id stringlengths 1 4 | media images listlengths 1 1 | messages stringlengths 84 1.58k | answer stringclasses 371
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199 | [{"role": "user", "question": "The segment $A E$ is divided into four equal parts and semicircles are drawn taking $A E, A D$ and $D E$ as diameters, creating two paths from $A$ to $E$ as shown. Determine the ratio of the length of the upper path to the length of the lower path.\n<image1>", "hint": "", "options": {"A":... | E | mcq | 5 | metric geometry - length | |
1990 | [{"role": "user", "question": "In how many different ways can the word BANANA be read from the following table by moving from one cell to another cell with which it shares an edge? Cells may be visited more than once. <image1>", "hint": "", "options": {}, "choices": []}] | 84 | numeric | 4 | combinatorics | |
1991 | [{"role": "user", "question": "The diagram shows a map of a park. The park is divided into regions. The number inside each region gives its perimeter, in $\\mathrm{km}$. What is the outer perimeter of the park? <image1>", "hint": "", "options": {"A": "$22 \\mathrm{~km}$", "B": "$26 \\mathrm{~km}$", "C": "$28 \\mathrm{~... | B | mcq | 4 | metric geometry - length | |
1992 | [{"role": "user", "question": "Vumos wants to write the integers 1 to 9 in the nine boxes shown so that the sum of the integers in any three adjacent boxes is a multiple of 3 . In how many ways can he do this? <image1>", "hint": "", "options": {"A": "$6 \\times 6 \\times 6 \\times 6$", "B": "$6 \\times 6 \\times 6$", "... | A | mcq | 4 | combinatorics | |
1993 | [{"role": "user", "question": "The diagram shows a solid with six triangular faces and five vertices. Andrew wants to write an integer at each of the vertices so that the sum of the numbers at the three vertices of each face is the same. He has already written the numbers 1 and 5 as shown.\n<image1>\nWhat is the sum of... | 11 | numeric | 5 | algebra | |
1994 | [{"role": "user", "question": "The diagram shows two circles and a square with sides of length $10 \\mathrm{~cm}$. One vertex of the square is at the centre of the large circle and two sides of the square are tangents to both circles. The small circle touches the large circle. The radius of the small circle is $(a-b \\... | 50 | numeric | 5 | metric geometry - length | |
1995 | [{"role": "user", "question": "The diagram shows a triangle $A B C$ with area $12 \\mathrm{~cm}^{2}$. The sides of the triangle are extended to points $P, Q, R, S, T$ and $U$ as shown so that $P A=A B=B S, Q A=A C=C T$ and $R B=B C=C U$.\n<image1>\nWhat is the area (in $\\mathrm{cm}^{2}$ ) of hexagon $P Q R S T U$ ?", ... | 156 | numeric | 5 | metric geometry - area | |
1996 | [{"role": "user", "question": "A ball is propelled from corner $A$ of a square snooker table of side 2 metres. After bouncing off three cushions as shown, the ball goes into a pocket at $B$. The total distance travelled by the ball is $\\sqrt{k}$ metres. What is the value of $k$ ?\n<image1>\n(Note that when the ball bo... | 52 | numeric | 5 | metric geometry - length | |
1997 | [{"role": "user", "question": "In rectangle $J K L M$, the bisector of angle $K J M$ cuts the diagonal $K M$ at point $N$ as shown. The distances between $N$ and sides $L M$ and $K L$ are $8 \\mathrm{~cm}$ and $1 \\mathrm{~cm}$ respectively. The length of $K L$ is $(a+\\sqrt{b}) \\mathrm{cm}$. What is the value of $a+b... | 16 | numeric | 5 | metric geometry - length | |
1998 | [{"role": "user", "question": "In quadrilateral $A B C D, \\angle A B C=\\angle A D C=90^{\\circ}, A D=D C$ and $A B+B C=20 \\mathrm{~cm}$.\n<image1>\nWhat is the area in $\\mathrm{cm}^{2}$ of quadrilateral $A B C D$ ?", "hint": "", "options": {}, "choices": []}] | 100 | numeric | 5 | metric geometry - area | |
1999 | [{"role": "user", "question": "Using this picture we can observe that\n$1+3+5+7=4 \\times 4$.\nWhat is the value of\n$1+3+5+7+9+11+13+15+17+19+21$ ?\n<image1>", "hint": "", "options": {}, "choices": []}] | 121 | numeric | 5 | algebra | |
2 | [{"role": "user", "question": "Which bike is most expensive?\n<image1>", "hint": "", "options": {"A": "A", "B": "B", "C": "C", "D": "D", "E": "E"}, "choices": []}] | A | mcq | 2 | arithmetic | |
20 | [{"role": "user", "question": "How many plums (see the picture) weigh as much as an apple?\n<image1>", "hint": "", "options": {}, "choices": []}] | 3 | numeric | 2 | arithmetic | |
200 | [{"role": "user", "question": "A mathematically skilled spider spins a cobweb and some of the strings have lengths as shown in the picture. If $x$ is an integer, determine the value of $x$.\n<image1>", "hint": "", "options": {}, "choices": []}] | 13 | numeric | 5 | metric geometry - length | |
2000 | [{"role": "user", "question": "Both rows of the following grid have the same sum. What is the value of $*$ ?\n<image1>", "hint": "", "options": {}, "choices": []}] | 950 | numeric | 1 | arithmetic | |
2001 | [{"role": "user", "question": "<image1>\nIn the diagram, $P T$ and $P S$ are tangents to a circle with centre $O$. The point $Y$ lies on the circumference of the circle; and the point $Z$ is where the line $P Y$ meets the radius $O S$.\nAlso, $\\angle S P Z=10^{\\circ}$ and $\\angle T O S=150^{\\circ}$.\nHow many degre... | 160 | numeric | 4 | metric geometry - angle | |
2002 | [{"role": "user", "question": "The diagram shows two concentric circles. Chord $A B$ of the larger circle is tangential to the smaller circle.\nThe length of $A B$ is $32 \\mathrm{~cm}$ and the area of the shaded region is $k \\pi \\mathrm{cm}^{2}$.\nWhat is the value of $k$ ?\n<image1>", "hint": "", "options": {}, "ch... | 256 | numeric | 5 | metric geometry - area | |
2003 | [{"role": "user", "question": "Delia is joining three vertices of a square to make four right-angled triangles.\nShe can create four triangles doing this, as shown.\n<image1>\nHow many right-angled triangles can Delia make by joining three vertices of a regular polygon with 18 sides?", "hint": "", "options": {}, "choic... | 144 | numeric | 5 | combinatorial geometry | |
2004 | [{"role": "user", "question": "The large equilateral triangle shown consists of 36 smaller equilateral triangles. Each of the smaller equilateral triangles has area $10 \\mathrm{~cm}^{2}$.\nThe area of the shaded triangle is $K \\mathrm{~cm}^{2}$. Find $K$.\n<image1>", "hint": "", "options": {}, "choices": []}] | 110 | numeric | 5 | combinatorial geometry | |
2005 | [{"role": "user", "question": "A barcode of the type shown in the two examples is composed of alternate strips of black and white, where the leftmost and rightmost strips are always black. Each strip (of either colour) has a width of 1 or 2 . The total width of the barcode is 12 . The barcodes are always read from left... | 116 | numeric | 5 | combinatorics | |
2006 | [{"role": "user", "question": "An integer is to be written in each circle of the network shown. The integers must be written so that the sum of the numbers at the end of each line segment is the same. Two of the integers have already been written. What is the total of all the integers in the completed diagram?\n<image1... | 132 | numeric | 5 | algebra | |
2007 | [{"role": "user", "question": "The figure shows a shape consisting of a regular hexagon of side $18 \\mathrm{~cm}$, six triangles and six squares. The outer perimeter of the shape is $P \\mathrm{~cm}$. What is the value of $P$ ?\n<image1>", "hint": "", "options": {}, "choices": []}] | 216 | numeric | 5 | metric geometry - length | |
2008 | [{"role": "user", "question": "The figure shows a quadrilateral $A B C D$ in which $A D=D C$ and $\\angle A D C=\\angle A B C=90^{\\circ}$. The point $E$ is the foot of the perpendicular from $D$ to $A B$. The length $D E$ is 25 . What is the area of quadrilateral $A B C D$ ?\n<image1>", "hint": "", "options": {}, "cho... | 625 | numeric | 5 | metric geometry - area | |
2009 | [{"role": "user", "question": "Priti is learning a new language called Tedio. During her one hour lesson, which started at midday, she looks at the clock and notices that the hour hand and the minute hand make exactly the same angle with the vertical, as shown in the diagram. How many whole seconds remain until the end... | 276 | numeric | 4 | metric geometry - angle | |
201 | [{"role": "user", "question": "Two semicircles are drawn as shown in the figure. The chord $C D$, of length 4 , is parallel to the diameter $A B$ of the greater semicircle and touches the smaller semicircle. Then the area of the shaded region equals\n<image1>", "hint": "", "options": {"A": "$\\pi$", "B": "$1.5 \\pi$", ... | C | mcq | 5 | metric geometry - area | |
2010 | [{"role": "user", "question": "Robin shoots three arrows at a target. He earns points for each shot as shown in the figure. However, if any of his arrows miss the target or if any two of his arrows hit adjacent regions of the target, he scores a total of zero. How many different scores can he obtain?\n<image1>", "hint"... | 13 | numeric | 5 | combinatorics | |
2011 | [{"role": "user", "question": "At each of the vertices of a cube sits a Bunchkin. Two Bunchkins are said to be adjacent if and only if they sit at either end of one of the cube's edges. Each Bunchkin is either a 'truther', who always tells the truth, or a 'liar', who always lies. All eight Bunchkins say 'I am adjacent ... | 4 | numeric | 1 | logic | |
2012 | [{"role": "user", "question": "The pattern shown in the diagram is constructed using semicircles. Each semicircle has a diameter that lies on the horizontal axis shown and has one of the black dots at either end. The distance between each pair of adjacent black dots is $1 \\mathrm{~cm}$. The area, in $\\mathrm{cm}^{2}$... | 121 | numeric | 5 | metric geometry - area | |
2013 | [{"role": "user", "question": "Each square in this cross-number can be filled with a non-zero digit such that all of the conditions in the clues are fulfilled. The digits used are not necessarily distinct.\n<image1>\nACROSS\n1. A square\n3. The answer to this Kangaroo question\n5. A square\nDOWN\n1. 4 down minus eleven... | 829 | numeric | 1 | logic | |
2014 | [{"role": "user", "question": "The diagram shows a semicircle with diameter $P Q$ inscribed in a rhombus $A B C D$. The rhombus is tangent to the arc of the semicircle in two places. Points $P$ and $Q$ lie on sides $B C$ and $C D$ of the rhombus respectively. The line of symmetry of the semicircle is coincident with th... | 603 | numeric | 5 | metric geometry - area | |
2015 | [{"role": "user", "question": "The line segments $P Q R S$ and $W X Y S$ intersect circle $C_{1}$ at points $P, Q, W$ and $X$.\n<image1>\nThe line segments intersect circle $C_{2}$ at points $Q, R, X$ and $Y$. The lengths $Q R, R S$ and $X Y$ are 7, 9 and 18 respectively. The length $W X$ is six times the length $Y S$.... | 150 | numeric | 5 | metric geometry - length | |
2016 | [{"role": "user", "question": "The diagram shows a 16 metre by 16 metre wall. Three grey squares are painted on the wall as shown.\n<image1>\nThe two smaller grey squares are equal in size and each makes an angle of $45^{\\circ}$ with the edge of the wall. The grey squares cover a total area of $B$ metres squared.\nWha... | 128 | numeric | 5 | metric geometry - area | |
2017 | [{"role": "user", "question": "Identical regular pentagons are arranged in a ring. The partially completed ring is shown in the diagram. Each of the regular pentagons has a perimeter of 65 . The regular polygon formed as the inner boundary of the ring has a perimeter of $P$. What is the value of $P$ ?\n<image1>", "hint... | 130 | numeric | 5 | metric geometry - length | |
2018 | [{"role": "user", "question": "The function $J(x)$ is defined by:\n$$\nJ(x)= \\begin{cases}4+x & \\text { for } x \\leq-2 \\\\ -x & \\text { for }-2<x \\leq 0 \\\\ x & \\text { for } x>0\\end{cases}\n$$\n<image1>\nHow many distinct real solutions has the equation $J(J(J(x)))=0$ ?", "hint": "", "options": {}, "choices":... | 4 | numeric | 4 | analytic geometry | |
2019 | [{"role": "user", "question": "In the triangle $A B C$ the points $M$ and $N$ lie on the side $A B$ such that $A N=A C$ and $B M=B C$.\nWe know that $\\angle M C N=43^{\\circ}$.\nFind the size in degrees of $\\angle A C B$.\n<image1>", "hint": "", "options": {}, "choices": []}] | 94 | numeric | 4 | metric geometry - angle | |
202 | [{"role": "user", "question": "Which is the graph of the function $y=\\sqrt{|(1+x)(1-|x|)|}$?\n<image1>", "hint": "", "options": {"A": "A", "B": "B", "C": "C", "D": "D", "E": "E"}, "choices": []}] | D | mcq | 4 | analytic geometry | |
2020 | [{"role": "user", "question": "Each square in this cross-number can be filled with a non-zero digit such that all of the conditions in the clues are fulfilled. The digits used are not necessarily distinct. What is the answer to 3 ACROSS?\n<image1>\n\\section*{ACROSS}\n1. A composite factor of 1001\n3. Not a palindrome\... | 295 | numeric | 1 | logic | |
2021 | [{"role": "user", "question": "Two identical cylindrical sheets are cut open along the dotted lines and glued together to form one bigger cylindrical sheet, as shown. The smaller sheets each enclose a volume of 100. What volume is enclosed by the larger\n<image1>", "hint": "", "options": {}, "choices": []}] | 400 | numeric | 3 | solid geometry | |
2022 | [{"role": "user", "question": "Margot writes the numbers $1,2,3,4,5,6,7$ and 8 in the top row of a table, as shown. In the second row she plans to write the same set of numbers, in any order.\nEach number in the third row is obtained by finding the sum of the two numbers above it.\n<image1>\nIn how many different ways ... | 576 | numeric | 5 | combinatorics | |
2023 | [{"role": "user", "question": "A square fits snugly between the horizontal line and two touching circles of radius 1000, as shown. The line is tangent to the circles.\nWhat is the side-length of the square?\n<image1>", "hint": "", "options": {}, "choices": []}] | 400 | numeric | 5 | metric geometry - length | |
2024 | [{"role": "user", "question": "Each square in this cross-number can be filled with a non-zero digit such that all of the conditions in the clues are fulfilled. The digits used are not necessarily distinct.\nWhat is the answer to 3 ACROSS?\n<image1>\n\\section*{ACROSS}\n1. A multiple of 7\n3. The answer to this Question... | 961 | numeric | 1 | logic | |
2025 | [{"role": "user", "question": "How many of the figures shown can be drawn with one continuous line without drawing a segment twice?\n<image1>", "hint": "", "options": {}, "choices": []}] | 3 | numeric | 5 | graph theory | |
2026 | [{"role": "user", "question": "Five cards have the numbers $101,102,103,104$ and 105 on their fronts.\n<image1>\nOn the reverse, each card has one of five different positive integers: $a, b, c, d$ and $e$ respectively.\nWe know that $c=b e, a+b=d$ and $e-d=a$.\nFrankie picks up the card which has the largest integer on... | 103 | numeric | 5 | algebra | |
2027 | [{"role": "user", "question": "In the figure shown there are three concentric circles and two perpendicular diameters. The three shaded regions have equal area. The radius of the small circle is 2 . The product of the three radii is $Y$.\nWhat is the value of $Y^{2}$ ?\n<image1>", "hint": "", "options": {}, "choices": ... | 384 | numeric | 5 | metric geometry - length | |
2028 | [{"role": "user", "question": "Five cards have the numbers $101,102,103,104$ and 105 on their fronts.\n<image1>\nOn the reverse, each card has one of five different positive integers: $a, b, c, d$ and $e$ respectively. We know that $a+2=b-2=2 c=\\frac{d}{2}=e^{2}$.\nGina picks up the card which has the largest integer ... | 104 | numeric | 5 | algebra | |
2029 | [{"role": "user", "question": "The perimeter of the square in the figure is 40 . The perimeter of the larger equilateral triangle in the figure is $a+b \\sqrt{p}$, where $p$ is a prime number. What is the value of $7 a+5 b+3 p$ ?\n<image1>", "hint": "", "options": {}, "choices": []}] | 269 | numeric | 5 | metric geometry - length | |
203 | [{"role": "user", "question": "We see in the diagram at the right a piece of the graphic of the function\n$$\nf(x)=a x^{3}+b x^{2}+c x+d.\n$$\nWhat is the value of $b$?\n<image1>", "hint": "", "options": {}, "choices": []}] | -2 | numeric | 4 | analytic geometry | |
2030 | [{"role": "user", "question": "A particular flag is in the shape of a rectangle divided into five smaller congruent rectangles as shown. When written in its lowest terms, the ratio of the side lengths of the smaller rectangle is $\\lambda: 1$, where $\\lambda<1$. What is the value of $360 \\lambda$ ? <image1>", "hint":... | 120 | numeric | 5 | combinatorial geometry | |
2031 | [{"role": "user", "question": "Five cards have the numbers $101,102,103,104$ and 105 on their fronts. <image1>\nOn the reverse, each card has a statement printed as follows:\n101: The statement on card 102 is false\n102: Exactly two of these cards have true statements\n103: Four of these cards have false statements\n10... | 206 | numeric | 1 | logic | |
2032 | [{"role": "user", "question": "The smallest four two-digit primes are written in different squares of a $2 \\times 2$ table.\n\nThe sums of the numbers in each row and column are calculated.\n\nTwo of these sums are 24 and 28.\n\nThe other two sums are $c$ and $d$, where $c<d$.\n\nWhat is the value of $5 c+7 d$ ?\n\n<i... | 412 | numeric | 5 | algebra | |
2033 | [{"role": "user", "question": "Stephen's calculator displays only one digit, as shown in the diagram. Unfortunately, the calculator is broken. Each time he switches it on, each of the seven bars will either illuminate (show up) or not, with probability 0.5 . The resultant display correctly shows one of the ten digits $... | 173 | numeric | 5 | combinatorics | |
2034 | [{"role": "user", "question": "Each cell in this cross-number can be filled with a non-zero digit so that all of the conditions in the clues are satisfied. The digits used are not necessarily distinct.\n<image1>\n\\section*{ACROSS}\n1. Four less than a factor of 105.\n3. One more than a palindrome.\n5. The square-root ... | 841 | numeric | 1 | logic | |
2035 | [{"role": "user", "question": "A machine-shop cutting tool has the shape of a notched circle, as shown. The radius of the circle is $\\sqrt{50}$ cm, the length of $AB$ is 6 cm, and that of $BC$ is 2 cm. The angle $ABC$ is a right angle. Find the square of the distance (in centimeters) from $B$ to the center of the c... | 26 | numeric | 5 | metric geometry - length | |
2036 | [{"role": "user", "question": "The solid shown has a square base of side length $s$. The upper edge is parallel to the base and has length $2s$. All other edges have length $s$. Given that $s = 6 \\sqrt{2}$, what is the volume of the solid?\n<image1>", "hint": "", "options": {}, "choices": []}] | 288 | numeric | 3 | solid geometry | |
2037 | [{"role": "user", "question": "In the adjoining figure, two circles of radii 6 and 8 are drawn with their centers 12 units apart. At $P$, one of the points of intersection, a line is drawn in such a way that the chords $QP$ and $PR$ have equal length. Find the square of the length of $QP$.\n\n<image1>", "hint": "", "... | 130 | numeric | 5 | metric geometry - length | |
2038 | [{"role": "user", "question": "The adjoining figure shows two intersecting chords in a circle, with $B$ on minor arc $AD$. Suppose that the radius of the circle is 5, that $BC = 6$, and that $AD$ is bisected by $BC$. Suppose further that $AD$ is the only chord starting at $A$ which is bisected by $BC$. It follows th... | 175 | numeric | 5 | metric geometry - length | |
2039 | [{"role": "user", "question": "A point $P$ is chosen in the interior of $\\triangle ABC$ so that when lines are drawn through $P$ parallel to the sides of $\\triangle ABC$, the resulting smaller triangles, $t_1$, $t_2$, and $t_3$ in the figure, have areas 4, 9, and 49, respectively. Find the area of $\\triangle ABC$.\n... | 144 | numeric | 5 | metric geometry - area | |
204 | [{"role": "user", "question": "The digits of the sequence $123451234512345 \\ldots$ fill the cells on a sheet of paper in a spiral-like manner beginning with the marked cell (see the figure). Which digit is written in the cell being 100 cells above the marked one?\n<image1>", "hint": "", "options": {}, "choices": []}] | 1 | numeric | 5 | algebra | |
2040 | [{"role": "user", "question": "A small square is constructed inside a square of area 1 by dividing each side of the unit square into $n$ equal parts, and then connecting the vertices to the division points closest to the opposite vertices. Find the value of $n$ if the the area of the small square is exactly 1/1985.\n<... | 32 | numeric | 5 | metric geometry - area | |
2041 | [{"role": "user", "question": "As shown in the figure, triangle $ABC$ is divided into six smaller triangles by lines drawn from the vertices through a common interior point. The areas of four of these triangles are as indicated. Find the area of triangle $ABC$.\n\n<image1>", "hint": "", "options": {}, "choices": []}] | 315 | numeric | 5 | metric geometry - area | |
2042 | [{"role": "user", "question": "Three 12 cm $\\times$ 12 cm squares are each cut into two pieces $A$ and $B$, as shown in the first figure below, by joining the midpoints of two adjacent sides. These six pieces are then attached to a regular hexagon, as shown in the second figure, so as to fold into a polyhedron. What... | 864 | numeric | 3 | solid geometry | |
2043 | [{"role": "user", "question": "Rectangle $ABCD$ is divided into four parts of equal area by five segments as shown in the figure, where $XY = YB + BC + CZ = ZW = WD + DA + AX$, and $PQ$ is parallel to $AB$. Find the length of $AB$ (in cm) if $BC = 19$ cm and $PQ = 87$ cm.\n\n<image1>", "hint": "", "options": {}, "choi... | 193 | numeric | 5 | metric geometry - length | |
2044 | [{"role": "user", "question": "Triangle $ABC$ has right angle at $B$, and contains a point $P$ for which $PA = 10$, $PB = 6$, and $\\angle APB = \\angle BPC = \\angle CPA$. Find $PC$.\n\n<image1>", "hint": "", "options": {}, "choices": []}] | 33 | numeric | 5 | metric geometry - length | |
2045 | [{"role": "user", "question": "Squares $S_1$ and $S_2$ are inscribed in right triangle $ABC$, as shown in the figures below. Find $AC + CB$ if area$(S_1) = 441$ and area$(S_2) = 440$.\n\n<image1>", "hint": "", "options": {}, "choices": []}] | 462 | numeric | 5 | metric geometry - length | |
2046 | [{"role": "user", "question": "One commercially available ten-button lock may be opened by depressing -- in any order -- the correct five buttons. The sample shown below has $\\{1, 2, 3, 6, 9\\}$ as its combination. Suppose that these locks are redesigned so that sets of as many as nine buttons or as few as one butto... | 770 | numeric | 5 | combinatorics | |
2047 | [{"role": "user", "question": "It is possible to place positive integers into the vacant twenty-one squares of the $5 \\times 5$ square shown below so that the numbers in each row and column form arithmetic sequences. Find the number that must occupy the vacant square marked by the asterisk (*).\n\n<image1>", "hint": ... | 142 | numeric | 5 | algebra | |
2048 | [{"role": "user", "question": "Let $P$ be an interior point of triangle $ABC$ and extend lines from the vertices through $P$ to the opposite sides. Let $a$, $b$, $c$, and $d$ denote the lengths of the segments indicated in the figure. Find the product $abc$ if $a + b + c = 43$ and $d = 3$.\n\n<image1>", "hint": "", "... | 441 | numeric | 5 | metric geometry - length | |
2049 | [{"role": "user", "question": "Two skaters, Allie and Billie, are at points $A$ and $B$, respectively, on a flat, frozen lake. The distance between $A$ and $B$ is $100$ meters. Allie leaves $A$ and skates at a speed of $8$ meters per second on a straight line that makes a $60^\\circ$ angle with $AB$. At the same time A... | 160 | numeric | 5 | algebra | |
205 | [{"role": "user", "question": "Numbers 3,4 and two other unknown numbers are written in the cells of the $2 \\times 2$ table. It is known that the sums of numbers in the rows are equal to 5 and 10, and the sum of numbers in one of the columns is equal to 9. The larger number of the two unknown ones is\n<image1>", "hint... | 6 | numeric | 5 | algebra | |
2050 | [{"role": "user", "question": "Let $ABCD$ be a tetrahedron with $AB=41$, $AC=7$, $AD=18$, $BC=36$, $BD=27$, and $CD=13$, as shown in the figure. Let $d$ be the distance between the midpoints of edges $AB$ and $CD$. Find $d^{2}$.\n\n<image1>", "hint": "", "options": {}, "choices": []}] | 137 | numeric | 3 | solid geometry | |
2051 | [{"role": "user", "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 $\\triangle ABC$.\n\n<image1>", "hint": "", "options": {... | 108 | numeric | 5 | metric geometry - area | |
2052 | [{"role": "user", "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... | 594 | numeric | 3 | solid geometry | |
2053 | [{"role": "user", "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 ... | 135 | numeric | 5 | metric geometry - area | |
2054 | [{"role": "user", "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) ... | 792 | numeric | 5 | combinatorics | |
2055 | [{"role": "user", "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 unt... | 118 | numeric | 5 | algebra | |
2056 | [{"role": "user", "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... | 71 | numeric | 5 | transformation geometry | |
2057 | [{"role": "user", "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 $... | 255 | numeric | 5 | algebra | |
2058 | [{"role": "user", "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 $a+b$... | 616 | numeric | 5 | metric geometry - length | |
2059 | [{"role": "user", "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 $x$.\n\n<image1>", "hint": "", "options": {}, "choices": []}] | 200 | numeric | 5 | algebra | |
206 | [{"role": "user", "question": "A river starts at point $A$. As it flows the river splits into two. The first branch takes $\\frac{2}{3}$ of the water and the second takes the rest. Later the first branch splits into three, one taking $\\frac{1}{8}$ of the branch's water, the second $\\frac{5}{8}$ and the third one the ... | D | mcq | 5 | graph theory | |
2060 | [{"role": "user", "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 $m+n$.\n<image1>", "hint": "", "options": {}, "choices": []}] | 185 | numeric | 5 | transformation geometry | |
2061 | [{"role": "user", "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.\n\n<image1>", "hint": "", "options": {}, "choices": []}] | 260 | numeric | 5 | metric geometry - length | |
2062 | [{"role": "user", "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 ... | 154 | numeric | 5 | metric geometry - length | |
2063 | [{"role": "user", "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 clo... | 148 | numeric | 5 | metric geometry - area | |
2064 | [{"role": "user", "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 $n=5$.\n<image1>\nIf $n=202,$ then the area of the garden enclosed by the path, not incl... | 803 | numeric | 5 | metric geometry - area | |
2065 | [{"role": "user", "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 $m$ a... | 293 | numeric | 5 | transformation geometry | |
2066 | [{"role": "user", "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 $\\mathcal{A}$.\n\n<image1>", "h... | 408 | numeric | 5 | metric geometry - area | |
2067 | [{"role": "user", "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 intege... | 89 | numeric | 5 | metric geometry - area | |
2068 | [{"role": "user", "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... | 65 | numeric | 4 | analytic geometry | |
2069 | [{"role": "user", "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$.\n<image1>", "h... | 860 | numeric | 5 | combinatorics | |
207 | [{"role": "user", "question": "Given an isosceles triangle $A B C, C A=C B, A D=$ $=A C, D B=D C$ (see the fig.). Find the value of the angle $A C B$.\n<image1>", "hint": "", "options": {"A": "$98^{\\circ}$", "B": "$100^{\\circ}$", "C": "$104^{\\circ}$", "D": "$108^{\\circ}$", "E": "$110^{\\circ}$"}, "choices": []}] | D | mcq | 4 | metric geometry - angle | |
2070 | [{"role": "user", "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 $EF^{2}$.\n<image1>", "hint": "", "options": {}, "choices": []}] | 578 | numeric | 5 | metric geometry - length | |
2071 | [{"role": "user", "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 e... | 640 | numeric | 5 | combinatorics | |
2072 | [{"role": "user", "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 ... | 17 | numeric | 5 | algebra | |
2073 | [{"role": "user", "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). T... | 871 | numeric | 5 | metric geometry - length | |
2074 | [{"role": "user", "question": "The diagram below shows a $ 4\\times4$ rectangular array of points, each of which is $ 1$ unit away from its nearest neighbors.\n<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... | 240 | numeric | 5 | combinatorics | |
2075 | [{"role": "user", "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$. Ci... | 32 | numeric | 5 | metric geometry - length | |
2076 | [{"role": "user", "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 addit... | 318 | numeric | 5 | metric geometry - length | |
2077 | [{"role": "user", "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 s... | 280 | numeric | 5 | graph theory | |
2078 | [{"role": "user", "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 ... | 89 | numeric | 3 | solid geometry |
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