id stringlengths 1 4 | question stringlengths 31 709 | options listlengths 6 6 | image_path stringlengths 12 15 | image imagewidth (px) 56 2.52k | answer stringclasses 5
values | solution stringclasses 259
values | level int64 1 5 | subject stringclasses 16
values | Answer(Option) stringclasses 5
values |
|---|---|---|---|---|---|---|---|---|---|
1778 | In the diagram $P Q R S$ is a rhombus. Point $T$ is the mid-point of $P S$ and point $W$ is the mid-point of $S R$.
<image1>
What is the ratio of the unshaded area to the shaded area? | [
"$1: 1$",
"$2: 3$",
"$3: 5$",
"$4: 7$",
"$5: 9$",
"I don't know"
] | images/1778.jpg | A | First draw in the line segment $Q S$ as shown. Since we are told that $T$ is the mid-point of $P S$, the triangles $P T Q$ and $T S Q$ have equal bases. Since they also have the same perpendicular height, their areas are equal. Similarly, since $W$ is the mid-point of $S R$, we have ![](https://cdn.mathpix.com/croppe... | 2 | metric geometry - area | A | |
1780 | Claudette has eight dice, each with one of the letters $P, Q, R$ and $S$ written on all six faces. She builds the block shown in the diagram so that dice with faces which touch have different letters written on them.
What letter is written on the faces of the one dice which is not shown on the picture? <image1> | [
"P",
"Q",
"R",
"S",
"It is impossible to say",
"I don't know"
] | images/1780.jpg | B | The dice above the hidden one has $\mathrm{R}$ marked on it, while the dice on the bottom level that touch the hidden dice have $\mathrm{P}$ and $\mathrm{S}$ marked on them. Hence the missing dice can have none of these letters on it. Therefore the missing dice has $\mathrm{Q}$ marked on its faces. | 2 | logic | B | |
1783 | In the diagram, lines $Q T$ and $R S$ are parallel and $P Q$ and $Q T$ are equal. Angle $S T Q$ is $154^{\circ}$. What is the size of angle $S R Q$ ? <image1> | [
"$120^{\\circ}$",
"$122^{\\circ}$",
"$124^{\\circ}$",
"$126^{\\circ}$",
"$128^{\\circ}$",
"I don't know"
] | images/1783.jpg | E | Since $\angle S T Q=154^{\circ}$ and angles on a straight line add to $180^{\circ}, \angle P T Q=26^{\circ}$. Hence, since $P Q$ and $Q T$ are equal, triangle $P T Q$ is isosceles and so $\angle T P Q=26^{\circ}$. Therefore, since angles in a triangle add to $180^{\circ}, \angle T Q P=128^{\circ}$. Finally, since $Q T$... | 3 | metric geometry - angle | E | |
1784 | A regular octagon is folded exactly in half three times until a triangle is obtained. The bottom corner of the triangle is then cut off with a cut perpendicular to one side of the triangle as shown.
<image1>
Which of the following will be seen when the triangle is unfolded?
<image2> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1784.jpg | C | Let the angles $x, y$ and $z$ be as shown in the diagram. Since the triangle containing angle $x$ at its vertex is formed by folding the octagon in half three times, $x=\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times 360^{\circ}=45^{\circ}$. Therefore, since angles in a triangle add to $180^{\circ}$ and the cu... | 1 | transformation geometry | C | |
1786 | Joseph writes the numbers 1 to 12 in the circles so that the numbers in adjacent circles differ by either 1 or 2 . Which pair of numbers does he write in adjacent circles? <image1> | [
"3 and 4",
"5 and 6",
"6 and 7",
"8 and 9",
"8 and 10",
"I don't know"
] | images/1786.jpg | E | Consider first the placement of the number 12. Since the only numbers that differ from 12 by 1 or 2 are 10 and 11 , the two numbers on either side of 12 are 10 and 11 . Now consider the number 11. The numbers that differ by 1 or 2 from 11 are 12, 10 and 9 and hence, since 12 and 10 are already placed, the number on the... | 3 | combinatorics | E | |
1787 | Patricia painted some of the cells of a $4 \times 4$ grid. Carl counted how many red cells there were in each row and in each column and created a table to show his answers.
Which of the following tables could Carl have created?
<image1> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1787.jpg | D | The top row of the table in option A indicates that all four cells in that row need to be painted but this is not possible if the left hand column is to have no cells painted. In each table, the sum of the row totals and the sum of column totals represents the total number of cells painted. The table in option B has a ... | 2 | logic | D | |
1789 | A square piece of paper of area $64 \mathrm{~cm}^{2}$ is folded twice, as shown in the diagram. What is the sum of the areas of the two shaded rectangles?
<image1> | [
"$10 \\mathrm{~cm}^{2}$",
"$14 \\mathrm{~cm}^{2}$",
"$15 \\mathrm{~cm}^{2}$",
"$16 \\mathrm{~cm}^{2}$",
"$24 \\mathrm{~cm}^{2}$",
"I don't know"
] | images/1789.jpg | D | Consider the square divided up into 16 congruent smaller squares, as shown. Since the original square has area $64 \mathrm{~cm}^{2}$, each of the smaller squares has area $4 \mathrm{~cm}^{2}$. When the square is folded twice in the manner shown, it can be seen that each of the shaded rectangles is made up of two small ... | 1 | transformation geometry | D | |
1795 | Christopher has made a building out of blocks. The grid on the right shows the number of blocks in each part of the building, when viewed from above. Which of the following gives the view you see when you look at Christopher's building from the front?
\begin{tabular}{|l|l|l|l|}
\hline 4 & 2 & 3 & 2 \\
\hline 3 & 3 & 1 ... | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1795.jpg | E | When you look at Christopher's building from the front, you will see towers of height 4, 3, 3 and 2 as these are the largest numbers of blocks indicated in each of the four columns of the grid. Hence the view that will be seen is E. | 5 | descriptive geometry | E | |
1800 | One slice of a circular cake is $15 \%$ of the whole cake. What is the size of the angle marked with the question mark? <image1> | [
"$30^{\\circ}$",
"$45^{\\circ}$",
"$54^{\\circ}$",
"$15^{\\circ}$",
"$20^{\\circ}$",
"I don't know"
] | images/1800.jpg | C | $15 \%$ of the cake creates a slice whose angle forms $15 \%$ of $360^{\circ}$, which is $54^{\circ}$. | 2 | metric geometry - angle | C | |
1801 | In the picture, the three strips labelled 1,2,3 have the same horizontal width $a$. These three strips connect two parallel lines. Which of these statements is true? <image1> | [
"All three strips have the same area.",
"Strip 1 has the largest area.",
"Strip 2 has the largest area.",
"Strip 3 has the largest area.",
"It is impossible to say which has the largest area without knowing $a$.",
"I don't know"
] | images/1801.jpg | A | Strip 1 is a rectangle with area $a b$, where $b$ is the distance between the two parallel lines. Strip 2 is a parallelogram with area $a b$. Cutting strip 3 as shown creates two parallelograms. The area of each is $a \times$ its vertical height. Since the two heights add to give $b$, strip 3 also has area $a b$. ![](h... | 4 | metric geometry - area | A | |
1802 | In figure 1, alongside, the area of the square equals $a$. The area of each circle in both figures equals $b$. Three circles are lined up as shown in figure 2. An elastic band is placed around these three circles without moving them. What is the area inside the elastic band?
<image1> | [
"$3 b$",
"$2 a+b$",
"$a+2 b$",
"$3 a$",
"$a+b$",
"I don't know"
] | images/1802.jpg | B | The grey regions give an area equal to $b$. The remaining region can be split into two equal squares, which both have area $a$. Hence the total area is $2 a+b$.  | 4 | metric geometry - area | B | |
1803 | The cuboid shown has been built using four shapes, each made from four small cubes. Three of the shapes can be completely seen, but the dark one is only partly visible. Which of the following shapes could be the dark one? <image1>
<image2> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1803.jpg | C | There are three small cubes not visible in the diagram and all belonging to the dark shape. They form a straight line along the back of the base. We then need a shape which has three cubes in a straight line and an extra cube on the middle of those three cubes. | 3 | combinatorial geometry | C | |
1805 | What is the ratio of the areas of the triangles $A D E$ and $A B C$ in the picture? <image1> | [
"$9: 4$",
"$7: 3$",
"$4: 5$",
"$15: 10$",
"$26: 9$",
"I don't know"
] | images/1805.jpg | A | Triangles $A D E$ and $A B C$ are similar since $\angle D A E=\angle C A B$ and $\frac{A D}{A C}=\frac{A E}{A B}$. Since $\frac{A D}{A C}=\frac{3}{2}$, the scale factor for the respective areas is $\left(\frac{3}{2}\right)^{2}=\frac{9}{4}$. | 4 | metric geometry - area | A | |
1806 | A rectangular sheet of paper which measures $6 \mathrm{~cm} \times 12 \mathrm{~cm}$ is folded along its diagonal (Diagram A). The shaded areas in Diagram B are then cut off and the paper is unfolded leaving the rhombus shown in Diagram C. What is the length of the side of the rhombus? <image1> | [
"$\\frac{7}{2} \\sqrt{5} \\mathrm{~cm}$",
"$7.35 \\mathrm{~cm}$",
"$7.5 \\mathrm{~cm}$",
"$7.85 \\mathrm{~cm}$",
"$8.1 \\mathrm{~cm}$",
"I don't know"
] | images/1806.jpg | C | Consider Diagram $B$ from the question, as shown, and let the rhombus have side $x \mathrm{~cm}$. Thus $A C=C D=x, B C=12-x$ and $A B=6$. Applying Pythagoras' Theorem to $\triangle A B C$ gives $x^{2}=6^{2}+(12-x)^{2} \Rightarrow x^{2}=36+144-24 x+x^{2} \Rightarrow$ $24 x=180$ so the side of the rhombus is $7.5 \mathrm... | 4 | transformation geometry | C | |
1807 | In the diagram, $Q R=P S$. What is the size of $\angle P S R$ ? <image1> | [
"$30^{\\circ}$",
"$50^{\\circ}$",
"$55^{\\circ}$",
"$65^{\\circ}$",
"$70^{\\circ}$",
"I don't know"
] | images/1807.jpg | D | Since $\angle Q P R=180^{\circ}-\left(75^{\circ}+30^{\circ}\right)=75^{\circ}$, triangle $P Q R$ is isosceles with $Q R=P R=P S . \quad$ Hence triangle $P R S$ is isosceles so that $\angle P S R=\frac{1}{2}\left(180^{\circ}-50^{\circ}\right)=65^{\circ}$. | 2 | metric geometry - angle | D | |
1808 | Roo has a very unusual chessboard of side 7, in which only the squares which lie on the diagonals are shaded. Kanga then asks the question "What would be the total white area of your chessboard if each side was 2003 squares long?" What is the correct answer? <image1> | [
"$2002^{2}$",
"$2002 \\times 2001$",
"$2003^{2}$",
"$2003 \\times 2004$",
"$2004^{2}$",
"I don't know"
] | images/1808.jpg | A | In the large board there would be $2003+2002$ shaded squares. If instead this number of squares were shaded along two adjacent edges, the white region would then be a square of side 2002 . | 4 | metric geometry - area | A | |
1809 | 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> | [
"$2: 1$",
"$3: 1$",
"$4: 1$",
"$5: 1$",
"$6: 1$",
"I don't know"
] | images/1809.jpg | D | Let the radius of the inner circle be $R$ so its area is then $\pi R^{2}$. The white ring and the inner black circle together have area $\pi(2 R)^{2}=4 \pi R^{2}$. The whole target has area $\pi(3 R)^{2}=9 \pi R^{2}$. Hence the outer black circle has area $9 \pi R^{2}-4 \pi R^{2}=5 \pi R^{2}$ which is 5 times the area ... | 4 | metric geometry - area | D | |
1811 | Sol is having fun playing with water in two tanks. Tank $X$ has a base of area of $200 \mathrm{~cm}^{2}$.
Tank $Y$ 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, on the bottom of Tank X. The water... | [
"$1 \\mathrm{~cm}$",
"$2 \\mathrm{~cm}$",
"$3 \\mathrm{~cm}$",
"$4 \\mathrm{~cm}$",
"$5 \\mathrm{~cm}$",
"I don't know"
] | images/1811.jpg | C | The water has base of area $200 \mathrm{~cm}^{2}$ and volume $1000 \mathrm{~cm}^{3}$. The empty tank has base of area $100 \mathrm{~cm}^{2}$ and volume $700 \mathrm{~cm}^{3}$. The water displaced by putting the empty tank in Tank $X$ is then $700 \mathrm{~cm}^{3}$. The water in the empty tank is $1000-700=300 \mathrm{~... | 2 | solid geometry | C | |
1812 | 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> | [
"25",
"$20 \\sqrt{ } 2$",
"28.5",
"30",
"$10 \\sqrt{ } 10$",
"I don't know"
] | images/1812.jpg | B | Drawing in the triangle as shown we have $h^{2}=30^{2}-10^{2}=900-100=800$ and $h=\sqrt{800}=20 \sqrt{2}$.  | 4 | metric geometry - length | B | |
1818 | 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>
The ratio of the area of the shaded parts of all five circles to the area of the unshaded parts of all five circles is | [
"$5: 4$",
"$2: 3$",
"$2: 5$",
"$1: 4$",
"$1: 3$",
"I don't know"
] | images/1818.jpg | B | There are 5 circles in the diagram, of which $1+4 \times \frac{1}{4}=2$ are shaded. Hence the shaded to unshaded ratio is $2: 3$. | 4 | metric geometry - area | B | |
1820 | Kanga and Roo are hopping around a stadium with a perimeter of $330 \mathrm{~m}$. Each of them makes one jump every second.
Kanga'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 tired and stops after 25 seconds whil... | [
"15 seconds",
"24 seconds",
"51 seconds",
"66 seconds",
"76 seconds",
"I don't know"
] | images/1820.jpg | C | After 25 seconds Kanga is $25 \times 3 \mathrm{~m}=75 \mathrm{~m}$ ahead of Roo. Hence Kanga has a further $255 \mathrm{~m}$ to hop before she is beside Roo once more. This takes her $255 \div 5=51$ seconds. | 3 | algebra | C | |
1822 | 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> | [
"$2 \\pi+2 \\mathrm{~cm}^{2}$",
"$8 \\mathrm{~cm}^{2}$",
"$2 \\pi+1 \\mathrm{~cm}^{2}$",
"$7 \\mathrm{~cm}^{2}$",
"$2 \\pi \\mathrm{cm}^{2}$",
"I don't know"
] | images/1822.jpg | B | Dividing the grey-shaded area as shown we see that the upper semicircle is equal in area to the two quarter circles shaded black. Therefore the whole grey-shaded area is equal to the area of the rectangle $A B F E$, which is $8 \mathrm{~cm}^{2}$. ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-045... | 4 | metric geometry - area | B | |
1823 | What is the sum of the 10 angles marked on the diagram on the right? <image1> | [
"$300^{\\circ}$",
"$450^{\\circ}$",
"$360^{\\circ}$",
"$600^{\\circ}$",
"$720^{\\circ}$",
"I don't know"
] | images/1823.jpg | E | The sum of the fifteen angles in the five triangles is $5 \times 180^{\circ}=900^{\circ}$. The sum of the unmarked central angles in the five triangles is $180^{\circ}$, since each can be paired with the angle between the two triangles opposite. Thus the sum of the marked angles is $900^{\circ}-180^{\circ}=720^{\circ}$... | 2 | metric geometry - angle | E | |
1826 | 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> | [
"$\\frac{1}{2}$",
"$\\frac{2}{3}$",
"$\\frac{3}{5}$",
"$\\frac{4}{7}$",
"$\\frac{5}{9}$",
"I don't know"
] | images/1826.jpg | D | 4 | arithmetic | D | ||
1831 | 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> | [
"$\\sqrt{2}-1$",
"$\\frac{\\sqrt{2}}{2}$",
"$\\frac{\\sqrt{2}+1}{2}$",
"$\\sqrt{2}+1$",
"$\\sqrt{3}-\\sqrt{2}$",
"I don't know"
] | images/1831.jpg | A | 4 | transformation geometry | A | ||
1832 | 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 travelled?
<image1> | [
"$10 \\pi$",
"$5 \\pi+5 \\pi \\sqrt{2}$",
"$10 \\pi+5 \\pi \\sqrt{2}$",
"$5 \\pi+10 \\pi \\sqrt{2}$",
"$10 \\pi+10 \\pi \\sqrt{2}$",
"I don't know"
] | images/1832.jpg | C | 4 | transformation geometry | C | ||
1834 | 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> | [
"$X=0, Y=0$",
"$X=0, Y=1$",
"$X=1, Y=0$",
"$X=1, Y=1$",
"It is impossible to complete.",
"I don't know"
] | images/1834.jpg | A | The table can be filled in just by looking for a row or column with two identical entries already. Notice that the top row has two 0s so the missing entries are both 1. The completed table is \begin{tabular}{|l|l|l|l|} \hline 0 & 1 & 0 & 1 \\ \hline 1 & 1 & 0 & 0 \\ \hline 0 & 0 & 1 & 1 \\ \hline 1 & 0 & 1 & 0 \\ \hli... | 5 | logic | A | |
1837 | 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> | [
"$20 \\pi$",
"$22 \\pi$",
"$40 \\pi$",
"$88 \\pi$",
"$100 \\pi$",
"I don't know"
] | images/1837.jpg | A | The dog can trace out $\frac{3}{4}$ of a circle with radius $10 \mathrm{~m} ; \frac{1}{4}$ of a circle with radius $6 \mathrm{~m}$; and $\frac{1}{4}$ of a circle with radius $4 \mathrm{~m}$. The total perimeter is $\frac{3}{4} \times 2 \pi \times 10+\frac{1}{4} \times 2 \pi \times 6+\frac{1}{4} \times 2 \pi \times 4=20... | 4 | metric geometry - length | A | |
1838 | 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> | [
"$6+\\pi / 2$",
"$12+\\pi$",
"$6+\\pi$",
"$12+2 \\pi$",
"$6+2 \\pi$",
"I don't know"
] | images/1838.jpg | C | Along the six edges the centre moves $1 \mathrm{~cm}$ (parallel to the edges). Around the six vertices it traces out an arc of radius $\frac{1}{2} \mathrm{~cm}$ and angle $60^{\circ}$, which has length $$ \frac{60}{360} \times 2 \pi \times \frac{1}{2}=\frac{\pi}{6} $$ The total distance $=6 \times\left(1+\frac{\pi}{6... | 4 | transformation geometry | C | |
1839 | 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> | [
"$L=H+3 S$",
"$H=L S$",
"$H=\\frac{1}{2}(L+S)$",
"$H=L-S$",
"$H=\\sqrt{L S}$",
"I don't know"
] | images/1839.jpg | E | By splitting area $S$ into three small triangles $T$, we see that $S=3 T ; L=12 T ;$ and $H=6 T$. Substituting these into the given expressions, we can see that only $\mathrm{E}$ is always true. ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-075.jpg?height=228&width=214&top_left_y=226&top_left_x=... | 4 | metric geometry - area | E | |
1840 | 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> | [
"$\\frac{1}{2}$",
"$\\frac{1}{\\sqrt{2}}$",
"$\\sqrt{2}-1$",
"$2-\\sqrt{2}$",
"It depends on the relative sizes of the circles.",
"I don't know"
] | images/1840.jpg | D | The three straight lines shown have lengths $R$, $R+r$ and $r$. Taking vertical components we have $R+(R+r) \cos 45^{\circ}+r=1$. That is $R+\frac{1}{\sqrt{2}}(R+r)+r=1$ or simply $(R+r)(\sqrt{ } 2+1)=\sqrt{ } 2$, so $R+r=\frac{\sqrt{2}}{\sqrt{2}+1}=2-\sqrt{ } 2$. ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2... | 4 | metric geometry - length | D | |
1841 | 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 ?
<image1> | [
"S",
"I",
"M",
"O",
"N",
"I don't know"
] | images/1841.jpg | D | Box 4 must keep the ' $M$ ' so the ' $M$ ' must be removed from box 5 and the 'I' left there. Similarly, 'I' must be removed from box 3 and ' $N$ ' remain. Then ' $I$ ', ' $M$ ', ' $\mathrm{N}$ ' must be removed from box 2 and ' $\mathrm{O}$ ' remains. | 4 | combinatorics | D | |
1842 | Four unit squares are placed edge to edge as shown. What is the length of the line $P Q$ ? <image1> | [
"5",
"$\\sqrt{13}$",
"$\\sqrt{5}+\\sqrt{2}$",
"$\\sqrt{5}$",
"13",
"I don't know"
] | images/1842.jpg | B | The horizontal and vertical distances between $P$ and $Q$ are 3 and 2 units respectively, so by Pythagoras' Theorem, the distance is $\sqrt{3^{2}+2^{2}}=\sqrt{13}$. | 4 | metric geometry - length | B | |
1843 | One face of a cardboard cube is cut along its diagonals, as shown.
Which of the following are not nets for this cube?
<image1>
<image2> | [
"1 and 3",
"1 and 5",
"2 and 4",
"2 and 4",
"3 and 5",
"I don't know"
] | images/1843.jpg | E | In diagram 3 the triangles at the top and bottom will fold up in such a way as to overlap with one of the square faces as indicated. In diagram 5 the triangles at the bottom will fit together but the triangle at the top will overlap with one of the square faces. ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed27879... | 2 | solid geometry | E | |
1844 | 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> | [
"$\\frac{2}{3}$",
"$\\frac{1}{2}$",
"$\\frac{1}{3}$",
"$\\frac{1}{4}$",
"$\\frac{3}{5}$",
"I don't know"
] | images/1844.jpg | B | The regular hexagons have interior angles of $120^{\circ}$ and fit into the corners of the parallelogram, so the parallelogram must have interior angles of $120^{\circ}$ and $60^{\circ}$.  ... | 3 | combinatorial geometry | B | |
1845 | 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 ?
<image1> | [
"$F$ and $K$",
"$G$ and $J$",
"$H$ and $I$",
"all six numbers",
"only one of them",
"I don't know"
] | images/1845.jpg | A | To be divisible by three, at least two integers must differ by a multiple of three, which is true for all of $F, G$, $J$, and $K$, so they must all be divisible by 3 . To be divisible by five, there must be at least two integers that differ by a multiple of 5 , which is true for all of $F, H, I$, and $K$, so they must ... | 3 | algebra | A | |
1846 | 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> | [
"$5^{\\circ}$",
"$10^{\\circ}$",
"$15^{\\circ}$",
"$20^{\\circ}$",
"$25^{\\circ}$",
"I don't know"
] | images/1846.jpg | A | Since $\angle K P L=120^{\circ}, \angle K P J=60^{\circ}$ and, as $\angle J K P=50^{\circ}$, $\angle K J P=70^{\circ}$ (by angle sum of a triangle). So, since triangle $J K L$ is isosceles, $\angle J K L=\frac{1}{2}(180-70)^{\circ}=55^{\circ}$ giving $\angle P K L=\angle J K L-\angle J K P=5^{\circ}$. ![](https://cdn.... | 2 | metric geometry - angle | A | |
1847 | 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 $Q$ ? <image1> | [
"$\\frac{5 \\pi}{4}$",
"$\\frac{5 \\pi}{3}$",
"$\\frac{\\pi}{2}$",
"$\\frac{2 \\pi}{3}$",
"$\\frac{3 \\pi}{2}$",
"I don't know"
] | images/1847.jpg | E | The triangle that joins up the centres of the circles has sides of length $3 \mathrm{~cm}, 4 \mathrm{~cm}, 5 \mathrm{~cm}$ so must be a right-angled triangle by the converse to Pythagoras' Theorem. Therefore the length of the longer arc of circle $C_{1}$ is $\frac{3}{4} \times 2 \pi \times 1=\frac{3 \pi}{2} \mathrm{~cm... | 4 | metric geometry - length | E | |
1851 | 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> | [
"$2-\\sqrt{2}$",
"$\\frac{3}{4}$",
"$\\sqrt{5}-\\sqrt{2}$",
"$\\frac{\\sqrt{3}}{3}$",
"$ \\sqrt{3}-1$",
"I don't know"
] | images/1851.jpg | E | Extend $P Q$ and let $X, Y$ be the intersections with $M N$ and $K L$ respectively as shown. The length of $P M$ is 1 because $P$ lies on an arc centred at $M$. By Pythagoras' Theorem, $P X^{2}=P M^{2}-M X^{2}=1^{2}-\left(\frac{1}{2}\right)^{2}=\frac{3}{4}$. So $P X=\frac{\sqrt{3}}{2}$. As $K L M N$ is a unit squar... | 4 | metric geometry - length | E | |
1852 | 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> | [
"76",
"$80-2 \\pi$",
"$40-4 \\pi$",
"$80-\\pi$",
"$78 \\pi$",
"I don't know"
] | images/1852.jpg | B | The three angles of the triangle add to $180^{\circ}$, so the combined area of the three sectors of the circles that are inside the triangle add up to half a circle with area $\frac{1}{2} \times \pi \times 2^{2}=\frac{4 \pi}{2}=2 \pi$. So the grey area is $(80-2 \pi) \mathrm{cm}^{2}$. ![](https://cdn.mathpix.com/cropp... | 4 | metric geometry - area | B | |
1853 | 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> | [
"$120^{\\circ}$",
"$124^{\\circ}$",
"$128^{\\circ}$",
"$132^{\\circ}$",
"$136^{\\circ}$",
"I don't know"
] | images/1853.jpg | B | Let $\angle O L M=\angle O L N=a^{\circ}, \angle O M L=\angle O M N=b^{\circ}$ and $\angle L O M=c^{\circ}$. Angles in a triangle add up to $180^{\circ}$, so from $\triangle L M N, 2 a^{\circ}+2 b^{\circ}+68^{\circ}=180^{\circ}$ which gives $2\left(a^{\circ}+b^{\circ}\right)=112^{\circ}$ i.e. $a+b=56$. Also, from $\tri... | 2 | metric geometry - angle | B | |
1854 | 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 large circle in centimetres? <image1... | [
"$1+\\sqrt{2}$",
"$\\sqrt{5}$",
"$\\sqrt{2}$",
"$\\frac{5}{2}$",
"$\\frac{4}{5} \\pi$",
"I don't know"
] | images/1854.jpg | A | Let $R$ be the radius of each of the larger circles. The sides of the square are equal to $R+1$, the sum of the two radii. The diagonal of the square is $2 R$. By Pythagoras, $(R+1)^{2}+(R+1)^{2}=(2 R)^{2}$. Simplifying gives $2(R+1)^{2}=4 R^{2}$, i.e. $(R+1)^{2}=2 R^{2}$, so $R+1=\sqrt{ } 2 R$ $[-\sqrt{2} R$ is not po... | 4 | metric geometry - length | A | |
1858 | 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> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1858.jpg | D | Imagine refolding these strips once, as is shown in the diagram on the left. The peaks on one side of the fold must match with hollows on the other side (which they all do!). We obtain the half-size strips shown in the diagram on the right. Now imagine refolding these strips about their mid-points. We can see that, in ... | 4 | transformation geometry | D | |
1859 | 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> | [
"parallelogram",
"trapezium",
"right-angled triangle",
"obtuse-angled triangle",
"all the shapes $\\mathrm{A}-\\mathrm{D}$ can be made",
"I don't know"
] | images/1859.jpg | E | The shape $R S U V$ is a parallelogram; $R S T U$ is a trapezium; $R S U$ is a right-angled triangle; $R S V$ is an obstuse-angled triangle. Therefore all the shapes can be made.  | 3 | combinatorial geometry | E | |
1860 | 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> | [
"$\\sqrt{2}$",
"$\\frac{\\sqrt{3}}{2}$",
"$\\sqrt{3}$",
"$\\sqrt{5}-1$",
"$\\sqrt{6}-1$",
"I don't know"
] | images/1860.jpg | A | The angles in equilateral triangles are all $60^{\circ}$ so $\angle P S U=90^{\circ}-60^{\circ}=30^{\circ}$, $\angle T S U=30^{\circ}+60^{\circ}=90^{\circ}$ and $U S=1=S T$. Using Pythagoras' theorem on the right-angled triangle $T S U$ we have $T U^{2}=1^{2}+1^{2}=2$ so $T U=\sqrt{2}$. | 4 | metric geometry - length | A | |
1861 | 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> | [
"$10^{\\circ}$",
"$20^{\\circ}$",
"$30^{\\circ}$",
"$40^{\\circ}$",
"$50^{\\circ}$",
"I don't know"
] | images/1861.jpg | D | Angle $U P T$ is $360^{\circ}-330^{\circ}=30^{\circ}$ and the reflex angle $Q R S$ is $270^{\circ}$. Since the angles in the quadrilateral $P Q R S$ add to $360^{\circ}$, we have $\angle R S P=360^{\circ}-\left(270^{\circ}+30^{\circ}+20^{\circ}\right)=40^{\circ}$. | 2 | metric geometry - angle | D | |
1862 | 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> | [
"$2 \\pi$",
"$4 \\pi$",
"$6 \\pi$",
"$8 \\pi$",
"$12 \\pi$",
"I don't know"
] | images/1862.jpg | C | Each of the four congruent parts has three arcs on its perimeter: Two semicircles of radius $2 \mathrm{~cm}$ (which have total length $2 \times \pi \times 2=4 \pi \mathrm{cm}$ ) and a quarter-arc of radius $4 \mathrm{~cm}$ (length $\frac{1}{4} \times 2 \times \pi \times 4=2 \pi \mathrm{cm}$ ). Therefore the perimeter h... | 4 | metric geometry - length | C | |
1863 | 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> | [
"Alicia",
"Bea",
"Carlos",
"Dani",
"Ernesto",
"I don't know"
] | images/1863.jpg | D | For each runner, the gradient of the line joining the origin to his or her plotted point is equal to the total distance divided by the time taken, which is also his or her average speed. Hence the fastest runner has the steepest line, so it is Dani. | 5 | analytic geometry | D | |
1867 | 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 square and which each touch two s... | [
"$4 \\pi(3-2 \\sqrt{2})$",
"$4 \\pi \\sqrt{2}$",
"$\\frac{16}{9} \\pi$",
"$\\pi$",
"$\\frac{4}{\\sqrt{2}} \\pi$",
"I don't know"
] | images/1867.jpg | A | The diagram shows one of the four shaded circles. The point $A$ is a vertex of the original square and $O$ is its centre. So $A Y=Y O=1$, and $A X=A O=\sqrt{2}$ by Pythagoras. Also $X Y=A X-A Y=\sqrt{2}-1$. So each shaded circle has radius $\sqrt{2}-1$. Hence the area of the four shaded circles is $4 \times \pi(\sqrt{2... | 4 | metric geometry - area | A | |
1871 | 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> | [
"only $T$",
"only $U$",
"either of $U$ and $W$",
"any of $U, V$ and $W$",
"any of $T, U, V$ and $W$",
"I don't know"
] | images/1871.jpg | C | Label the horizontal line segment $P Q$, and the vertical line segment $R S$. A rotation of $90^{\circ}$ anticlockwise about $U$, or $90^{\circ}$ clockwise about $W$ would map $P Q$ onto $R S$. When a rotation is performed, the distance of any point from the centre of rotation is preserved. Hence $T$ cannot be a centr... | 4 | transformation geometry | C | |
1872 | 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> | [
"$6(1+\\sqrt{2})$",
"$6\\left(1+\\frac{1}{2} \\sqrt{3}\\right)$",
"$12$",
"$6+3 \\sqrt{2}$",
"$9$",
"I don't know"
] | images/1872.jpg | C | The interior angles of a regular hexagon are all $120^{\circ}$. At any vertex of the hexagon, there are two squares and a triangle, so the angle of the triangle at that point must be $360^{\circ}-120^{\circ}-90^{\circ}-90^{\circ}=60^{\circ}$. Hence the other two angles of the triangle must add to $180^{\circ}-60^{\circ... | 4 | metric geometry - length | C | |
1876 | 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 Street, Nail Street and Ruler Stree... | [
"Adeline",
"Benjamin",
"Carole",
"David",
"It is impossible to tell without a better map",
"I don't know"
] | images/1876.jpg | A | A pair of straight lines intersects at most once, but Adeline's and Carole's roads intersect twice so one of them must be Curvy Street; similarly Adeline's and Benjamin's roads intersect twice so one of them must also be Curvy Street. Therefore Adeline lives on Curvy Street. | 2 | topology | A | |
1879 | 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>
<image2> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1879.jpg | C |  The diagram shows the back eight cubes. So the white piece has shape C. | 3 | combinatorial geometry | C | |
1881 | 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> | [
"5",
"$2 \\sqrt{3}+2$",
"$3 \\sqrt{2}$",
"$3 \\sqrt{3}$",
"6",
"I don't know"
] | images/1881.jpg | B | The width of three circles across the top is $6 \mathrm{~cm}$, so each circle has diameter $2 \mathrm{~cm}$ and radius $1 \mathrm{~cm}$. The triangles joining the centres of three circles as shown are equilateral with edge lengths $2 \mathrm{~cm}$. By Pythagoras, the heights of the triangles are $\sqrt{2^{2}-1^{2}}=\sq... | 4 | metric geometry - length | B | |
1882 | 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> | [
"$8 / 3$",
"$10 / 3$",
"$11 / 3$",
"4",
"$13 / 3$",
"I don't know"
] | images/1882.jpg | B | Let $H, I, J$ be the vertices of the triangle, $C$ the centre of the circle, and $K$ the point where the semicircle touches the edge $H I$ as shown. The angle $C K H$ is a right angle because $H I$ is tangent to the circle and so perpendicular to the radius CK. The two triangles HKC and $H J I$ are similar since they e... | 4 | metric geometry - length | B | |
1888 | 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?
<image1>
<image2> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1888.jpg | E | The net of the cube consists of six large squares, each of which is split into four $2 \times 2$ squares. Each of these large squares must have 2 black squares and 2 white squares in alternating colours. This eliminates nets A, B, D. Around each of the 8 vertices of the cube, there are either 3 black squares or 3 whit... | 2 | solid geometry | E | |
1889 | 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>
<image2> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1889.jpg | A | After rotation $90^{\circ}$ anticlockwise, we obtain shape E. When reflected in the $x$-axis this gives shape A. | 5 | analytic geometry | A | |
1890 | 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> | [
"$20^{\\circ}$",
"$25^{\\circ}$",
"$30^{\\circ}$",
"$35^{\\circ}$",
"$40^{\\circ}$",
"I don't know"
] | images/1890.jpg | D | Since triangle $S T R$ is equilateral, $\angle S T R=60^{\circ}$. Hence $\angle S T V=130^{\circ}$. Triangle $S T V$ is isosceles (since $S T=T V$ ), so $\angle T S V=\frac{1}{2}\left(180^{\circ}-130^{\circ}\right)=25^{\circ}$. Thus $\angle R S V=60^{\circ}-25^{\circ}=35^{\circ}$. | 4 | transformation geometry | D | |
1893 | 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 calculation? <image1> | [
"A",
"C",
"C",
"D",
"it depends on the size of the rectangle",
"I don't know"
] | images/1893.jpg | A | Since all the coordinates are negative, each of the calculations will yield a positive value. The smallest value will come from the least negative $y$-coordinate divided by the most negative $x$-coordinate; this comes from point $A$. | 5 | analytic geometry | A | |
1894 | 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> | [
"$P Q$",
"$P R$",
"$P S$",
"$Q R$",
"$R S$",
"I don't know"
] | images/1894.jpg | A | The triangles are similar because they both contain angles of $59^{\circ}, 60^{\circ}, 61^{\circ}$. The smallest side of a triangle is always opposite the smallest angle, so line segment $P R$ is the smallest edge of triangle $P Q R$, though it is not the smallest edge of triangle $P R S$; hence triangle $P Q R$ is lar... | 4 | metric geometry - length | A | |
1897 | If $r, s$, and $t$ denote the lengths of the 'lines' in the picture, then which of the following inequalities is correct? <image1> | [
"$r<s<t$",
"$r<t<s$",
"$s<r<t$",
"$s<t<r$",
"$t<s<r$",
"I don't know"
] | images/1897.jpg | E | If each square has side-length one unit, then the length $r$ is 16 units. The length $s$ consists of 8 straight unit lengths and two semicircles with radius 1 unit, so $s=8+2 \pi$. The length $t$ consists of 8 straight unit lengths and two diagonals (that together make the hypotenuse of a right-angled triangle with sho... | 3 | combinatorial geometry | E | |
1903 | In the picture, $P T$ is a tangent to the circle with centre $O$ and $P S$ is the angle bisector of angle $R P T$.
What is the size of angle TSP? <image1> | [
"$30^{\\circ}$",
"$45^{\\circ}$",
"$50^{\\circ}$",
"$60^{\\circ}$",
"It depends on the position of point $P$.",
"I don't know"
] | images/1903.jpg | B | Denote $\angle S P T$ by $x$. Since $T P$ is a tangent and $O T$ is a radius, $\angle O T P=90^{\circ}$. So $\angle T O P=180^{\circ}-\angle O T P-\angle O P T=180^{\circ}-90^{\circ}-2 x=90^{\circ}-2 x$. Then $\angle T O R=90^{\circ}+2 x$ (angles on a straight line). But triangle TOR is isosceles (OT and $O R$ are both... | 2 | metric geometry - angle | B | |
1904 | The diagram shows a triangle $F H G$ with $F H=6, G H=8$ and $F G=10$. The point $I$ is the midpoint of $F G$, and HIJK is a square. The line segment $I J$ intersects $G H$ at $L$. What is the area of the shaded quadrilateral HLJK? <image1> | [
"$\\frac{124}{8}$",
"$\\frac{125}{8}$",
"$\\frac{126}{8}$",
"$\\frac{127}{8}$",
"$\\frac{128}{8}$",
"I don't know"
] | images/1904.jpg | B | Triangle $F G H$ is right-angled with the right angle at $H$ because its sides $6,8,10$ form a Pythagorean triple. Using the converse of 'angles in a semicircle are right angles', we deduce that $F G$ is the diameter of a circle with centre at $I$ (midpoint of $F G$ ) and radius 5 (half of the length $F G$ ). Thus $I H... | 4 | metric geometry - area | B | |
1905 | The diagram shows a square with sides of length $a$. The shaded part of the square is bounded by a semicircle and two quarter-circle arcs. What is the shaded area? <image1> | [
"$\\frac{\\pi a^{2}}{8}$",
"$\\frac{a^{2}}{2}$",
"$\\frac{\\pi a^{2}}{2}$",
"$\\frac{a^{2}}{4}$",
"$\\frac{\\pi a^{2}}{4}$",
"I don't know"
] | images/1905.jpg | B | If the semicircle is cut into two quarter-circles, these can be placed next to the other shaded region to fill up half the square. Hence the shaded area is half of the area of the square, namely $\frac{1}{2} a^{2}$. | 4 | metric geometry - area | B | |
1906 | Mr Hyde can't remember exactly where he has hidden his treasure. He knows it is at least $5 \mathrm{~m}$ from his hedge, and at most $5 \mathrm{~m}$ from his tree. Which of the following shaded areas could represent the largest region where his treasure could lie?
<image1> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1906.jpg | A | Points which are at most $5 \mathrm{~m}$ from the tree lie on or inside a circle of radius $5 \mathrm{~m}$ with its centre at the tree. However, not all of the inside of the circle will be shaded because the treasure is at least $5 \mathrm{~m}$ from the hedge, so we should have an unshaded rectangular strip next to the... | 4 | metric geometry - area | A | |
1907 | The picture shows the same die in three different positions. When the die is rolled, what is the probability of rolling a 'YES'?
<image1> | [
"$\\frac{1}{3}$",
"$\\frac{1}{2}$",
"$\\frac{5}{9}$",
"$\\frac{2}{3}$",
"$\\frac{5}{6}$",
"I don't know"
] | images/1907.jpg | B | We will show that the word "YES" appears exactly three times, giving the probability $3 / 6$ or $1 / 2$. Firstly note that "YES" appears twice on the second die. The third die also shows "YES" and this cannot be the same as either "YES" on the second die: Under the first "YES" is "MAYBE", but on the third die the word ... | 2 | solid geometry | B | |
1908 | In the grid, each small square has side of length 1 . What is the minimum distance from 'Start' to 'Finish' travelling only on edges or diagonals of the squares? <image1> | [
"$2 \\sqrt{2}$",
"$\\sqrt{10}+\\sqrt{2}$",
"$2+2 \\sqrt{2}$",
"$4 \\sqrt{2}$",
"$6$",
"I don't know"
] | images/1908.jpg | C | The shortest routes consist of two diagonals (right and down) each of length $\sqrt{2}$, and two sides of length 1 , giving a total length $2+2 \sqrt{2}$. | 3 | combinatorial geometry | C | |
1910 | A large set of weighing scales has two identical sets of scales placed on it, one on each pan. Four weights $W, X, Y, Z$ are placed on the weighing scales as shown in the left diagram.
<image1>
Then two of these weights are swapped, and the pans now appear as shown in the diagram on the right. Which two weights were sw... | [
"$W$ and $Z$",
"$W$ and $Y$",
"$W$ and $X$",
"$X$ and $Z$",
"$X$ and $Y$",
"I don't know"
] | images/1910.jpg | A | From the first picture, we can see: \begin{tabular}{lrl} From the right scale: & $Z$ & $>Y$ \\ From the left scale: & $X$ & $>W$ \\ From the large scale: & $Y+Z>W+X$ \end{tabular} It follows from (1), (2) and (3) that $Z+Z>Y+Z>W+X>W+W$, so $2 Z>2 W$, and hence $Z>W \ldots(4)$. We can show that most swaps give a con... | 3 | algebra | A | |
1912 | In the triangle $F G H$, we can draw a line parallel to its base $F G$, through point $X$ or $Y$. The areas of the shaded regions are the same. The ratio $H X: X F=4: 1$. What is the ratio $H Y: Y F$ ? <image1> | [
"$1: 1$",
"$2: 1$",
"$3: 1$",
"$3: 2$",
"$4: 3$",
"I don't know"
] | images/1912.jpg | D | In the triangle on the left, the unshaded triangle is similar to triangle $F G H$, and is obtained from it by a scale factor of $\frac{4}{5}$. Hence its area is $\left(\frac{4}{5}\right)^{2}=\frac{16}{25}$ of the area of $F G H$. The shaded area is therefore $\frac{9}{25}$ of the area of $F G H$. Hence $H Y: H F=3: 5$ ... | 4 | metric geometry - length | D | |
1913 | A square is split into nine identical squares, each with sides of length one unit. Circles are inscribed in two of these squares, as shown. What is the shortest distance between the two circles? <image1> | [
"$2 \\sqrt{2}-1$",
"$\\sqrt{2}+1$",
"$2 \\sqrt{2}$",
"2",
"3",
"I don't know"
] | images/1913.jpg | A | Each square has side-length 1 unit, so by Pythagoras' Theorem the diagonals have length $\sqrt{1^{2}+1^{2}}=\sqrt{2}$. The distance between the two circles consists of a whole diagonal and two part diagonals (from the corner of a square to the circle). This is the same as ![](https://cdn.mathpix.com/cropped/2023_12_27_... | 4 | metric geometry - length | A | |
1914 | The large triangle shown has sides of length 5 units. What percentage of the area of the triangle is shaded? <image1> | [
"$80 \\%$",
"$85 \\%$",
"$88 \\%$",
"$90 \\%$",
"impossible to determine",
"I don't know"
] | images/1914.jpg | C | By dissecting the triangle into smaller, identical triangles, we see that the shaded area is 22/25 of the larger triangle, which corresponds to $88 / 100=88 \%$. | 4 | metric geometry - area | C | |
1917 | The picture shows a cube with four marked angles, $\angle W X Y$, $\angle X Y Z, \angle Y Z W$ and $\angle Z W X$. What is the sum of these angles? <image1> | [
"$315^{\\circ}$",
"$330^{\\circ}$",
"$345^{\\circ}$",
"$360^{\\circ}$",
"$375^{\\circ}$",
"I don't know"
] | images/1917.jpg | B | The lengths of $W X, X Z$ and $Z W$ are all equal (each being the diagonal of a square face), hence triangle $W X Z$ is equilateral and angle $Z W X$ is $60^{\circ}$. The other angles are all $90^{\circ}$, so the total of all four angles is $90^{\circ}+90^{\circ}+90^{\circ}+60^{\circ}=330^{\circ}$. ![](https://cdn.ma... | 2 | solid geometry | B | |
1920 | Which of the following diagrams shows the locus of the midpoint of the wheel when the wheel rolls along the zig-zag curve shown?
<image1> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1920.jpg | E | As the wheel goes over the top it pivots around the peak so the midpoint travels through a circular arc. At the troughs the wheel changes directions in an instant from down-right to up-right, so the midpoint travels through a sharp change of direction. This gives the locus in diagram $\mathrm{E}$. | 4 | transformation geometry | E | |
1921 | A circle of radius 1 rolls along a straight line from the point $K$ to the point $L$, where $K L=11 \pi$. Which of the following pictures shows the correct appearance of the circle when it reaches $L$ ?
<image1>
<image2> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1921.jpg | D | The circumference is $2 \pi$, so every time the circle rolls $2 \pi$ it has turned $360^{\circ}$ and looks the same as it did at $K$. After $11 \pi$, it has turned $5 \frac{1}{2}$ turns, which is picture D. | 4 | transformation geometry | D | |
1923 | A belt drive system consists of the wheels $K, L$ and $M$, which rotate without any slippage. The wheel $L$ makes 4 full turns when $K$ makes 5 full turns; also $L$ makes 6 full turns when $M$ makes 7 full turns.
<image1>
The perimeter of wheel $M$ is $30 \mathrm{~cm}$. What is the perimeter of wheel $K$ ? | [
"$27 \\mathrm{~cm}$",
"$28 \\mathrm{~cm}$",
"$29 \\mathrm{~cm}$",
"$30 \\mathrm{~cm}$",
"$31 \\mathrm{~cm}$",
"I don't know"
] | images/1923.jpg | B | To compare wheels $K$ and $M$, we need to find the lowest common multiple of 4 and 6 , which is 12 . When wheel $L$ makes 12 turns, wheel $K$ makes 15 turns and wheel $M$ makes 14 turns. When wheel $L$ makes 24 turns, wheel $K$ makes 30 turns and wheel $M$ makes 28 turns, so the ratio of the circumferences of wheel $K$... | 4 | metric geometry - length | B | |
1926 | On a balance scale, three different masses were put at random on each pan and the result is shown in the picture. The masses are of 101, 102, 103, 104, 105 and 106 grams. What is the probability that the 106 gram mass stands on the heavier pan?
<image1> | [
"$75 \\%$",
"$80 \\%$",
"$90 \\%$",
"$95 \\%$",
"$100 \\%$",
"I don't know"
] | images/1926.jpg | B | The total mass is $621 \mathrm{~g}$ so any three masses with total mass over $310.5 \mathrm{~g}$ could be in the heavier pan. There are eight of these triples that include the $106 \mathrm{~g}$ mass: $(106,105,104)$, $(106,105,103),(106,105,102),(106,105,101),(106,104,103),(106,104,102),(106$, $104,101)$, and $(106,103... | 4 | combinatorics | B | |
1928 | The diagram shows a triangle $F H I$, and a point $G$ on $F H$ such that $G H=F I$. The points $M$ and $N$ are the midpoints of $F G$ and $H I$ respectively. Angle $N M H=\alpha^{\circ}$. Which of the following gives an expression for $\angle I F H$ ?
<image1> | [
"$2 \\alpha^{\\circ}$",
"$(90-\\alpha)^{\\circ}$",
"$45+\\alpha^{\\circ}$",
"$\\left(90-\\frac{1}{2} \\alpha\\right)^{\\circ}$",
"$60^{\\circ}$",
"I don't know"
] | images/1928.jpg | A | We start by drawing the line segment $I G$. Let $P$ be the point on $I G$ such that $P N$ is parallel to $F H$. The angle $P N M$ is alternate to $N M H$ so $\angle P N M=\alpha$. Also, the triangle PNI is similar to the triangle $G H I$ (the angles of each triangle are clearly the same); moreover since $N$ is the ![]... | 2 | metric geometry - angle | A | |
1929 | The distance from the top of the can on the floor to the top of the bottle on the table is $150 \mathrm{~cm}$. The distance from the top of the bottle on the floor to the top of the can on the table is $110 \mathrm{~cm}$. What is the height of the table?
<image1> | [
"$110 \\mathrm{~cm}$",
"$120 \\mathrm{~cm}$",
"$130 \\mathrm{~cm}$",
"$140 \\mathrm{~cm}$",
"$150 \\mathrm{~cm}$",
"I don't know"
] | images/1929.jpg | C | Let $t$ be the height of the table, $b$ the height of the bottle and $c$ the height of the can (all measured in $\mathrm{cm}$ ). The first diagram shows $t+b=c+150$; the second diagram shows $t+c=b+110$. Adding these equations gives $2 t+b+c=260+b+c$ so $2 t=260$. So the table has height $130 \mathrm{~cm}$. | 3 | algebra | C | |
1930 | The diagram shows three congruent regular hexagons. Some diagonals have been drawn, and some regions then shaded. The total shaded areas of the hexagons are $X, Y, Z$ as shown. Which of the following statements is true? <image1> | [
"$X, Y$ and $Z$ are all the same",
"$\\quad Y$ and $Z$ are equal, but $X$ is different",
"$X$ and $Z$ are equal, but $Y$ is different",
"$X$ and $Y$ are equal, but $Z$ is different",
"$X, Y, Z$ are all different",
"I don't know"
] | images/1930.jpg | A | By joining the vertices of the inner triangle to the centre of the hexagon in both the first and third diagrams, it can be seen that each hexagon has been dissected into six equal parts, three of which are shaded. Therefore, $X, Y$ and $Z$ are each half of the hexagon and hence they are all the same. ![](https://cdn.ma... | 4 | metric geometry - area | A | |
1937 | Two chords $P Q$ and $P R$ are drawn in a circle with diameter $P S$. The point $T$ lies on $P R$ and $Q T$ is perpendicular to $P R$. The angle $Q P R=60^{\circ}, P Q=24 \mathrm{~cm}, R T=3 \mathrm{~cm}$. What is the length of the chord $Q S$ in $\mathrm{cm}$ ? <image1> | [
"$\\sqrt{3}$",
"2",
"3",
"$2 \\sqrt{3}$",
"$3 \\sqrt{2}$",
"I don't know"
] | images/1937.jpg | D | $P S$ is a diameter, so $\angle P R S=90^{\circ}$ (angle in a semicircle). Let $U$ be the point on $Q T$ for which $S U$ is perpendicular to $Q T$. Hence RSUT is a rectangle, and $S U=T R=3$. In triangle $T P Q$, we have $\angle P T Q=90^{\circ}$ and $\angle Q P T=60^{\circ}$, so $\angle T Q P=30^{\circ}$. Also, $\ang... | 4 | metric geometry - length | D | |
1938 | Two angles are marked on the $3 \times 3$ grid of squares.
<image1>
Which of the following statements about the angles is correct? | [
"$\\alpha=\\beta$",
"$2 \\alpha+\\beta=90$",
"$\\alpha+\\beta=60$",
"$2 \\beta+\\alpha=90$",
"$\\alpha+\\beta=45$",
"I don't know"
] | images/1938.jpg | B | The triangle $P Q R$ is congruent to the triangle $T Q S$ since they are right-angled triangles with sides of length 3 and 2 . Hence the angle $P Q R$ is also $\alpha^{\circ}$ and then $\alpha+\beta+\alpha=90$. One can check that the other statements are false. ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed27879... | 2 | metric geometry - angle | B | |
1939 | Inside each unit square a certain part has been shaded. In which square is the total shaded area the largest?
<image1> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1939.jpg | A | In each square, each triangle has height one unit. In each of the squares B, C, D, E, the sum of the bases of these triangles is one unit since they cover one side of the unit square, so the shaded area is half of each unit square. However, in square A, there is a rectangle which covers double the area of a triangle of... | 4 | metric geometry - area | A | |
1940 | On each of three pieces of paper a five-digit number is written as shown. Three of the digits are covered. The sum of the three numbers is 57263 . What are the covered digits? <image1> | [
"0,2 and 2",
"1,2 and 9",
"2, 4 and 9",
"2,7 and 8",
"5,7 and 8",
"I don't know"
] | images/1940.jpg | B | Let the missing digits be $P, Q, R$. Placing the numbers in a column addition, we get: $$ \begin{aligned} & \begin{array}{lllll} 1 & 5 & 7 & 2 & 8 \end{array} \\ & \begin{array}{lllll} 2 & 2 & P & 0 & 4 \end{array} \\ & +\begin{array}{ccccc} Q & R & 3 & 3 & 1 \\ \hline 5 & 7 & 2 & 6 & 3 \end{array} \end{aligned} $$ T... | 3 | algebra | B | |
1941 | The flag of Kangaria is a rectangle with side-lengths in the ratio $3: 5$. The flag is divided into four rectangles of equal area as shown. What is the ratio of the length of the shorter sides of the white rectangle to the length of its longer sides? <image1> | [
"$1: 3$",
"$1: 4$",
"$2: 7$",
"$3: 10$",
"$4: 15$",
"I don't know"
] | images/1941.jpg | E | Let the length of the white rectangle be $x$ and its height $y$. Then the height of the flag is $3 y$ and hence its width is $5 y$. The four rectangles which make up the flag are equal in area, so we have $3 y \times 5 y=4 x y$. This simplifies to $15 y=4 x$ (since $y$ is non-zero) and hence $y: x=4: 15$. | 4 | metric geometry - length | E | |
1942 | The diagram shows a shape made of arcs of three circles, each with radius $R$. The centres of the circles lie on the same straight line, and the middle circle passes through the centres of the other two circles. What is the perimeter of the shape? <image1> | [
"$\\frac{2 \\pi R \\sqrt{3}}{3}$",
"$\\frac{5 \\pi R}{3}$",
"$\\frac{10 \\pi R}{3}$",
"$2 \\pi R \\sqrt{3}$",
"$4 \\pi R$",
"I don't know"
] | images/1942.jpg | C | By drawing radii and chords as shown, we can see that the triangles are equilateral and therefore each of the angles is $60^{\circ}$. Hence the left and right circles have each lost $120^{\circ}$ (one third) of their circumferences, and the central circle has one-third $\left(\frac{1}{6}+\frac{1}{6}\right)$ of its circ... | 4 | metric geometry - length | C | |
1943 | The diagram shows a net of an octahedron. When this is folded to form the octahedron, which of the labelled line segments will coincide with the line segment labelled $x$ ? <image1> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1943.jpg | E | The four triangles on the left will fold to form one square-based pyramid (without the base). The four triangles on the right will fold to make another pyramid, with the two pyramids hinged at the dashed edge. When these two pyramids are folded at this edge, the bottom end of $x$ will coincide with the right-hand end o... | 2 | solid geometry | E | |
1944 | A square has two of its vertices on a semicircle and the other two on the diameter of the semicircle as shown. The radius of the circle is 1 . What is the area of the square? <image1> | [
"$\\frac{4}{5}$",
"$\\frac{\\pi}{4}$",
"1",
"$\\frac{4}{3}$",
"$\\frac{2}{\\sqrt{3}}$",
"I don't know"
] | images/1944.jpg | A | Let $O$ be the centre of the circle, and $P, Q, P^{\prime}$ and $Q^{\prime}$ the vertices of the square. The triangles $O P Q$ and $O P^{\prime} Q^{\prime}$ are congruent since they are right-angled and have two equal sides $\left(P Q=P^{\prime} Q^{\prime}\right.$ since they are edges of a square, and $O P=O P^{\prime}... | 4 | metric geometry - area | A | |
1945 | A network consists of 16 vertices and 24 edges that connect them, as shown. An ant begins at the vertex labelled Start. Every minute, it walks from one vertex to a neighbouring vertex, crawling along a connecting edge. At which of the vertices labelled $P, Q, R, S, T$ can the ant be after 2019 minutes? <image1> | [
"only $P, R$ or $S$,",
"not $Q$",
"only $Q$",
"only $T$",
"all of the vertices are possible",
"I don't know"
] | images/1945.jpg | C | Labelling vertices alternately $0 / 1$ leads to the labelling shown. After an odd number of steps, the ant is always on a vertex labelled 1 . The only such vertex labelled with a letter is $Q$. ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-268.jpg?height=437&width=394&top_left_y=1141&top_left_x=... | 3 | algebra | C | |
1946 | The diagram shows triangle $J K L$ of area $S$. The point $M$ is the midpoint of $K L$. The points $P, Q, R$ lie on the extended lines $L J, M J, K J$, respectively, such that $J P=2 \times J L, J Q=3 \times J M$ and $J R=4 \times J K$.
What is the area of triangle $P Q R$ ? <image1> | [
"$S$",
"$2 S$",
"$3 S$",
"$\\frac{1}{2} S$",
"$\\frac{1}{3} S$",
"I don't know"
] | images/1946.jpg | A | Let $u$ be the area of triangle $J L M$, and $v$ be the area of triangle $J M K$. Since $M$ is the midpoint of $K L$, the triangle $J L M$ has half the area of the triangle $J L K$, so $u=v=\frac{1}{2} S$. Note that $\angle Q J R=\angle K J M, J R=4 \times J K$ and $J Q=3 \times J M$. Hence, using the formula "Area of... | 4 | metric geometry - area | A | |
1949 | The shortest path from Atown to Cetown runs through Betown. Two of the signposts that can be seen on this path are shown, but one of them is broken and a number missing. What distance was written on the broken sign? <image1> | [
"$2 \\mathrm{~km}$",
"$3 \\mathrm{~km}$",
"$4 \\mathrm{~km}$",
"$5 \\mathrm{~km}$",
"$6 \\mathrm{~km}$",
"I don't know"
] | images/1949.jpg | A | The first signpost shows that Betown is $4 \mathrm{~km}$ from Atown. The second signpost is $6 \mathrm{~km}$ from Atown, so must be $2 \mathrm{~km}$ from Betown. | 4 | metric geometry - length | A | |
1951 | Two squares of different sizes are drawn inside an equilateral triangle. One side of one of these squares lies on one of the sides of the triangle as shown. What is the size of the angle marked by the question mark? <image1> | [
"$25^{\\circ}$",
"$30^{\\circ}$",
"$35^{\\circ}$",
"$45^{\\circ}$",
"$50^{\\circ}$",
"I don't know"
] | images/1951.jpg | E | On the diagram is a pentagon outlined, and four of its angles are known. The sum of the angles in a pentagon is $540^{\circ}$. The missing angle is then $540^{\circ}-270^{\circ}-70^{\circ}-60^{\circ}-90^{\circ}=50^{\circ}$. . A fly is sitting on the exact spot where the six triangles meet. How far from the bottom of the window is the fly sitting? <image1> | [
"$3 \\mathrm{~cm}$",
"$5 \\mathrm{~cm}$",
"$5.5 \\mathrm{~cm}$",
"$6 \\mathrm{~cm}$",
"$7.5 \\mathrm{~cm}$",
"I don't know"
] | images/1952.jpg | D | Let $h$ be the height of the fly above the base of the window. Each side-length of the square window of area $81 \mathrm{~cm}^{2}$ is $9 \mathrm{~cm}$. The two triangles that form the bottom part of the window have total area equal to a third of the whole window, namely $27 \mathrm{~cm}^{2}$. Hence $\frac{1}{2} \times ... | 4 | metric geometry - length | D | |
1953 | Two identical rectangles with sides of length $3 \mathrm{~cm}$ and $9 \mathrm{~cm}$ are overlapping as in the diagram. What is the area of the overlap of the two rectangles? <image1> | [
"$12 \\mathrm{~cm}^{2}$",
"$13.5 \\mathrm{~cm}^{2}$",
"$14 \\mathrm{~cm}^{2}$",
"$15 \\mathrm{~cm}^{2}$",
"$16 \\mathrm{~cm}^{2}$",
"I don't know"
] | images/1953.jpg | D | First we prove that the four white triangles are congruent. Note that they each have a right angle. Also angles $H G A$ and $F G E$ are equal (vertically opposite). The quadrilateral $A C E G$ is a parallelogram since each of its sides come from the rectangles. Hence angles $F G E$ and $G A C$ are equal (corresponding)... | 4 | transformation geometry | D | |
1955 | A zig-zag line starts at the point $P$, at one end of the diameter $P Q$ of a circle. Each of the angles between the zig-zag line and the diameter $P Q$ is equal to $\alpha$ as shown. After four peaks, the zig-zag line ends at the point $Q$. What is the size of angle $\alpha$ ? <image1> | [
"$60^{\\circ}$",
"$72^{\\circ}$",
"$75^{\\circ}$",
"$80^{\\circ}$",
"$86^{\\circ}$",
"I don't know"
] | images/1955.jpg | B | After four peaks the zig-zag is at the end of the diameter, so after two peaks it must be at the centre $O$ of the circle. The triangle $O P R$ is isosceles since $O P$ and $O R$ are both radii, hence angle $P R O=\alpha$ and angle $P O R=180^{\circ}-2 \alpha$. Angle OTR $=180^{\circ}-\alpha$ (angles on a straight line... | 2 | metric geometry - angle | B | |
1956 | A rectangle with perimeter $30 \mathrm{~cm}$ is divided by two lines, forming a square of area $9 \mathrm{~cm}^{2}$, as shown in the figure.
<image1>
What is the perimeter of the shaded rectangle? | [
"$14 \\mathrm{~cm}$",
"$16 \\mathrm{~cm}$",
"$18 \\mathrm{~cm}$",
"$21 \\mathrm{~cm}$",
"$24 \\mathrm{~cm}$",
"I don't know"
] | images/1956.jpg | C | The square has sides of length $\sqrt{9} \mathrm{~cm}=3 \mathrm{~cm}$. Let $x$ and $y$ be the width and length in $\mathrm{cm}$ of the shaded rectangle. Then the large rectangle has perimeter $2(3+x+3+y)=30$. This gives $6+x+y=15$ so $x+y=9$. Hence the perimeter in $\mathrm{cm}$ of the shaded rectangle is $2(x+y)=2 \ti... | 4 | metric geometry - length | C | |
1957 | Six rectangles are arranged as shown. The number inside each rectangle gives the area, in $\mathrm{cm}^{2}$, of that rectangle. The rectangle on the top left has height $6 \mathrm{~cm}$.
<image1>
What is the height of the bottom right rectangle? | [
"$4 \\mathrm{~cm}$",
"$5 \\mathrm{~cm}$",
"$6 \\mathrm{~cm}$",
"$7.5 \\mathrm{~cm}$",
"$10 \\mathrm{~cm}$",
"I don't know"
] | images/1957.jpg | B | To obtain an area of $18 \mathrm{~cm}^{2}$, the width of the top left rectangle must be $(18 \div 6) \mathrm{cm}=3 \mathrm{~cm}$. Then the bottom left rectangle must have height $(12 \div 3) \mathrm{cm}=4 \mathrm{~cm}$. Similarly the bottom middle rectangle must have width $(16 \div 4) \mathrm{cm}=4 \mathrm{~cm}$, the ... | 3 | algebra | B | |
1958 | Five line segments are drawn inside a rectangle as shown.
<image1>
What is the sum of the six marked angles? | [
"$360^{\\circ}$",
"$720^{\\circ}$",
"$900^{\\circ}$",
"$1080^{\\circ}$",
"$1120^{\\circ}$",
"I don't know"
] | images/1958.jpg | D | The six marked angles, together with the 4 right angles of the rectangle, are the 10 interior angles of a decagon. Since angles in a decagon add up to $(10-2) \times 180^{\circ}=8 \times 180^{\circ}$, the six marked angles add up to $(8 \times 180-4 \times 90)^{\circ}=6 \times 180^{\circ}=1080^{\circ}$. | 2 | metric geometry - angle | D | |
1965 | A caterpillar crawled up a smooth slope from $A$ to $B$, and crept down the stairs from $B$ to $C$. What is the ratio of the distance the caterpillar travelled from $B$ to $C$ to the distance it travelled from $A$ to $B$ ? <image1> | [
"$1: 1$",
"2:1",
"3:1",
"$\\sqrt{2}: 1$",
"$\\sqrt{3}: 1$",
"I don't know"
] | images/1965.jpg | E | Let $h$ be the height of the slope. By dropping the perpendicular from $B$ to the base $A C$, we create two right-angled triangles $A B D$ and $B C D$. Angle $A B D=$ $(180-60-90)^{\circ}=30^{\circ}$ so triangle $A B D$ is half of an equilateral triangle and length $A B$ is twice $A D$. Let length $A D=x$ so that $A B=... | 4 | metric geometry - length | E | |
1969 | On Nadya's smartphone, the diagram shows how much time she spent last week on four of her apps. This week she halved the time spent on two of these apps, but spent the same amount of time as the previous week on the other two apps.
<image1>
Which of the following could be the diagram for this week?
<image2> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1969.jpg | E | In Diagram $\mathrm{E}$ the times for the first and third apps have been halved, while the other two are unchanged. It can be easily checked that the other diagrams do not work. | 1 | statistics | E | |
1973 | The centres of the seven circles shown all lie on the same line. The four smaller circles have radius $1 \mathrm{~cm}$. The circles touch, as shown.
<image1>
What is the total area of the shaded regions? | [
"$\\pi \\mathrm{cm}^{2}$",
"$2 \\pi \\mathrm{cm}^{2}$",
"$3 \\pi \\mathrm{cm}^{2}$",
"$4 \\pi \\mathrm{cm}^{2}$",
"$5 \\pi \\mathrm{cm}^{2}$",
"I don't know"
] | images/1973.jpg | B | If the shaded pieces on the right-hand side are reflected in a central vertical line, the total shaded area is then the area of one large circle minus the areas of two small circles. The radius of each large circle is $2 \mathrm{~cm}$ so the shaded area, in $\mathrm{cm}^{2}$, equals $\pi \times 2^{2}-2 \times \pi \time... | 4 | metric geometry - area | B |
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