n$$\n\nintersects the axes at three points. The vertex of this parabola is the point $V$. Determine the value of $k$ and the coordinates of $V$.\n\n<image_1>", "hint": "", "options": {}, "choices": [], "n_images": 1, "context":... | $4,(4,16)$ | open | Geometry | Competition | Numerical,Tuple | |
2273 | [{"role": "user", "question": "A school has a row of $n$ open lockers, numbered 1 through $n$. After arriving at school one day, Josephine starts at the beginning of the row and closes every second locker until reaching the end of the row, as shown in the example below. Then on her way back, she closes every second loc... | 33 | numeric | Combinatorics | Competition | Numerical | |
2278 | [{"role": "user", "question": "In the diagram, $P Q R S$ is a quadrilateral. What is its perimeter?\n\n<image_1>", "hint": "", "options": {}, "choices": [], "n_images": 1, "context": "", "subject": "Math", "language": "English"}] | 52 | numeric | Geometry | Competition | Numerical | |
2279 | [{"role": "user", "question": "In the diagram, $A$ has coordinates $(0,8)$. Also, the midpoint of $A B$ is $M(3,9)$ and the midpoint of $B C$ is $N(7,6)$. What is the slope of $A C$ ?\n\n<image_1>", "hint": "", "options": {}, "choices": [], "n_images": 1, "context": "", "subject": "Math", "language": "English"}] | $-\frac{3}{4}$ | open | Geometry | Competition | Numerical | |
2284 | [{"role": "user", "question": "In the diagram, $A B D E$ is a rectangle, $\\triangle B C D$ is equilateral, and $A D$ is parallel to $B C$. Also, $A E=2 x$ for some real number $x$.\n\n<image_1>\nDetermine the length of $A B$ in terms of $x$.", "hint": "", "options": {}, "choices": [], "n_images": 1, "context": "", "su... | $2 \sqrt{3} x$ | open | Geometry | Competition | Expression | |
2285 | [{"role": "user", "question": "In the diagram, $A B D E$ is a rectangle, $\\triangle B C D$ is equilateral, and $A D$ is parallel to $B C$. Also, $A E=2 x$ for some real number $x$.\n\n<image_1>\nDetermine positive integers $r$ and $s$ for which\n\n$$\n\\frac{A C}{A D}=\\sqrt{\\frac{r}{s}}\n$$", "hint": "", "options": ... | 7,4 | numeric | Geometry | Competition | Numerical | |
2290 | [{"role": "user", "question": "Five distinct integers are to be chosen from the set $\\{1,2,3,4,5,6,7,8\\}$ and placed in some order in the top row of boxes in the diagram. Each box that is not in the top row then contains the product of the integers in the two boxes connected to it in the row directly above. Determine... | 8 | numeric | Number Theory | Competition | Numerical | |
2295 | [{"role": "user", "question": "In the diagram, eleven circles of four different radius 1, each circle labelled $X$ has radius 2, the circle labelled $Y$ has radius 4 , and the circle labelled $Z$ has radius $r$. Each of the circles labelled $W$ or $X$ is tangent to three other circles. The circle labelled $Y$ is tangen... | $25538$,$2053$ | open | Geometry | Competition | Numerical | |
2297 | [{"role": "user", "question": "A circular disc is divided into 36 sectors. A number is written in each sector. When three consecutive sectors contain $a, b$ and $c$ in that order, then $b=a c$. If the number 2 is placed in one of the sectors and the number 3 is placed in one of the adjacent sectors, as shown, what is t... | 48 | numeric | Algebra | Competition | Numerical | |
2299 | [{"role": "user", "question": "In the diagram, $A C D F$ is a rectangle with $A C=200$ and $C D=50$. Also, $\\triangle F B D$ and $\\triangle A E C$ are congruent triangles which are right-angled at $B$ and $E$, respectively. What is the area of the shaded region?\n\n<image_1>", "hint": "", "options": {}, "choices": []... | 2500 | numeric | Geometry | Competition | Numerical | |
2302 | [{"role": "user", "question": "In the diagram, $\\triangle X Y Z$ is isosceles with $X Y=X Z=a$ and $Y Z=b$ where $b<2 a$. A larger circle of radius $R$ is inscribed in the triangle (that is, the circle is drawn so that it touches all three sides of the triangle). A smaller circle of radius $r$ is drawn so that it touc... | $\frac{2 a+b}{2 a-b}$ | open | Geometry | Competition | Expression | |
2307 | [{"role": "user", "question": "In the diagram, what is the area of figure $A B C D E F$ ?\n\n<image_1>", "hint": "", "options": {}, "choices": [], "n_images": 1, "context": "", "subject": "Math", "language": "English"}] | 48 | numeric | Geometry | Competition | Numerical | |
2308 | [{"role": "user", "question": "In the diagram, $A B C D$ is a rectangle with $A E=15, E B=20$ and $D F=24$. What is the length of $C F$ ?\n\n<image_1>", "hint": "", "options": {}, "choices": [], "n_images": 1, "context": "", "subject": "Math", "language": "English"}] | 7 | numeric | Geometry | Competition | Numerical | |
2309 | [{"role": "user", "question": "In the diagram, $A B C D$ is a square of side length 6. Points $E, F, G$, and $H$ are on $A B, B C, C D$, and $D A$, respectively, so that the ratios $A E: E B, B F: F C$, $C G: G D$, and $D H: H A$ are all equal to $1: 2$.\n\nWhat is the area of $E F G H$ ?\n\n<image_1>", "hint": "", "op... | 20 | numeric | Geometry | Competition | Numerical | |
2310 | [{"role": "user", "question": "In the diagram, line $A$ has equation $y=2 x$. Line $B$ is obtained by reflecting line $A$ in the $y$-axis. Line $C$ is perpendicular to line $B$. What is the slope of line $C$ ?\n\n<image_1>", "hint": "", "options": {}, "choices": [], "n_images": 1, "context": "", "subject": "Math", "lan... | $\frac{1}{2}$ | open | Geometry | Competition | Numerical | |
2311 | [{"role": "user", "question": "Three squares, each of side length 1 , are drawn side by side in the first quadrant, as shown. Lines are drawn from the origin to $P$ and $Q$. Determine, with explanation, the length of $A B$.\n\n<image_1>", "hint": "", "options": {}, "choices": [], "n_images": 1, "context": "", "subject"... | $\frac{1}{6}$ | open | Geometry | Competition | Numerical | |
2314 | [{"role": "user", "question": "In the diagram, the parabola with equation $y=x^{2}+t x-2$ intersects the $x$-axis at points $P$ and $Q$.\n\nAlso, the line with equation $y=3 x+3$ intersects the parabola at points $P$ and $R$. Determine the value of $t$ and the area of triangle $P Q R$.\n\n<image_1>", "hint": "", "optio... | -1,27 | numeric | Geometry | Competition | Numerical | |
2316 | [{"role": "user", "question": "In the diagram, $A C=B C, A D=7, D C=8$, and $\\angle A D C=120^{\\circ}$. What is the value of $x$ ?\n\n<image_1>", "hint": "", "options": {}, "choices": [], "n_images": 1, "context": "", "subject": "Math", "language": "English"}] | $13 \sqrt{2}$ | open | Geometry | Competition | Numerical | |
2324 | [{"role": "user", "question": "Donna has a laser at $C$. She points the laser beam at the point $E$. The beam reflects off of $D F$ at $E$ and then off of $F H$ at $G$, as shown, arriving at point $B$ on $A D$. If $D E=E F=1 \\mathrm{~m}$, what is the length of $B D$, in metres?\n\n<image_1>", "hint": "", "options": {}... | 3 | numeric | Geometry | Competition | Numerical | |
2335 | [{"role": "user", "question": "An L shape is made by adjoining three congruent squares. The L is subdivided into four smaller L shapes, as shown. Each of the resulting L's is subdivided in this same way. After the third round of subdivisions, how many L's of the smallest size are there?\n<image_1>", "hint": "", "option... | 64 | numeric | Number Theory | Competition | Numerical | |
2337 | [{"role": "user", "question": "Jimmy is baking two large identical triangular cookies, $\\triangle A B C$ and $\\triangle D E F$. Each cookie is in the shape of an isosceles right-angled triangle. The length of the shorter sides of each of these triangles is $20 \\mathrm{~cm}$. He puts the cookies on a rectangular baki... | $(20+4 \sqrt{2})$ | open | Geometry | Competition | Numerical | |
2347 | [{"role": "user", "question": "In the diagram, $\\angle A C B=\\angle A D E=90^{\\circ}$. If $A B=75, B C=21, A D=20$, and $C E=47$, determine the exact length of $B D$.\n\n<image_1>", "hint": "", "options": {}, "choices": [], "n_images": 1, "context": "", "subject": "Math", "language": "English"}] | 65 | numeric | Geometry | Competition | Numerical | |
2353 | [{"role": "user", "question": "A circle, with diameter $A B$ as shown, intersects the positive $y$-axis at point $D(0, d)$. Determine $d$.\n\n<image_1>", "hint": "", "options": {}, "choices": [], "n_images": 1, "context": "", "subject": "Math", "language": "English"}] | 4 | numeric | Geometry | Competition | Numerical | |
2354 | [{"role": "user", "question": "A square $P Q R S$ with side of length $x$ is subdivided into four triangular regions as shown so that area (A) + area $(B)=\\text{area}(C)$. If $P T=3$ and $R U=5$, determine the value of $x$.\n\n<image_1>", "hint": "", "options": {}, "choices": [], "n_images": 1, "context": "", "subject... | 15 | numeric | Geometry | Competition | Numerical | |
2357 | [{"role": "user", "question": "In the diagram, $A D=D C, \\sin \\angle D B C=0.6$ and $\\angle A C B=90^{\\circ}$. What is the value of $\\tan \\angle A B C$ ?\n\n<image_1>", "hint": "", "options": {}, "choices": [], "n_images": 1, "context": "", "subject": "Math", "language": "English"}] | $\frac{3}{2}$ | open | Geometry | Competition | Numerical | |
2358 | [{"role": "user", "question": "On a cross-sectional diagram of the Earth, the $x$ and $y$-axes are placed so that $O(0,0)$ is the centre of the Earth and $C(6.40,0.00)$ is the location of Cape Canaveral. A space shuttle is forced to land on an island at $A(5.43,3.39)$, as shown. Each unit represents $1000 \\mathrm{~km}... | 3570 | numeric | Geometry | Competition | Numerical | |
2360 | [{"role": "user", "question": "The parabola $y=-x^{2}+4$ has vertex $P$ and intersects the $x$-axis at $A$ and $B$. The parabola is translated from its original position so that its vertex moves along the line $y=x+4$ to the point $Q$. In this position, the parabola intersects the $x$-axis at $B$ and $C$. Determine the... | $(8,0)$ | open | Geometry | Competition | Tuple | |
2362 | [{"role": "user", "question": "In the isosceles trapezoid $A B C D$, $A B=C D=x$. The area of the trapezoid is 80 and the circle with centre $O$ and radius 4 is tangent to the four sides of the trapezoid. Determine the value of $x$.\n\n<image_1>", "hint": "", "options": {}, "choices": [], "n_images": 1, "context": "", ... | 10 | numeric | Geometry | Competition | Numerical | |
2368 | [{"role": "user", "question": "In the diagram, points $P(p, 4), B(10,0)$, and $O(0,0)$ are shown. If $\\triangle O P B$ is right-angled at $P$, determine all possible values of $p$.\n\n<image_1>", "hint": "", "options": {}, "choices": [], "n_images": 1, "context": "", "subject": "Math", "language": "English"}] | 2,8 | numeric | Geometry | Competition | Numerical | |
2373 | [{"role": "user", "question": "A snail's shell is formed from six triangular sections, as shown. Each triangle has interior angles of $30^{\\circ}, 60^{\\circ}$ and $90^{\\circ}$. If $A B$ has a length of $1 \\mathrm{~cm}$, what is the length of $A H$, in $\\mathrm{cm}$ ?\n\n<image_1>", "hint": "", "options": {}, "choi... | $\frac{64}{27}$ | open | Geometry | Competition | Numerical | |
2374 | [{"role": "user", "question": "In rectangle $A B C D$, point $E$ is on side $D C$. Line segments $A E$ and $B D$ are perpendicular and intersect at $F$. If $A F=4$ and $D F=2$, determine the area of quadrilateral $B C E F$.\n\n<image_1>", "hint": "", "options": {}, "choices": [], "n_images": 1, "context": "", "subject"... | 19 | numeric | Geometry | Competition | Numerical | |
2378 | [{"role": "user", "question": "In the diagram, points $B, P, Q$, and $C$ lie on line segment $A D$. The semi-circle with diameter $A C$ has centre $P$ and the semi-circle with diameter $B D$ has centre $Q$. The two semi-circles intersect at $R$. If $\\angle P R Q=40^{\\circ}$, determine the measure of $\\angle A R D$.\... | $110$ | open | Geometry | Competition | Numerical | |
2385 | [{"role": "user", "question": "In the diagram, a line is drawn through points $P, Q$ and $R$. If $P Q=Q R$, what are the coordinates of $R$ ?\n\n<image_1>", "hint": "", "options": {}, "choices": [], "n_images": 1, "context": "", "subject": "Math", "language": "English"}] | (3,8) | open | Geometry | Competition | Tuple | |
2386 | [{"role": "user", "question": "In the diagram, $O A=15, O P=9$ and $P B=4$. Determine the equation of the line through $A$ and $B$. Explain how you got your answer.\n\n<image_1>", "hint": "", "options": {}, "choices": [], "n_images": 1, "context": "", "subject": "Math", "language": "English"}] | $y=-3 x+39$ | open | Geometry | Competition | Expression | |
2387 | [{"role": "user", "question": "In the diagram, $\\triangle A B C$ is right-angled at $B$ and $A B=10$. If $\\cos (\\angle B A C)=\\frac{5}{13}$, what is the value of $\\tan (\\angle A C B)$ ?\n\n<image_1>", "hint": "", "options": {}, "choices": [], "n_images": 1, "context": "", "subject": "Math", "language": "English"}... | $\frac{5}{12}$ | open | Geometry | Competition | Numerical | |
2389 | [{"role": "user", "question": "In the diagram, $A B=B C=2 \\sqrt{2}, C D=D E$, $\\angle C D E=60^{\\circ}$, and $\\angle E A B=75^{\\circ}$. Determine the perimeter of figure $A B C D E$. Explain how you got your answer.\n\n<image_1>", "hint": "", "options": {}, "choices": [], "n_images": 1, "context": "", "subject": "... | $4+4 \sqrt{2}+2 \sqrt{3}$ | open | Geometry | Competition | Numerical | |
2393 | [{"role": "user", "question": "In the diagram, the parabola intersects the $x$-axis at $A(-3,0)$ and $B(3,0)$ and has its vertex at $C$ below the $x$-axis. The area of $\\triangle A B C$ is 54 . Determine the equation of the parabola. Explain how you got your answer.\n\n<image_1>", "hint": "", "options": {}, "choices":... | $y=2 x^{2}-18$ | open | Geometry | Competition | Expression | |
2394 | [{"role": "user", "question": "In the diagram, $A(0, a)$ lies on the $y$-axis above $D$. If the triangles $A O B$ and $B C D$ have the same area, determine the value of $a$. Explain how you got your answer.\n\n<image_1>", "hint": "", "options": {}, "choices": [], "n_images": 1, "context": "", "subject": "Math", "langua... | 4 | numeric | Geometry | Competition | Numerical | |
2395 | [{"role": "user", "question": "The Little Prince lives on a spherical planet which has a radius of $24 \\mathrm{~km}$ and centre $O$. He hovers in a helicopter $(H)$ at a height of $2 \\mathrm{~km}$ above the surface of the planet. From his position in the helicopter, what is the distance, in kilometres, to the furthes... | 10 | numeric | Geometry | Competition | Numerical | |
2396 | [{"role": "user", "question": "In the diagram, points $A$ and $B$ are located on islands in a river full of rabid aquatic goats. Determine the distance from $A$ to $B$, to the nearest metre. (Luckily, someone has measured the angles shown in the diagram as well as the distances $C D$ and $D E$.)\n\n<image_1>", "hint": ... | 66 | numeric | Geometry | Competition | Numerical | |
2399 | [{"role": "user", "question": "In the $4 \\times 4$ grid shown, three coins are randomly placed in different squares. Determine the probability that no two coins lie in the same row or column.\n\n<image_1>", "hint": "", "options": {}, "choices": [], "n_images": 1, "context": "", "subject": "Math", "language": "English"... | $\frac{6}{35}$ | open | Geometry | Competition | Numerical | |
2400 | [{"role": "user", "question": "In the diagram, the area of $\\triangle A B C$ is 1 . Trapezoid $D E F G$ is constructed so that $G$ is to the left of $F, D E$ is parallel to $B C$, $E F$ is parallel to $A B$ and $D G$ is parallel to $A C$. Determine the maximum possible area of trapezoid $D E F G$.\n\n<image_1>", "hint... | $\frac{1}{3}$ | open | Geometry | Competition | Numerical | |
2404 | [{"role": "user", "question": "In the diagram, $\\triangle P Q R$ has $P Q=a, Q R=b, P R=21$, and $\\angle P Q R=60^{\\circ}$. Also, $\\triangle S T U$ has $S T=a, T U=b, \\angle T S U=30^{\\circ}$, and $\\sin (\\angle T U S)=\\frac{4}{5}$. Determine the values of $a$ and $b$.\n<image_1>", "hint": "", "options": {}, "c... | $24,15$ | open | Geometry | Competition | Numerical | |
2405 | [{"role": "user", "question": "A triangle of area $770 \\mathrm{~cm}^{2}$ is divided into 11 regions of equal height by 10 lines that are all parallel to the base of the triangle. Starting from the top of the triangle, every other region is shaded, as shown. What is the total area of the shaded regions?\n\n<image_1>", ... | $420$ | open | Geometry | Competition | Numerical | |
2406 | [{"role": "user", "question": "A square lattice of 16 points is constructed such that the horizontal and vertical distances between adjacent points are all exactly 1 unit. Each of four pairs of points are connected by a line segment, as shown. The intersections of these line segments are the vertices of square $A B C D... | $\frac{9}{10}$ | open | Geometry | Competition | Numerical | |
2412 | [{"role": "user", "question": "At the Canadian Eatery with Multiple Configurations, there are round tables, around which chairs are placed. When a table has $n$ chairs around it for some integer $n \\geq 3$, the chairs are labelled $1,2,3, \\ldots, n-1, n$ in order around the table. A table is considered full if no mor... | $k+1$ | open | Combinatorics | Competition | Expression | |
2413 | [{"role": "user", "question": "At the Canadian Eatery with Multiple Configurations, there are round tables, around which chairs are placed. When a table has $n$ chairs around it for some integer $n \\geq 3$, the chairs are labelled $1,2,3, \\ldots, n-1, n$ in order around the table. A table is considered full if no mor... | 209 | numeric | Combinatorics | Competition | Numerical | |
2417 | [{"role": "user", "question": "In the diagram, $\\triangle A B C$ is right-angled at $B$ and $\\triangle A C D$ is right-angled at $A$. Also, $A B=3, B C=4$, and $C D=13$. What is the area of quadrilateral $A B C D$ ?\n\n<image_1>", "hint": "", "options": {}, "choices": [], "n_images": 1, "context": "", "subject": "Mat... | 36 | numeric | Geometry | Competition | Numerical | |
2418 | [{"role": "user", "question": "Three identical rectangles $P Q R S$, WTUV and $X W V Y$ are arranged, as shown, so that $R S$ lies along $T X$. The perimeter of each of the three rectangles is $21 \\mathrm{~cm}$. What is the perimeter of the whole shape?\n<image_1>", "hint": "", "options": {}, "choices": [], "n_images"... | 42 | numeric | Geometry | Competition | Numerical | |
2424 | [{"role": "user", "question": "The diagram shows two hills that meet at $O$. One hill makes a $30^{\\circ}$ angle with the horizontal and the other hill makes a $45^{\\circ}$ angle with the horizontal. Points $A$ and $B$ are on the hills so that $O A=O B=20 \\mathrm{~m}$. Vertical poles $B D$ and $A C$ are connected by... | $(16-10 \sqrt{2})$ | open | Geometry | Competition | Numerical | |
2428 | [{"role": "user", "question": "In the diagram, line segments $A C$ and $D F$ are tangent to the circle at $B$ and $E$, respectively. Also, $A F$ intersects the circle at $P$ and $R$, and intersects $B E$ at $Q$, as shown. If $\\angle C A F=35^{\\circ}, \\angle D F A=30^{\\circ}$, and $\\angle F P E=25^{\\circ}$, determ... | $32.5$ | open | Geometry | Competition | Numerical | |
2429 | [{"role": "user", "question": "In the diagram, $A B C D$ and $P N C D$ are squares of side length 2, and $P N C D$ is perpendicular to $A B C D$. Point $M$ is chosen on the same side of $P N C D$ as $A B$ so that $\\triangle P M N$ is parallel to $A B C D$, so that $\\angle P M N=90^{\\circ}$, and so that $P M=M N$. De... | $\frac{16}{3}$ | open | Geometry | Competition | Numerical | |
2436 | [{"role": "user", "question": "In the diagram, $\\triangle A B C$ is right-angled at $B$ and $A C=20$. If $\\sin C=\\frac{3}{5}$, what is the length of side $B C$ ?\n\n<image_1>", "hint": "", "options": {}, "choices": [], "n_images": 1, "context": "", "subject": "Math", "language": "English"}] | 16 | numeric | Geometry | Competition | Numerical | |
2437 | [{"role": "user", "question": "A helicopter is flying due west over level ground at a constant altitude of $222 \\mathrm{~m}$ and at a constant speed. A lazy, stationary goat, which is due west of the helicopter, takes two measurements of the angle between the ground and the helicopter. The first measurement the goat m... | 123 | numeric | Geometry | Competition | Numerical | |
2446 | [{"role": "user", "question": "A regular hexagon is a six-sided figure which has all of its angles equal and all of its side lengths equal. In the diagram, $A B C D E F$ is a regular hexagon with an area of 36. The region common to the equilateral triangles $A C E$ and $B D F$ is a hexagon, which is shaded as shown. Wh... | 12 | numeric | Geometry | Competition | Numerical | |
2447 | [{"role": "user", "question": "At the Big Top Circus, Herc the Human Cannonball is fired out of the cannon at ground level. (For the safety of the spectators, the cannon is partially buried in the sand floor.) Herc's trajectory is a parabola until he catches the vertical safety net, on his way down, at point $B$. Point... | 48 | numeric | Geometry | Competition | Numerical | |
2455 | [{"role": "user", "question": "In the diagram, $V$ is the vertex of the parabola with equation $y=-x^{2}+4 x+1$. Also, $A$ and $B$ are the points of intersection of the parabola and the line with equation $y=-x+1$. Determine the value of $A V^{2}+B V^{2}-A B^{2}$.\n\n<image_1>", "hint": "", "options": {}, "choices": []... | 60 | numeric | Geometry | Competition | Numerical | |
2456 | [{"role": "user", "question": "In the diagram, $A B C$ is a quarter of a circular pizza with centre $A$ and radius $20 \\mathrm{~cm}$. The piece of pizza is placed on a circular pan with $A, B$ and $C$ touching the circumference of the pan, as shown. What fraction of the pan is covered by the piece of pizza?\n\n<image_... | $\frac{1}{2}$ | open | Geometry | Competition | Numerical | |
2457 | [{"role": "user", "question": "The deck $A B$ of a sailboat is $8 \\mathrm{~m}$ long. Rope extends at an angle of $60^{\\circ}$ from $A$ to the top $(M)$ of the mast of the boat. More rope extends at an angle of $\\theta$ from $B$ to a point $P$ that is $2 \\mathrm{~m}$ below $M$, as shown. Determine the height $M F$ o... | $\frac{8 \sqrt{3} \tan \theta+2 \sqrt{3}}{\tan \theta+\sqrt{3}}$ | open | Geometry | Competition | Expression | |
2465 | [{"role": "user", "question": "In the diagram, triangle ABC is right-angled at B. MT is the perpendicular bisector of $B C$ with $M$ on $B C$ and $T$ on $A C$. If $A T=A B$, what is the size of $\\angle A C B$ ?\n\n<image_1>", "hint": "", "options": {}, "choices": [], "n_images": 1, "context": "", "subject": "Math", "l... | $30^{\circ}$ | open | Geometry | Competition | Numerical | |
2468 | [{"role": "user", "question": "In the diagram, $A B C D E F$ is a regular hexagon with a side\nlength of 10 . If $X, Y$ and $Z$ are the midpoints of $A B, C D$ and $E F$, respectively, what is the length of $X Z$ ?\n\n<image_1>", "hint": "", "options": {}, "choices": [], "n_images": 1, "context": "", "subject": "Math",... | 15 | numeric | Geometry | Competition | Numerical | |
2470 | [{"role": "user", "question": "In the diagram, $A C=2 x, B C=2 x+1$ and $\\angle A C B=30^{\\circ}$. If the area of $\\triangle A B C$ is 18 , what is the value of $x$ ?\n\n<image_1>", "hint": "", "options": {}, "choices": [], "n_images": 1, "context": "", "subject": "Math", "language": "English"}] | 4 | numeric | Geometry | Competition | Numerical | |
2471 | [{"role": "user", "question": "A ladder, $A B$, is positioned so that its bottom sits on horizontal ground and its top rests against a vertical wall, as shown. In this initial position, the ladder makes an angle of $70^{\\circ}$ with the horizontal. The bottom of the ladder is then pushed $0.5 \\mathrm{~m}$ away from t... | 26 | numeric | Geometry | Competition | Numerical | |
2480 | [{"role": "user", "question": "In the diagram, $P Q R S$ is an isosceles trapezoid with $P Q=7, P S=Q R=8$, and $S R=15$. Determine the length of the diagonal $P R$.\n\n<image_1>", "hint": "", "options": {}, "choices": [], "n_images": 1, "context": "", "subject": "Math", "language": "English"}] | 13 | numeric | Geometry | Competition | Numerical | |
2483 | [{"role": "user", "question": "In the diagram, $\\triangle A B C$ has $A B=A C$ and $\\angle B A C<60^{\\circ}$. Point $D$ is on $A C$ with $B C=B D$. Point $E$ is on $A B$ with $B E=E D$. If $\\angle B A C=\\theta$, determine $\\angle B E D$ in terms of $\\theta$.\n\n<image_1>", "hint": "", "options": {}, "choices": [... | $3 \theta$ | open | Geometry | Competition | Expression | |
2484 | [{"role": "user", "question": "In the diagram, the ferris wheel has a diameter of $18 \\mathrm{~m}$ and rotates at a constant rate. When Kolapo rides the ferris wheel and is at its lowest point, he is $1 \\mathrm{~m}$ above the ground. When Kolapo is at point $P$ that is $16 \\mathrm{~m}$ above the ground and is rising... | 9 | numeric | Geometry | Competition | Numerical | |
2485 | [{"role": "user", "question": "On Saturday, Jimmy started painting his toy helicopter between 9:00 a.m. and 10:00 a.m. When he finished between 10:00 a.m. and 11:00 a.m. on the same morning, the hour hand was exactly where the minute hand had been when he started, and the minute hand was exactly where the hour hand had... | $\frac{12}{13}$ | open | Algebra | Competition | Numerical | |
2491 | [{"role": "user", "question": "In the diagram, the circle with centre $C(1,1)$ passes through the point $O(0,0)$, intersects the $y$-axis at $A$, and intersects the $x$-axis at $B(2,0)$. Determine, with justification, the coordinates of $A$ and the area of the part of the circle that lies in the first quadrant.\n\n<ima... | $(0,2),\pi+2$ | open | Geometry | Competition | Tuple,Numerical | |
2495 | [{"role": "user", "question": "Survivors on a desert island find a piece of plywood $(A B C)$ in the shape of an equilateral triangle with sides of length $2 \\mathrm{~m}$. To shelter their goat from the sun, they place edge $B C$ on the ground, lift corner $A$, and put in a vertical post $P A$ which is $h \\mathrm{~m}... | 163 | numeric | Geometry | Competition | Numerical | |
2503 | [{"role": "user", "question": "Points $A_{1}, A_{2}, \\ldots, A_{N}$ are equally spaced around the circumference of a circle and $N \\geq 3$. Three of these points are selected at random and a triangle is formed using these points as its vertices.\n\nThrough this solution, we will use the following facts:\n\nWhen an ac... | $\frac{2}{5}$ | open | Geometry | Competition | Numerical | |
2505 | [{"role": "user", "question": "In the diagram, $\\triangle P Q S$ is right-angled at $P$ and $\\triangle Q R S$ is right-angled at $Q$. Also, $P Q=x, Q R=8, R S=x+8$, and $S P=x+3$ for some real number $x$. Determine all possible values of the perimeter of quadrilateral $P Q R S$.\n\n<image_1>", "hint": "", "options": ... | 22,46 | numeric | Geometry | Competition | Numerical | |
2511 | [{"role": "user", "question": "In the diagram, $\\triangle A B D$ has $C$ on $B D$. Also, $B C=2, C D=1, \\frac{A C}{A D}=\\frac{3}{4}$, and $\\cos (\\angle A C D)=-\\frac{3}{5}$. Determine the length of $A B$.\n\n<image_1>", "hint": "", "options": {}, "choices": [], "n_images": 1, "context": "", "subject": "Math", "la... | $\frac{13}{7}$ | open | Geometry | Competition | Numerical | |
2512 | [{"role": "user", "question": "Suppose that $a>\\frac{1}{2}$ and that the parabola with equation $y=a x^{2}+2$ has vertex $V$. The parabola intersects the line with equation $y=-x+4 a$ at points $B$ and $C$, as shown. If the area of $\\triangle V B C$ is $\\frac{72}{5}$, determine the value of $a$.\n\n<image_1>", "hint... | $\frac{5}{2}$ | open | Algebra | Competition | Numerical | |
2514 | [{"role": "user", "question": "Suppose that $m$ and $n$ are positive integers with $m \\geq 2$. The $(m, n)$-sawtooth sequence is a sequence of consecutive integers that starts with 1 and has $n$ teeth, where each tooth starts with 2, goes up to $m$ and back down to 1 . For example, the $(3,4)$-sawtooth sequence is\n\n... | 31 | numeric | Number Theory | Competition | Numerical | |
2515 | [{"role": "user", "question": "Suppose that $m$ and $n$ are positive integers with $m \\geq 2$. The $(m, n)$-sawtooth sequence is a sequence of consecutive integers that starts with 1 and has $n$ teeth, where each tooth starts with 2, goes up to $m$ and back down to 1 . For example, the $(3,4)$-sawtooth sequence is\n\n... | $3 m^{2}-2$ | open | Number Theory | Competition | Expression | |
2517 | [{"role": "user", "question": "At Pizza by Alex, toppings are put on circular pizzas in a random way. Every topping is placed on a randomly chosen semicircular half of the pizza and each topping's semi-circle is chosen independently. For each topping, Alex starts by drawing a diameter whose angle with the horizonal is ... | $\frac{1}{2}$ | open | Combinatorics | Competition | Numerical | |
2518 | [{"role": "user", "question": "At Pizza by Alex, toppings are put on circular pizzas in a random way. Every topping is placed on a randomly chosen semicircular half of the pizza and each topping's semi-circle is chosen independently. For each topping, Alex starts by drawing a diameter whose angle with the horizonal is ... | $\frac{3}{4}$ | open | Combinatorics | Competition | Numerical | |
2519 | [{"role": "user", "question": "At Pizza by Alex, toppings are put on circular pizzas in a random way. Every topping is placed on a randomly chosen semicircular half of the pizza and each topping's semi-circle is chosen independently. For each topping, Alex starts by drawing a diameter whose angle with the horizonal is ... | $\frac{N}{2^{N-1}}$ | open | Combinatorics | Competition | Expression | |
2530 | [{"role": "user", "question": "In rectangle $A B C D, F$ is on diagonal $B D$ so that $A F$ is perpendicular to $B D$. Also, $B C=30, C D=40$ and $A F=x$. Determine the value of $x$.\n\n<image_1>", "hint": "", "options": {}, "choices": [], "n_images": 1, "context": "", "subject": "Math", "language": "English"}] | 24 | numeric | Geometry | Competition | Numerical | |
2533 | [{"role": "user", "question": "In the diagram, $\\triangle A B C$ is right-angled at $C$. Also, $2 \\sin B=3 \\tan A$. Determine the measure of angle $A$.\n\n<image_1>", "hint": "", "options": {}, "choices": [], "n_images": 1, "context": "", "subject": "Math", "language": "English"}] | $30^{\circ}$ | open | Geometry | Competition | Numerical | |
2535 | [{"role": "user", "question": "Alice drove from town $E$ to town $F$ at a constant speed of $60 \\mathrm{~km} / \\mathrm{h}$. Bob drove from $F$ to $E$ along the same road also at a constant speed. They started their journeys at the same time and passed each other at point $G$.\n\n<image_1>\n\nAlice drove from $G$ to $... | 90 | numeric | Geometry | Competition | Numerical | |
2537 | [{"role": "user", "question": "In the diagram, $D$ is the vertex of a parabola. The parabola cuts the $x$-axis at $A$ and at $C(4,0)$. The parabola cuts the $y$-axis at $B(0,-4)$. The area of $\\triangle A B C$ is 4. Determine the area of $\\triangle D B C$.\n\n<image_1>", "hint": "", "options": {}, "choices": [], "n_i... | 3 | numeric | Geometry | Competition | Numerical | |
2540 | [{"role": "user", "question": "In the diagram, $P Q R S$ is a square with sides of length 4. Points $T$ and $U$ are on sides $Q R$ and $R S$ respectively such that $\\angle U P T=45^{\\circ}$. Determine the maximum possible perimeter of $\\triangle R U T$.\n\n<image_1>", "hint": "", "options": {}, "choices": [], "n_ima... | 8 | numeric | Geometry | Competition | Numerical | |
2541 | [{"role": "user", "question": "Suppose there are $n$ plates equally spaced around a circular table. Ross wishes to place an identical gift on each of $k$ plates, so that no two neighbouring plates have gifts. Let $f(n, k)$ represent the number of ways in which he can place the gifts. For example $f(6,3)=2$, as shown be... | 7 | numeric | Algebra | Competition | Numerical | |
2543 | [{"role": "user", "question": "Suppose there are $n$ plates equally spaced around a circular table. Ross wishes to place an identical gift on each of $k$ plates, so that no two neighbouring plates have gifts. Let $f(n, k)$ represent the number of ways in which he can place the gifts. For example $f(6,3)=2$, as shown be... | 54 | numeric | Algebra | Competition | Numerical | |
2546 | [{"role": "user", "question": "A regular pentagon covers part of another regular polygon, as shown. This regular polygon has $n$ sides, five of which are completely or partially visible. In the diagram, the sum of the measures of the angles marked $a^{\\circ}$ and $b^{\\circ}$ is $88^{\\circ}$. Determine the value of $... | 9 | numeric | Geometry | Competition | Numerical | |
2550 | [{"role": "user", "question": "A circle has centre $O$ and radius 1. Quadrilateral $A B C D$ has all 4 sides tangent to the circle at points $P, Q, S$, and $T$, as shown. Also, $\\angle A O B=\\angle B O C=\\angle C O D=\\angle D O A$. If $A O=3$, determine the length of $D S$.\n\n<image_1>", "hint": "", "options": {},... | $\frac{1}{2 \sqrt{2}}$ | open | Geometry | Competition | Numerical | |
2566 | [{"role": "user", "question": "In the diagram, $A B=21$ and $B C=16$. Also, $\\angle A B C=60^{\\circ}, \\angle C A D=30^{\\circ}$, and $\\angle A C D=45^{\\circ}$. Determine the length of $C D$, to the nearest tenth.\n\n<image_1>", "hint": "", "options": {}, "choices": [], "n_images": 1, "context": "", "subject": "Mat... | $\frac{19 \sqrt{2}}{\sqrt{3}+1}$ | open | Geometry | Competition | Numerical | |
2567 | [{"role": "user", "question": "In the diagram, the large circle has radius 9 and centre $C(15,0)$. The small circles have radius 4 and centres $A$ and $B$ on the horizontal line $y=12$. Each of the two small circles is tangent to the large circle. It takes a bug 5 seconds to walk at a constant speed from $A$ to $B$ alo... | 2 | numeric | Geometry | Competition | Numerical | |
2573 | [{"role": "user", "question": "In the diagram, $A B C$ is a right-angled triangle with $P$ and $R$ on $A B$. Also, $Q$ is on $A C$, and $P Q$ is parallel to $B C$. If $R P=2$, $B R=3, B C=4$, and the area of $\\triangle Q R C$ is 5 , determine the length of $A P$.\n\n<image_1>", "hint": "", "options": {}, "choices": []... | 1 | numeric | Geometry | Competition | Numerical | |
2579 | [{"role": "user", "question": "In the diagram, sector $A O B$ is $\\frac{1}{6}$ of an entire circle with radius $A O=B O=18$. The sector is cut into two regions with a single straight cut through $A$ and point $P$ on $O B$. The areas of the two regions are equal. Determine the length of $O P$.\n\n<image_1>", "hint": ""... | $2 \sqrt{3} \pi$ | open | Geometry | Competition | Numerical |
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