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1809 | [{"role": "user", "question": "The target shown consists of an inner black circle with two rings, one black and one white, around it. The width of each ring is equal to the radius of the black circle. What is the ratio of the area of the black ring to the area of the inner black circle? <image1>", "hint": "", "options"... | D | mcq | 4 | metric geometry - area | |
181 | [{"role": "user", "question": "We are given three semi-circles as shown. $A B E F$ is a rectangle and the radius of each of the semi-circles is $2 \\mathrm{~cm}$. $E$ and $F$ are the centers of the bottom semi-circles. The area of the shaded region (in $\\mathrm{cm}^{2}$) is:\n<image1>", "hint": "", "options": {}, "cho... | 8 | numeric | 5 | metric geometry - area | |
1810 | [{"role": "user", "question": "A chain is made from circular links with external radius $3 \\mathrm{~cm}$ and internal radius $2 \\mathrm{~cm}$. When the rings are linked together as shown in the diagram, the length of the chain is $1.7 \\mathrm{~m}$. How many rings are used? <image1>", "hint": "", "options": {}, "choi... | 42 | numeric | 4 | arithmetic | |
1811 | [{"role": "user", "question": "Sol is having fun playing with water in two tanks. Tank $X$ has a base of area of $200 \\mathrm{~cm}^{2}$.\nTank $Y$ has a base of area $100 \\mathrm{~cm}^{2}$ and height $7 \\mathrm{~cm}$. Sol has partly filled Tank X to a depth of $5 \\mathrm{~cm}$. He then places Tank Y, which is empty... | C | mcq | 2 | solid geometry | |
1812 | [{"role": "user", "question": "Zoli wants to make a bench for his garden from some tree trunks sawn in half, as shown in the picture. The diameters of the two bottom trunks are 20 centimetres, and the diameter of the top trunk is 40 centimetres. What is the height of the bench in centimetres? <image1>", "hint": "", "op... | B | mcq | 4 | metric geometry - length | |
1813 | [{"role": "user", "question": "Barney has 16 cards: 4 blue $(B), 4$ red $(R), 4$ green $(G)$ and 4 yellow (Y). He wants to place them in the square shown so that every row and every column has exactly one of each card. The diagram shows how he started. How many different ways can he finish?\n<image1>", "hint": "", "opt... | 4 | numeric | 4 | combinatorics | |
1814 | [{"role": "user", "question": "Andrew, the absent-minded mountain climber, climbed a mountain range, with a profile as shown in Figure 1 from point $P$ to point $Q$. From time to time he had to turn back to find bits of his equipment that he had dropped. The graph of the height $H$ of his position at time $t$ is shown ... | 3 | numeric | 1 | statistics | |
1815 | [{"role": "user", "question": "The diagram shows a square and an equilateral right-angled crossshaped dodecagon. The length of the perimeter of the dodecagon is $36 \\mathrm{~cm}$. What is the area of the square in $\\mathrm{cm}^{2}$ ? <image1>", "hint": "", "options": {}, "choices": []}] | 72 | numeric | 4 | metric geometry - area | |
1816 | [{"role": "user", "question": "The shaded area is equal to $2 \\pi$. What is the length of $P Q$ ? <image1>", "hint": "", "options": {}, "choices": []}] | 4 | numeric | 4 | metric geometry - length | |
1817 | [{"role": "user", "question": "In the grid on the right, there are eight kangaroos. A kangaroo may jump into any empty square. Find the least number of the kangaroos which have to jump into an empty square so that in each row and column there are exactly two kangaroos. <image1>", "hint": "", "options": {}, "choices": [... | 1 | numeric | 5 | logic | |
1818 | [{"role": "user", "question": "In the diagram alongside, the five circles have the same radii and touch as shown. The square joins the centres of the four outer circles. <image1>\nThe ratio of the area of the shaded parts of all five circles to the area of the unshaded parts of all five circles is", "hint": "", "option... | B | mcq | 4 | metric geometry - area | |
1819 | [{"role": "user", "question": "In the diagram there are 7 squares. What is the difference between the number of triangles and the number of squares in the diagram? <image1>", "hint": "", "options": {}, "choices": []}] | 3 | numeric | 3 | combinatorial geometry | |
182 | [{"role": "user", "question": "The diagram shows a rectangle $A B E F$ and a triangle $A B C$. We know that the angle $A C F$ equals angle $C B E$. If $F C=6$ and $C E=2$ then the area of $A B C$ is:\n<image1>", "hint": "", "options": {"A": "12", "B": "16", "C": "$8 \\sqrt{2}$", "D": "$8 \\sqrt{3}$", "E": "Another valu... | D | mcq | 5 | metric geometry - area | |
1820 | [{"role": "user", "question": "Kanga and Roo are hopping around a stadium with a perimeter of $330 \\mathrm{~m}$. Each of them makes one jump every second.\nKanga's jumps are $5 \\mathrm{~m}$ long, while Roo's jumps are $2 \\mathrm{~m}$ long. They both start at the same point and move in the same direction. Roo gets ti... | C | mcq | 3 | algebra | |
1821 | [{"role": "user", "question": "What entry should replace $x$ in the table so that the numbers in each row, each column and each diagonal form an arithmetic sequence?\n(In an arithmetic sequence, there is a constant difference between successive terms.) <image1>", "hint": "", "options": {}, "choices": []}] | 42 | numeric | 3 | algebra | |
1822 | [{"role": "user", "question": "The diagram shows 3 semicircular arcs with the endpoints $A, B$ of one arc and the centres $E, F$ of the other two arcs at the vertices of a rectangle. What is the area of the shaded region when the radius of each semicircle is $2 \\mathrm{~cm}$ ? <image1>", "hint": "", "options": {"A": "... | B | mcq | 4 | metric geometry - area | |
1823 | [{"role": "user", "question": "What is the sum of the 10 angles marked on the diagram on the right? <image1>", "hint": "", "options": {"A": "$300^{\\circ}$", "B": "$450^{\\circ}$", "C": "$360^{\\circ}$", "D": "$600^{\\circ}$", "E": "$720^{\\circ}$"}, "choices": []}] | E | mcq | 2 | metric geometry - angle | |
1824 | [{"role": "user", "question": "The numbers on each pair of opposite faces on a die add up to 7 . A die is rolled without slipping around the circuit shown. At the start the top face is 3 . What number will be displayed on the top face at the end point? <image1>", "hint": "", "options": {}, "choices": []}] | 6 | numeric | 2 | solid geometry | |
1825 | [{"role": "user", "question": "How many ways are there to choose a white square and a black square, such as those shown, from an $8 \\times 8$ chess board so that these squares do not lie in either the same row or the same column? <image1>", "hint": "", "options": {}, "choices": []}] | 768 | numeric | 4 | combinatorics | |
1826 | [{"role": "user", "question": "The flag shown in the diagram consists of three stripes, each of equal height, which are divided into two, three and four equal parts, respectively. What fraction of the area of the flag is shaded? <image1>", "hint": "", "options": {"A": "$\\frac{1}{2}$", "B": "$\\frac{2}{3}$", "C": "$\\f... | E | mcq | 4 | arithmetic | |
1827 | [{"role": "user", "question": "The circle shown in the diagram is divided into four arcs of length 2, 5, 6 and $x$ units. The sector with arc length 2 has an angle of $30^{\\circ}$ at the centre. Determine the value of $x$. <image1>", "hint": "", "options": {}, "choices": []}] | 11 | numeric | 4 | metric geometry - length | |
1828 | [{"role": "user", "question": "The rectangle shown is divided into six squares. The length of the sides of the smallest square is 1 . What is the length of the sides of the largest square? <image1>", "hint": "", "options": {}, "choices": []}] | 7 | numeric | 4 | metric geometry - length | |
1829 | [{"role": "user", "question": "Each letter in the sum shown represents a different digit and the digit for $\\mathrm{A}$ is odd. What digit does $\\mathrm{G}$ represent?\n<image1>", "hint": "", "options": {}, "choices": []}] | 9 | numeric | 3 | algebra | |
183 | [{"role": "user", "question": "In the quadrilateral $A B C D$ the diagonal $B D$ is the bisector of $\\angle A B C$ and $A C=B C$. Given $\\angle B D C=80^{\\circ}$ and $\\angle A C B=20^{\\circ}, \\angle B A D$ is equal to:\n<image1>", "hint": "", "options": {"A": "$90^{\\circ}$", "B": "$100^{\\circ}$", "C": "$110^{\\... | D | mcq | 4 | metric geometry - angle | |
1830 | [{"role": "user", "question": "Two identical equilateral triangles overlap with their sides parallel, so that the overlapping region is the hexagon shown shaded in the diagram. The perimeter length of each triangle is 18 . What is the perimeter length of the shaded hexagon? <image1>", "hint": "", "options": {}, "choice... | 12 | numeric | 4 | transformation geometry | |
1831 | [{"role": "user", "question": "Two squares with side 1 have a common vertex, and the edge of one of them lies along the diagonal of the other. What is the area of the overlap between the squares? <image1>", "hint": "", "options": {"A": "$\\sqrt{2}-1$", "B": "$\\frac{\\sqrt{2}}{2}$", "C": "$\\frac{\\sqrt{2}+1}{2}$", "D"... | A | mcq | 4 | transformation geometry | |
1832 | [{"role": "user", "question": "A square $P Q R S$ with sides of length 10 is rolled without slipping along a line. Initially $P$ and $Q$ are on the line and the first roll is around point $Q$ as shown in the diagram. The rolling stops when $P$ first returns to the line. What is the length of the curve that $P$ has trav... | C | mcq | 4 | transformation geometry | |
1833 | [{"role": "user", "question": "The diagram shows two ordinary dice. What is the total number of spots on all the faces that cannot be seen in the diagram? <image1>", "hint": "", "options": {}, "choices": []}] | 27 | numeric | 2 | solid geometry | |
1834 | [{"role": "user", "question": "To complete the table, each cell must contain either 0 or 1 , and the total of each row and column must be 2 . What are the values of the entries $X$ and $Y$ ? <image1>", "hint": "", "options": {"A": "$X=0, Y=0$", "B": "$X=0, Y=1$", "C": "$X=1, Y=0$", "D": "$X=1, Y=1$", "E": "It is imposs... | A | mcq | 5 | logic | |
1835 | [{"role": "user", "question": "In the calculation alongside, different letters represent different digits.\n<image1>\nFind the least possible answer to the subtraction shown.", "hint": "", "options": {}, "choices": []}] | 110 | numeric | 3 | algebra | |
1836 | [{"role": "user", "question": "The diagram shows a triangle $J K L$ where two lines are drawn from each of the vertices $J$ and $K$ to points on the opposite sides. This divides the triangle into nine nonoverlapping sections. If instead, eight lines are drawn to the opposite sides, four from $J$ and four from $K$, how ... | 25 | numeric | 3 | combinatorial geometry | |
1837 | [{"role": "user", "question": "A dog is tied to the outside corner of a house by a rope of length $10 \\mathrm{~m}$. The house is a rectangle with sides of length $6 \\mathrm{~m}$ and $4 \\mathrm{~m}$. What is the length (in metres) of the curved boundary of the area in which the dog can roam? <image1>", "hint": "", "o... | A | mcq | 4 | metric geometry - length | |
1838 | [{"role": "user", "question": "A coin with diameter $1 \\mathrm{~cm}$ rolls around the outside of a regular hexagon with edges of length $1 \\mathrm{~cm}$ until it returns to its original position. In centimetres, what is the length of the path traced out by the centre of the coin? <image1>", "hint": "", "options": {"A... | C | mcq | 4 | transformation geometry | |
1839 | [{"role": "user", "question": "An equilateral triangle and a regular hexagon are inscribed in a circle which is itself inscribed in an equilateral triangle. $L$ is the area of the large triangle, $S$ is the area of the smaller triangle and $H$ is the area of the hexagon. Which of these statements is true? <image1>", "h... | E | mcq | 4 | metric geometry - area | |
184 | [{"role": "user", "question": "We consider the perimeter and the area of the region corresponding to the grey squares. How many more squares can we colour grey for the grey area to increase without increasing its perimeter?\n<image1>", "hint": "", "options": {}, "choices": []}] | 16 | numeric | 5 | combinatorial geometry | |
1840 | [{"role": "user", "question": "Two circles have their centres on the same diagonal of a square. They touch each other and the sides of the square as shown. The square has side length $1 \\mathrm{~cm}$. What is the sum of the radii of the circles in centimetres? <image1>", "hint": "", "options": {"A": "$\\frac{1}{2}$", ... | D | mcq | 4 | metric geometry - length | |
1841 | [{"role": "user", "question": "Five boxes contain cards as shown. Simon removes cards so that each box contains exactly one card, and the five cards remaining in the boxes can be used to spell his name. Which card remains in box 2 ?\n<image1>", "hint": "", "options": {"A": "S", "B": "I", "C": "M", "D": "O", "E": "N"}, ... | D | mcq | 4 | combinatorics | |
1842 | [{"role": "user", "question": "Four unit squares are placed edge to edge as shown. What is the length of the line $P Q$ ? <image1>", "hint": "", "options": {"A": "5", "B": "$\\sqrt{13}$", "C": "$\\sqrt{5}+\\sqrt{2}$", "D": "$\\sqrt{5}$", "E": "13"}, "choices": []}] | B | mcq | 4 | metric geometry - length | |
1843 | [{"role": "user", "question": "One face of a cardboard cube is cut along its diagonals, as shown.\nWhich of the following are not nets for this cube?\n<image1>\n<image2>", "hint": "", "options": {"A": "1 and 3", "B": "1 and 5", "C": "2 and 4", "D": "2 and 4", "E": "3 and 5"}, "choices": []}] | E | mcq | 2 | solid geometry | |
1844 | [{"role": "user", "question": "A parallelogram contains two identical regular hexagons. The hexagons share a common side, and each has two sides touching the sides of the parallelogram. What fraction of the parallelogram's area is shaded? <image1>", "hint": "", "options": {"A": "$\\frac{2}{3}$", "B": "$\\frac{1}{2}$", ... | B | mcq | 3 | combinatorial geometry | |
1845 | [{"role": "user", "question": "On the number line below, each gap equals one unit. Six integers are marked as shown. At least two of the integers are divisible by 3 , and at least two of them are divisible by 5 . Which of the integers are divisible by 15 ?\n<image1>", "hint": "", "options": {"A": "$F$ and $K$", "B": "$... | A | mcq | 3 | algebra | |
1846 | [{"role": "user", "question": "In the diagram, triangle $J K L$ is isosceles with $J K=J L, P Q$ is perpendicular to $J K$, angle $K P L$ is $120^{\\circ}$ and angle $J K P$ is $50^{\\circ}$. What is the size of angle $P K L$ ? <image1>", "hint": "", "options": {"A": "$5^{\\circ}$", "B": "$10^{\\circ}$", "C": "$15^{\\c... | A | mcq | 2 | metric geometry - angle | |
1847 | [{"role": "user", "question": "Three circles $C_{1}, C_{2}$ and $C_{3}$ of radii $1 \\mathrm{~cm}, 2 \\mathrm{~cm}$ and $3 \\mathrm{~cm}$ respectively touch as shown. $C_{1}$ meets $C_{2}$ at $P$ and meets $C_{3}$ at $Q$. What is the length in centimetres of the longer arc of circle $C_{1}$ between $P$ and $Q$ ? <image... | E | mcq | 4 | metric geometry - length | |
1848 | [{"role": "user", "question": "The diagram shows the net of a regular octahedron. In a Magic Octahedron, the four numbers on the faces that meet at a vertex add up to make the same total for every vertex. If the letters $F, G, H, J$ and $K$ are replaced with the numbers $2,4,6,7$, and 8 , in some order, to make a Magic... | 6 | numeric | 3 | algebra | |
1849 | [{"role": "user", "question": "An $n$-pyramid is defined to be a stack of $n$ layers of balls, with each layer forming a triangular array. The layers of a 3-pyramid are shown in the diagram.\nAn 8-pyramid is now formed where all the balls on the outside of the 8 -pyramid are black (including the base layer) and the bal... | 4 | numeric | 2 | solid geometry | |
185 | [{"role": "user", "question": "There are four cards on the table as in the picture. Every card has a letter on one side and a number on the other side. Peter said: \"For every card on the table it is true that if there is a vowel on one side, there is an even number on the other side.\" What is the smallest number of c... | 2 | numeric | 1 | logic | |
1850 | [{"role": "user", "question": "Sixteen unit squares are arranged to form a square array as shown in the diagram. What is the maximum number of diagonals that can be drawn in these unit squares so that no two diagonals share a common point (including endpoints)?\n<image1>", "hint": "", "options": {}, "choices": []}] | 10 | numeric | 4 | combinatorics | |
1851 | [{"role": "user", "question": "In the diagram, $K L M N$ is a unit square. Arcs of radius one unit are drawn using each of the four corners of the square as centres. The arcs centred at $K$ and $L$ intersect at $Q$; the arcs centred at $M$ and $N$ intersect at $P$. What is the length of $P Q$ ? <image1>", "hint": "", "... | E | mcq | 4 | metric geometry - length | |
1852 | [{"role": "user", "question": "The diagram shows a triangle and three circles whose centres are at the vertices of the triangle. The area of the triangle is $80 \\mathrm{~cm}^{2}$ and each of the circles has radius $2 \\mathrm{~cm}$. What is the area, in $\\mathrm{cm}^{2}$, of the shaded area? <image1>", "hint": "", "o... | B | mcq | 4 | metric geometry - area | |
1853 | [{"role": "user", "question": "The three angle bisectors of triangle $L M N$ meet at a point $O$ as shown. Angle $L N M$ is $68^{\\circ}$. What is the size of angle $L O M$ ? <image1>", "hint": "", "options": {"A": "$120^{\\circ}$", "B": "$124^{\\circ}$", "C": "$128^{\\circ}$", "D": "$132^{\\circ}$", "E": "$136^{\\circ... | B | mcq | 2 | metric geometry - angle | |
1854 | [{"role": "user", "question": "The diagram shows two identical large circles and two identical smaller circles whose centres are at the corners of a square. The two large circles are touching, and they each touch the two smaller circles. The radius of the small circles is $1 \\mathrm{~cm}$. What is the radius of a larg... | A | mcq | 4 | metric geometry - length | |
1855 | [{"role": "user", "question": "Three circular hoops are joined together so that they intersect at rightangles as shown. A ladybird lands on an intersection and crawls around the outside of the hoops by repeating this procedure: she travels along a quarter-circle, turns $90^{\\circ}$ to the right, travels along a quarte... | 6 | numeric | 2 | solid geometry | |
1856 | [{"role": "user", "question": "Barbara wants to place draughts on a $4 \\times 4$ board in such a way that the number of draughts in each row and in each column are all different (she may place more than one draught in a square, and a square may be empty). What is the smallest number of draughts that she would need? <i... | 14 | numeric | 4 | combinatorics | |
1857 | [{"role": "user", "question": "Four cubes, each with surface area $24 \\mathrm{~cm}^{2}$, are placed together to form a cuboid as shown. What is the surface area of this cuboid, in $\\mathrm{cm}^{2}$ ? <image1>", "hint": "", "options": {}, "choices": []}] | 64 | numeric | 2 | solid geometry | |
1858 | [{"role": "user", "question": "A rectangular strip of paper is folded in half three times, with each fold line parallel to the short edges. It is then unfolded so that the seven folds up or down can all be seen. Which of the following strips, viewed from a long edge, could not be made in this way? <image1>", "hint": ""... | D | mcq | 4 | transformation geometry | |
1859 | [{"role": "user", "question": "Six points are marked on a sheet of squared paper as shown. Which of the following shapes cannot be made by connecting some of these points using straight lines? <image1>", "hint": "", "options": {"A": "parallelogram", "B": "trapezium", "C": "right-angled triangle", "D": "obtuse-angled tr... | E | mcq | 3 | combinatorial geometry | |
186 | [{"role": "user", "question": "Susan has two pendants made of the same material. They are equally thick and weigh the same. One of them has the shape of an annulus created from two concentric circles with the radii $6 \\mathrm{~cm}$ and $4 \\mathrm{~cm}$ (see the diagram). The second has the shape of a solid circle. Wh... | D | mcq | 5 | metric geometry - length | |
1860 | [{"role": "user", "question": "The diagram shows a square $P Q R S$ and two equilateral triangles $R S U$ and PST. PQ has length 1 . What is the length of $T U$ ? <image1>", "hint": "", "options": {"A": "$\\sqrt{2}$", "B": "$\\frac{\\sqrt{3}}{2}$", "C": "$\\sqrt{3}$", "D": "$\\sqrt{5}-1$", "E": "$\\sqrt{6}-1$"}, "choic... | A | mcq | 4 | metric geometry - length | |
1861 | [{"role": "user", "question": "In the diagram, angle $P Q R$ is $20^{\\circ}$, and the reflex angle at $P$ is $330^{\\circ}$. The line segments $Q T$ and $S U$ are perpendicular. What is the size of angle RSP? <image1>", "hint": "", "options": {"A": "$10^{\\circ}$", "B": "$20^{\\circ}$", "C": "$30^{\\circ}$", "D": "$40... | D | mcq | 2 | metric geometry - angle | |
1862 | [{"role": "user", "question": "A circle of radius $4 \\mathrm{~cm}$ is divided into four congruent parts by arcs of radius $2 \\mathrm{~cm}$ as shown. What is the length of the perimeter of one of the parts, in $\\mathrm{cm}$ ? <image1>", "hint": "", "options": {"A": "$2 \\pi$", "B": "$4 \\pi$", "C": "$6 \\pi$", "D": "... | C | mcq | 4 | metric geometry - length | |
1863 | [{"role": "user", "question": "The scatter graph shows the distance run and time taken by five students during a training session. Who ran with the fastest average speed? <image1>", "hint": "", "options": {"A": "Alicia", "B": "Bea", "C": "Carlos", "D": "Dani", "E": "Ernesto"}, "choices": []}] | D | mcq | 5 | analytic geometry | |
1864 | [{"role": "user", "question": "A triangular piece of paper is folded along the dotted line shown in the left-hand diagram to form the heptagon shown in the right-hand diagram. The total area of the shaded parts of the heptagon is $1 \\mathrm{~cm}^{2}$. The area of the original triangle is $1 \\frac{1}{2}$ times the are... | 3 | numeric | 4 | transformation geometry | |
1865 | [{"role": "user", "question": "In a supermarket trolley park, there are two lines of tightly-packed trolleys. The first line has ten trolleys and is $2.9 \\mathrm{~m}$ long. The second line has twenty trolleys and is $4.9 \\mathrm{~m}$ long. What is the length of one trolley, in $\\mathrm{m}$ ? <image1>", "hint": "", "... | 1.1 | numeric | 3 | algebra | |
1866 | [{"role": "user", "question": "The diagram shows a trapezium $F G H I$ with $F G$ parallel to $I H$. GH and FI both have length 2. The point $M$ is the midpoint of $F I$ and $\\angle H M G=90^{\\circ}$. What is the length of the perimeter of the trapezium? <image1>", "hint": "", "options": {}, "choices": []}] | 6 | numeric | 4 | metric geometry - length | |
1867 | [{"role": "user", "question": "The diagram shows a square with sides of length 2. Four semicircles are drawn whose centres are the four vertices of the square. These semicircles meet at the centre of the square, and adjacent semicircles meet at their ends. Four circles are drawn whose centres lie on the edges of the sq... | A | mcq | 4 | metric geometry - area | |
1868 | [{"role": "user", "question": "An oval is constructed from four arcs of circles. Arc $P Q$ is the same as arc $R S$, and has radius $1 \\mathrm{~cm}$. Arc $Q R$ is the same as arc PS. At the points $P, Q, R, S$ where the arcs touch, they have a common tangent. The oval touches the midpoints of the sides of a rectangle ... | 6 | numeric | 4 | metric geometry - length | |
1869 | [{"role": "user", "question": "A bar code of the type shown is composed of alternate strips of black and white, always beginning and ending with a black strip. Each strip in the bar code has width either 1 or 2 , and the total width of the bar code is 12 . Two bar codes are different if they read differently from left ... | 116 | numeric | 4 | combinatorics | |
187 | [{"role": "user", "question": "In the diagram, $A B$ has length $1 ; \\angle A B C=\\angle A C D=90^{\\circ}$; $\\angle C A B=\\angle D A C=\\theta$. What is the length of $A D$?\n<image1>", "hint": "", "options": {"A": "$\\cos \\beta+\\tg \\beta$", "B": "$\\frac{1}{\\cos (2 \\beta)}$", "C": "$\\cos ^{2} \\beta$", "D":... | E | mcq | 5 | metric geometry - length | |
1870 | [{"role": "user", "question": "The figure shows a hexagonal lattice. Numbers are to be placed at each of the dots $\\cdot$ in such a way that the sum of the two numbers at the ends of each segment is always the same. Two of the numbers are already given. What number is $x$ ? <image1>", "hint": "", "options": {}, "choic... | 1 | numeric | 4 | graph theory | |
1871 | [{"role": "user", "question": "One of the line segments shown on the grid is the image produced by a rotation of the other line segment. Which of the points $T, U, V$, $W$ could be the centre of such a rotation? <image1>", "hint": "", "options": {"A": "only $T$", "B": "only $U$", "C": "either of $U$ and $W$", "D": "any... | C | mcq | 4 | transformation geometry | |
1872 | [{"role": "user", "question": "The diagram shows a shape made from a regular hexagon of side one unit, six triangles and six squares. What is the perimeter of the shape? <image1>", "hint": "", "options": {"A": "$6(1+\\sqrt{2})$", "B": "$6\\left(1+\\frac{1}{2} \\sqrt{3}\\right)$", "C": "$12$", "D": "$6+3 \\sqrt{2}$", "E... | C | mcq | 4 | metric geometry - length | |
1873 | [{"role": "user", "question": "A marble with radius $15 \\mathrm{~cm}$ fits exactly under a cone as shown in the diagram. The slant height of the cone is equal to the diameter of its base. What is the height of the cone in $\\mathrm{cm}$ ? <image1>", "hint": "", "options": {}, "choices": []}] | 45 | numeric | 2 | solid geometry | |
1874 | [{"role": "user", "question": "Barbara wants to place draughts on a $4 \\times 4$ board in such a way that the number of draughts in each row is equal to the number shown at the end of the row, and the number of draughts in each column is equal to the number shown at the bottom of the column. No more than one draught i... | 5 | numeric | 4 | combinatorics | |
1875 | [{"role": "user", "question": "Nik wants to write integers in the cells of a $3 \\times 3$ table so that the sum of the numbers in any $2 \\times 2$ square is 10 . He has already written five numbers in the table as shown. What is the sum of the four missing numbers?\n<image1>", "hint": "", "options": {}, "choices": []... | 12 | numeric | 3 | algebra | |
1876 | [{"role": "user", "question": "During a rough sailing trip, Jacques tried to sketch a map of his village. He managed to draw the four streets, the seven places where they cross and the houses of his friends. The houses are marked on the correct streets, and the intersections are correct, however, in reality, Arrow Stre... | A | mcq | 2 | topology | |
1877 | [{"role": "user", "question": "Simone has a cube with sides of length $10 \\mathrm{~cm}$, and a pack of identical square stickers. She places one sticker in the centre of each face of the cube, and one across each edge so that the stickers meet at their corners, as shown in the diagram. What is the total area in $\\mat... | 225 | numeric | 2 | solid geometry | |
1878 | [{"role": "user", "question": "In triangle $P Q R$, a point $S$ is chosen on the line segment $Q R$, then a point $T$ is chosen on the line segment $P S$. Considering the nine marked angles, what is the smallest number of different values that these nine angles could take? <image1>", "hint": "", "options": {}, "choices... | 3 | numeric | 3 | combinatorial geometry | |
1879 | [{"role": "user", "question": "A cuboid is made of four pieces as shown. Each piece consists of four cubes and is a single colour. What is the shape of the white piece? <image1>\n<image2>", "hint": "", "options": {"A": "A", "B": "B", "C": "C", "D": "D", "E": "E"}, "choices": []}] | C | mcq | 3 | combinatorial geometry | |
188 | [{"role": "user", "question": "The radius of the traffic sign is $20 \\mathrm{~cm}$. Each of the dark pieces is a quarter of a circle. The area of all 4 quarters equals that of the light part of the sign. What is the radius of this circle in centimetres?\n<image1>", "hint": "", "options": {"A": "$10 \\sqrt{2}$", "B": "... | A | mcq | 5 | metric geometry - length | |
1880 | [{"role": "user", "question": "The triangle $H I J$ has the same area as the square $F G H I$, whose sides are of length $4 \\mathrm{~cm}$. What is the perpendicular distance, in $\\mathrm{cm}$, of the point $J$ from the line extended through $F$ and $G$ ? <image1>", "hint": "", "options": {}, "choices": []}] | 12 | numeric | 4 | metric geometry - length | |
1881 | [{"role": "user", "question": "Six identical circles fit together tightly in a rectangle of width $6 \\mathrm{~cm}$ as shown. What is the height, in $\\mathrm{cm}$, of the rectangle? <image1>", "hint": "", "options": {"A": "5", "B": "$2 \\sqrt{3}+2$", "C": "$3 \\sqrt{2}$", "D": "$3 \\sqrt{3}$", "E": "6"}, "choices": []... | B | mcq | 4 | metric geometry - length | |
1882 | [{"role": "user", "question": "The right-angled triangle shown has sides of length $5 \\mathrm{~cm}, 12$ $\\mathrm{cm}$ and $13 \\mathrm{~cm}$. What, in $\\mathrm{cm}$, is the radius of the inscribed semicircle whose diameter lies on the side of length $12 \\mathrm{~cm}$ ? <image1>", "hint": "", "options": {"A": "$8 / ... | B | mcq | 4 | metric geometry - length | |
1883 | [{"role": "user", "question": "Numbers are to be placed into the table shown, one number in each cell, in such a way that each row has the same total, and each column has the same total. Some of the numbers are already given. What number is $x$ ? <image1>", "hint": "", "options": {}, "choices": []}] | 4 | numeric | 3 | algebra | |
1884 | [{"role": "user", "question": "A rectangular piece of paper $F G H I$ with sides of length $4 \\mathrm{~cm}$ and $16 \\mathrm{~cm}$ is folded along the line $M N$ so that the vertex $G$ coincides with the vertex $I$ as shown. The outline of the paper now makes a pentagon $F M N H^{\\prime} I$. What is the area, in $\\m... | 47 | numeric | 4 | transformation geometry | |
1885 | [{"role": "user", "question": "The diagram shows a square with sides of length $4 \\mathrm{~mm}$, a square with sides of length $5 \\mathrm{~mm}$, a triangle with area $8 \\mathrm{~mm}^{2}$, and a parallelogram. What is the area, in $\\mathrm{mm}^{2}$, of the parallelogram? <image1>", "hint": "", "options": {}, "choice... | 16 | numeric | 4 | metric geometry - area | |
1886 | [{"role": "user", "question": "Carlos creates a game. The diagram shows the board for the game. At the start, the kangaroo is at the school $(\\mathrm{S})$. According to the rules of the game, from any position except home $(\\mathrm{H})$, the kangaroo can jump to either of the two neighbouring positions. When the kang... | 64 | numeric | 4 | graph theory | |
1887 | [{"role": "user", "question": "The diagram shows six identical squares, each containing a shaded region.\n<image1> How many of the regions have perimeter equal in length to the perimeter of one of the squares?", "hint": "", "options": {}, "choices": []}] | 4 | numeric | 4 | metric geometry - length | |
1888 | [{"role": "user", "question": "The outside of a $2 \\times 2 \\times 2$ cube is painted with black and white squares in such a way that it appears as if it was built using alternate black cubes and white cubes, as shown. Which of the following is a net of the painted cube?\n<image1>\n<image2>", "hint": "", "options": {... | E | mcq | 2 | solid geometry | |
1889 | [{"role": "user", "question": "The trapezium shown in the diagram is rotated anti-clockwise by $90^{\\circ}$ around the origin $O$, and then reflected in the $x$-axis. Which of the following shows the end result of these transformations? <image1>\n<image2>", "hint": "", "options": {"A": "A", "B": "B", "C": "C", "D": "D... | A | mcq | 5 | analytic geometry | |
189 | [{"role": "user", "question": "The ratio of the radii of the sector and the incircle in the picture is $3: 1$. Than the ratio of their areas is:\n<image1>", "hint": "", "options": {"A": "$3: 2$", "B": "$4: 3$", "C": "$\\sqrt{3}: 1$", "D": "$2: 1$", "E": "$9: 1$"}, "choices": []}] | A | mcq | 5 | metric geometry - area | |
1890 | [{"role": "user", "question": "The diagram shows an equilateral triangle RST and also the triangle $T U V$ obtained by rotating triangle $R S T$ about the point $T$. Angle $R T V=70^{\\circ}$. What is angle $R S V$ ? <image1>", "hint": "", "options": {"A": "$20^{\\circ}$", "B": "$25^{\\circ}$", "C": "$30^{\\circ}$", "D... | D | mcq | 4 | transformation geometry | |
1891 | [{"role": "user", "question": "The diagram shows a shape made from six squares, each measuring $1 \\mathrm{~cm}$ by $1 \\mathrm{~cm}$. The shape has perimeter of length $14 \\mathrm{~cm}$. The zigzag shape is then continued until it has 2013 squares. What is the length, in $\\mathrm{cm}$, of the perimeter of the new sh... | 4028 | numeric | 3 | combinatorial geometry | |
1892 | [{"role": "user", "question": "The points $P$ and $Q$ are opposite vertices of a regular hexagon and the points $R$ and $S$ are midpoints of opposite edges, as shown. The area of the hexagon is $60 \\mathrm{~cm}^{2}$. What is the product of the lengths, in cms, of $P Q$ and $R S$ ? <image1>", "hint": "", "options": {},... | 80 | numeric | 4 | metric geometry - length | |
1893 | [{"role": "user", "question": "The rectangle $A B C D$ lies below the $x$-axis, and to the left of the $y$-axis. The edges of the rectangle are parallel to the coordinate axes. For each point $A, B, C, D$, the $y$-coordinate is divided by the $x$-coordinate. Which of the points yields the smallest value from this calcu... | A | mcq | 5 | analytic geometry | |
1894 | [{"role": "user", "question": "In quadrilateral $P Q R S, \\angle P Q R=59^{\\circ}, \\angle R P Q=60^{\\circ}$, $\\angle P R S=61^{\\circ}$ and $\\angle R S P=60^{\\circ}$, as shown. Which of the following line segments is the longest? <image1>", "hint": "", "options": {"A": "$P Q$", "B": "$P R$", "C": "$P S$", "D": "... | A | mcq | 4 | metric geometry - length | |
1895 | [{"role": "user", "question": "How many different paths are there between points $P$ and $Q$, only travelling along the edges in the direction of the arrows shown? <image1>", "hint": "", "options": {}, "choices": []}] | 12 | numeric | 4 | combinatorics | |
1896 | [{"role": "user", "question": "Several non-overlapping isosceles triangles have vertex $O$ in common. Every triangle shares an edge with each immediate neighbour. The smallest of the angles at $O$ has size $m^{\\circ}$, where $m$ is a positive integer and the other triangles have angles at $O$ of size $2 m^{\\circ}, 3 ... | 3 | numeric | 4 | combinatorics | |
1897 | [{"role": "user", "question": "If $r, s$, and $t$ denote the lengths of the 'lines' in the picture, then which of the following inequalities is correct? <image1>", "hint": "", "options": {"A": "$r<s<t$", "B": "$r<t<s$", "C": "$s<r<t$", "D": "$s<t<r$", "E": "$t<s<r$"}, "choices": []}] | E | mcq | 3 | combinatorial geometry | |
1898 | [{"role": "user", "question": "The big wheel of this penny-farthing bicycle has perimeter 4.2 metres. The small wheel has perimeter 0.9 metres. At a certain moment, the valves of both wheels are at their lowest points. The bicycle begins to roll.\nHow many metres will the bicycle have rolled forward when both valves ar... | 12.6 | numeric | 4 | transformation geometry | |
1899 | [{"role": "user", "question": "The diagram shows a regular octagon, with a line drawn between two of its vertices. The shaded area measures $3 \\mathrm{~cm}^{2}$.\nWhat is the area of the octagon in square centimetres? <image1>", "hint": "", "options": {}, "choices": []}] | 12 | numeric | 4 | metric geometry - area |
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