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 |
|---|---|---|---|---|---|---|---|---|---|
1408 | A belt system is made up of wheels $A, B$ and $C$, which rotate without sliding. $B$ rotates 4 times around, while $A$ turns 5 times around, and $B$ rotates 6 times around, while $C$ turns 7 times around. The circumference of $C$ is $30 \mathrm{~cm}$. How big is the circumference of $A$?
<image1> | [
"$27 \\mathrm{~cm}$",
"$28 \\mathrm{~cm}$",
"$29 \\mathrm{~cm}$",
"$30 \\mathrm{~cm}$",
"$31 \\mathrm{~cm}$",
"I don't know"
] | images/1408.jpg | B | null | 4 | metric geometry - length | B | |
1410 | In a convex quadrilateral $A B C D$ the diagonals are perpendicular to each other. The length of the edges are $A B=2017, B C=2018$ and $C D=2019$ (diagram not to scale). How long is side $A D$?
<image1> | [
"2016",
"2018",
"$\\sqrt{2020^{2}-4}$",
"$\\sqrt{2018^{2}+2}$",
"2020",
"I don't know"
] | images/1410.jpg | D | null | 4 | metric geometry - length | D | |
1412 | Three weights are randomly placed on each tray of a beam balance. The balance dips to the right hand side as shown on the picture. The masses of the weights are 101, 102, 103, 104, 105 and 106 grams. For how many percent of the possible distributions is the 106grams-weight on the right (heavier) side?
<image1> | [
"$75 \\%$",
"$80 \\%$",
"$90 \\%$",
"$95 \\%$",
"$100 \\%$",
"I don't know"
] | images/1412.jpg | B | null | 4 | combinatorics | B | |
1415 | The distance between the top of the cat that is sitting on the table to the top of the cat that is sleeping on the floor is $150 \mathrm{~cm}$. The distance from the top of the cat that is sleeping on the table to the top of the cat that is sitting on the floor is $110 \mathrm{~cm}$. How high is the table?
<image1> | [
"$110 \\mathrm{~cm}$",
"$120 \\mathrm{~cm}$",
"$130 \\mathrm{~cm}$",
"$140 \\mathrm{~cm}$",
"$150 \\mathrm{~cm}$",
"I don't know"
] | images/1415.jpg | C | null | 3 | algebra | C | |
1416 | In the three regular hexagons shown, $X, Y$ and $Z$ describe in this order the areas of the grey shaded parts. Which of the following statements is true?
<image1> | [
"$X=Y=Z$",
"$Y=Z \\neq X$",
"$Z=X \\neq Y$",
"$X=Y \\neq Z$",
"Each of the areas has a different value.",
"I don't know"
] | images/1416.jpg | A | null | 4 | metric geometry - area | A | |
1423 | Seven little dice were removed from a $3 \times 3 \times 3$ die, as can be seen in the diagram. The remaining (completely symmetrical) figure is cut along a plane through the centre and perpendicular to one of the four space diagonals. What does the cross-section look like?
<image1>
<image2> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1423.jpg | A | null | 3 | descriptive geometry | A | |
1424 | Two chords $A B$ and $A C$ are drawn into a circle with diameter $\mathrm{AD}. \angle B A C=60^{\circ}$, $\overline{A B}=24 \mathrm{~cm}$, $\mathrm{E}$ lies on $\mathrm{AC}$ so that $\overline{E C}=3 \mathrm{~cm}$, and $\mathrm{BE}$ is perpendicular to $\mathrm{AC}$. How long is the chord $\mathrm{BD}$?
<image1> | [
"$\\sqrt{3} \\mathrm{~cm}$",
"$2 \\mathrm{~cm}$",
"$3 \\mathrm{~cm}$",
"$2 \\sqrt{3} \\mathrm{~cm}$",
"$3 \\sqrt{2} \\mathrm{~cm}$",
"I don't know"
] | images/1424.jpg | D | null | 4 | metric geometry - length | D | |
1425 | A barber wants to write the word SHAVE on a board so that a customer who sees the word in the mirror can read the word normally. How does he have to write the word on the board?
<image1> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1425.jpg | E | null | 4 | transformation geometry | E | |
1426 | Five identical measuring jugs are filled with water. Four of them contain exactly the same amount of water. Which measuring jug contains a different amount?
<image1> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1426.jpg | B | null | 1 | statistics | B | |
1427 | Which of the following statements is definitely true for the angle marked in the diagram which is made up of nine squares?
<image1> | [
"$\\alpha=\\beta$",
"$2 \\alpha+\\beta=90^{\\circ}$",
"$\\alpha+\\beta=60^{\\circ}$",
"$2 \\beta+\\alpha=90^{\\circ}$",
"$\\alpha+\\beta=45^{\\circ}$",
"I don't know"
] | images/1427.jpg | B | null | 2 | metric geometry - angle | B | |
1428 | Inside a unit square a certain area has been coloured in black. In which square is the black area biggest?
<image1> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1428.jpg | A | null | 4 | metric geometry - area | A | |
1429 | Three five-digit numbers are written onto three separate pieces of paper as shown. Three of the digits in the picture are hidden. The sum of the three numbers is 57263. Which are the hidden 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/1429.jpg | B | null | 3 | algebra | B | |
1430 | The flag of Kanguria is a rectangle whose side lengths are in the ratio $3: 5$. The flag is split into four rectangles of equal area as shown. In which ratio are the side lengths of the white rectangle?
<image1> | [
"$1: 3$",
"$1: 4$",
"$2: 7$",
"$3: 10$",
"$4: 15$",
"I don't know"
] | images/1430.jpg | E | null | 4 | metric geometry - length | E | |
1432 | The diagram consists of three circles of equal radius $R$. The centre of those circles lie on a common straight line where the middle circle goes through the centres of the other two circles (see diagram). How big is the perimeter of the figure?
<image1> | [
"$\\frac{10 \\pi R}{3}$",
"$\\frac{5 \\pi R}{3}$",
"$\\frac{2 \\pi R \\sqrt{3}}{3}$",
"$2 \\pi R \\sqrt{3}$",
"$4 \\pi R$",
"I don't know"
] | images/1432.jpg | A | null | 4 | metric geometry - length | A | |
1434 | Two vertices of a square lie on a semi-circle as shown, while the other two lie on its diameter. The radius of the circle is $1 \mathrm{~cm}$. How big is the area of the square?
<image1> | [
"$\\frac{4}{5} \\mathrm{~cm}^{2}$",
"$\\frac{\\pi}{4} \\mathrm{~cm}^{2}$",
"$1 \\mathrm{~cm}^{2}$",
"$\\frac{4}{3} \\mathrm{~cm}^{2}$",
"$\\frac{2}{\\sqrt{3}} \\mathrm{~cm}^{2}$",
"I don't know"
] | images/1434.jpg | A | null | 4 | metric geometry - area | A | |
1435 | A graph consists of 16 points and several connecting lines as shown in the diagram. An ant is at point $A$. With every move the ant can move from the point where it currently is, along one of the connecting lines, to an adjacent point. At which of the points $P, Q, R, S$ and $T$ can the ant be after 2019 moves?
<image1... | [
"only at $P, R$ or $S$, not at $Q$ or $T$",
"only at $P$, $R$, $S$ or $T$, not at $Q$",
"only at $Q$",
"only at $T$",
"At all of the points",
"I don't know"
] | images/1435.jpg | C | null | 3 | algebra | C | |
1437 | In the addition beside, different letters represent different numbers and equal letters represent equal numbers. The resulting sum is a number of four digits, B being different from zero. What is the sum of the numbers of this number?
<image1> | [
"AA",
"BB",
"AB",
"BE",
"EA",
"I don't know"
] | images/1437.jpg | B | null | 3 | algebra | B | |
1438 | There are several figures that can be formed by nine squares of $1 \mathrm{~cm}$ side by side (see an example beside) and one of them has the biggest perimeter. What is this perimeter?
<image1> | [
"$12 \\mathrm{~cm}$",
"$14 \\mathrm{~cm}$",
"$16 \\mathrm{~cm}$",
"$18 \\mathrm{~cm}$",
"$20 \\mathrm{~cm}$",
"I don't know"
] | images/1438.jpg | E | null | 3 | combinatorial geometry | E | |
1441 | The shortest way from Atown to Cetown is through Betown. Going back by this road from Cetown to Atown, we first find the signposts on the left side of the road. Further on we find the road signs on the right side of the road. How far is it from Betown to Atown?
<image1> | [
"$1 \\mathrm{~km}$",
"$2 \\mathrm{~km}$",
"$3 \\mathrm{~km}$",
"$4 \\mathrm{~km}$",
"$5 \\mathrm{~km}$",
"I don't know"
] | images/1441.jpg | D | null | 4 | metric geometry - length | D | |
1443 | Two circles are tangent to each other and also to two sides of a square. What is the measure of the $A \hat{O} B$ angle, determined by three of these points of tangency, as shown in the figure?
<image1> | [
"$110^{\\circ}$",
"$112^{\\circ}$",
"$120^{\\circ}$",
"$128^{\\circ}$",
"$135^{\\circ}$",
"I don't know"
] | images/1443.jpg | E | null | 2 | metric geometry - angle | E | |
1444 | Ana plays with $n \times n$ boards by placing a token in each of the cells with no common points with other cells containing tokens. In the picture beside we see how to place as many chips as possible on $5 \times 5$ and $6 \times 6$ boards. In this way, how many chips can Ana possibly put on a $2020 \times 2020$ board... | [
"2020",
"4039",
"$674^{2}$",
"$1010^{2}$",
"$2020^{2}$",
"I don't know"
] | images/1444.jpg | D | null | 3 | algebra | D | |
1445 | The next window is a square of area $1 \mathrm{~m}^{2}$ and is composed of four triangles, which areas, indicated in the figure, follow the ratios $3 A=4 B$ and $2 C=3 D$. A fly is placed exactly at the point where these four triangles touch each other. The fly flies directly to the side closest to the window. How much... | [
"$40 \\mathrm{~cm}$",
"$30 \\mathrm{~cm}$",
"$25 \\mathrm{~cm}$",
"$20 \\mathrm{~cm}$",
"$10 \\mathrm{~cm}$",
"I don't know"
] | images/1445.jpg | A | null | 4 | metric geometry - length | A | |
1446 | Julia puts the nine chips on the right in a box. She then takes one chip at a time, without looking, and notes down its digit, obtaining, at the end, a number of nine different digits. What is the probability that the number written by Julia is divisible by 45?
<image1> | [
"$\\frac{1}{9}$",
"$\\frac{2}{9}$",
"$\\frac{1}{3}$",
"$\\frac{4}{9}$",
"$\\frac{8}{9}$",
"I don't know"
] | images/1446.jpg | A | null | 4 | combinatorics | A | |
1447 | A rectangular sheet with one side of $12 \mathrm{~cm}$ is folded along its $20 \mathrm{~cm}$ diagonal. What is the overlapping area of the folded parts, indicated in gray in the picture beside?
<image1> | [
"$24 \\mathrm{~cm}^{2}$",
"$36 \\mathrm{~cm}^{2}$",
"$48 \\mathrm{~cm}^{2}$",
"$50 \\mathrm{~cm}^{2}$",
"$75 \\mathrm{~cm}^{2}$",
"I don't know"
] | images/1447.jpg | E | null | 4 | transformation geometry | E | |
1449 | A large rectangular plot is divided into two lots that are separated from each other by an $A B C D$ fence, as shown in the picture beside. The $A B, B C$ and $C D$ parts of this fence are parallel to the sides of the rectangle and have lengths of $30 \mathrm{~m}$, $24 \mathrm{~m}$ and $10 \mathrm{~m}$, respectively. T... | [
"$8 \\mathrm{~m}$",
"$10 \\mathrm{~m}$",
"$12 \\mathrm{~m}$",
"$14 \\mathrm{~m}$",
"$16 \\mathrm{~m}$",
"I don't know"
] | images/1449.jpg | C | null | 4 | metric geometry - length | C | |
1451 | Jenny looks at her weather app that shows the predicted weather and maximum temperatures for the next five days. Which of the following represents the corresponding graph of maximum temperatures?
<image1>
<image2> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1451.jpg | B | null | 1 | statistics | B | |
1452 | A park is shaped like an equilateral triangle. A cat wants to walk along one of the three indicated paths (thicker lines) from the upper corner to the lower right corner. The lengths of the paths are $P, Q$ and $R$, as shown. Which of the following statements about the lengths of the paths is true?
<image1> | [
"$P<Q<R$",
"$P<R<Q$",
"$P<Q=R$",
"$P=R<Q$",
"$P=Q=R$",
"I don't know"
] | images/1452.jpg | B | null | 4 | metric geometry - length | B | |
1453 | Six rectangles are arranged as shown. The top left-hand rectangle has height $6 \mathrm{~cm}$. The numbers within the rectangles indicate their areas in $\mathrm{cm}^{2}$. What is the height of the bottom right-hand rectangle?
<image1> | [
"$4 \\mathrm{~cm}$",
"$5 \\mathrm{~cm}$",
"$6 \\mathrm{~cm}$",
"$7.5 \\mathrm{~cm}$",
"$10 \\mathrm{~cm}$",
"I don't know"
] | images/1453.jpg | B | null | 4 | metric geometry - length | B | |
1454 | Six congruent rhombuses, each of area $5 \mathrm{~cm}^{2}$, form a star. The tips of the star are joined to draw a regular hexagon, as shown. What is the area of the hexagon?
<image1> | [
"$36 \\mathrm{~cm}^{2}$",
"$40 \\mathrm{~cm}^{2}$",
"$45 \\mathrm{~cm}^{2}$",
"$48 \\mathrm{~cm}^{2}$",
"$60 \\mathrm{~cm}^{2}$",
"I don't know"
] | images/1454.jpg | C | null | 4 | metric geometry - area | C | |
1455 | A rectangle with perimeter $30 \mathrm{~cm}$ is divided into four parts by a vertical line and a horizontal line. One of the parts is a square of area $9 \mathrm{~cm}^{2}$, as shown in the figure. What is the perimeter of rectangle $A B C D$?
<image1> | [
"$14 \\mathrm{~cm}$",
"$16 \\mathrm{~cm}$",
"$18 \\mathrm{~cm}$",
"$21 \\mathrm{~cm}$",
"$24 \\mathrm{~cm}$",
"I don't know"
] | images/1455.jpg | C | null | 4 | metric geometry - length | C | |
1456 | Ally drew 3 triangles on a grid. Exactly 2 of them have the same area, exactly 2 of them are isosceles, and exactly 2 are right-angled triangles. 2 of the triangles are shown. Which could be the third one?
<image1>
<image2> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1456.jpg | D | null | 3 | combinatorial geometry | D | |
1458 | The figure shows a semicircle with center $O$. Two of the angles are given. What is the size, in degrees, of the angle $\alpha$?
<image1> | [
"$9^{\\circ}$",
"$11^{\\circ}$",
"$16^{\\circ}$",
"$17.5^{\\circ}$",
"$18^{\\circ}$",
"I don't know"
] | images/1458.jpg | A | null | 2 | metric geometry - angle | A | |
1461 | What is the sum of the six marked angles in the picture?
<image1> | [
"$360^{\\circ}$",
"$900^{\\circ}$",
"$1080^{\\circ}$",
"$1120^{\\circ}$",
"$1440^{\\circ}$",
"I don't know"
] | images/1461.jpg | C | null | 2 | metric geometry - angle | C | |
1463 | An ant climbs from $C$ to $A$ on path $C A$ and descends from $A$ to $B$ on the stairs, as shown in the diagram. What is the ratio of the lengths of the ascending and descending paths?
<image1> | [
"1",
"$\\frac{1}{2}$",
"$\\frac{1}{3}$",
"$\\frac{\\sqrt{2}}{2}$",
"$\\frac{\\sqrt{3}}{3}$",
"I don't know"
] | images/1463.jpg | E | null | 4 | metric geometry - length | E | |
1468 | The midpoints of both longer sides of a rectangle are connected with the vertices (see diagram). Which fraction of the rectangle is shaded?
<image1> | [
"$\\frac{1}{5}$",
"$\\frac{1}{4}$",
"$\\frac{2}{7}$",
"$\\frac{1}{3}$",
"$\\frac{2}{5}$",
"I don't know"
] | images/1468.jpg | B | null | 4 | metric geometry - area | B | |
1469 | Sonja's smartphone displays the diagram on the right. It shows how long she has worked with four different apps in the previous week. This week he has spent only half the amount of time using two of the apps and the same amount of time as last week using the other two apps. Which of the following pictures could be the ... | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1469.jpg | C | null | 1 | statistics | C | |
1472 | The diagram shows three big circles of equal size and four small circles. Each small circle touches two big circles and has radius 1. How big is the shaded area?
<image1> | [
"$\\pi$",
"$2 \\pi$",
"$3 \\pi$",
"$4 \\pi$",
"$6 \\pi$",
"I don't know"
] | images/1472.jpg | B | null | 4 | metric geometry - area | B | |
1474 | The rectangle $A B C D$ is made up of 12 congruent rectangles (see diagram). How big is the ratio $\frac{A D}{D C}$?
<image1> | [
"$\\frac{8}{9}$",
"$\\frac{5}{6}$",
"$\\frac{7}{8}$",
"$\\frac{2}{3}$",
"$\\frac{9}{8}$",
"I don't know"
] | images/1474.jpg | A | null | 3 | combinatorial geometry | A | |
1476 | The diagram shows a square $P Q R S$ with side length 1. The point $U$ is the midpoint of the side $R S$ and the point $W$ is the midpoint of the square. The three line segments, $T W, U W$ and $V W$ split the square into three equally big areas. How long is the line segment $S V$?
<image1> | [
"$\\frac{1}{2}$",
"$\\frac{2}{3}$",
"$\\frac{3}{4}$",
"$\\frac{4}{5}$",
"$\\frac{5}{6}$",
"I don't know"
] | images/1476.jpg | E | null | 4 | metric geometry - length | E | |
1479 | Twelve weights have integer masses of $1 \mathrm{~g}, 2 \mathrm{~g}, 3 \mathrm{~g}, \ldots, 11 \mathrm{~g}$ and $12 \mathrm{~g}$ respectively. A vendor divides those weights up into 3 groups of 4 weights each. The total mass of the first group is $41 \mathrm{~g}$, the mass of the second group is $26 \mathrm{~g}$ (see d... | [
"$3 \\mathrm{~g}$",
"$5 \\mathrm{~g}$",
"$7 \\mathrm{~g}$",
"$8 \\mathrm{~g}$",
"$10 \\mathrm{~g}$",
"I don't know"
] | images/1479.jpg | C | null | 3 | algebra | C | |
1481 | Consider the five circles with midpoints $A, B, C, D$ and $E$ respectively, which touch each other as displayed in the diagram. The line segments, drawn in, connect the midpoints of adjacent circles. The distances between the midpoints are $A B=16, B C=14, C D=17, D E=13$ and $A E=14$ Which of the points is the midpoin... | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1481.jpg | A | null | 4 | metric geometry - length | A | |
1482 | A hemispheric hole is carved into each face of a wooden cube with sides of length 2. All holes are equally sized, and their midpoints are in the centre of the faces of the cube. The holes are as big as possible so that each hemisphere touches each adjacent hemisphere in exactly one point. How big is the diameter of the... | [
"1",
"2",
"$\\sqrt{2}$",
"$\\frac{3}{2}$",
"$\\sqrt{\\frac{3}{2}}$",
"I don't know"
] | images/1482.jpg | C | null | 2 | solid geometry | C | |
1483 | A dark disc with two holes is placed on the dial of a watch as shown in the diagram. The dark disc is now rotated so that the number 10 can be seen through one of the two holes. Which of the numbers could one see through the other hole now? <image1> | [
"2 and 6",
"3 and 7",
"3 and 6",
"1 and 9",
"2 and 7",
"I don't know"
] | images/1483.jpg | A | null | 4 | transformation geometry | A | |
1484 | On her way to school Maria first had to run to the underground, she exited from that after two stops and subsequently walked the rest of the way by foot all the way to school. Which of the following speed-time-diagrams best describes her journey to school?
<image1> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1484.jpg | D | null | 5 | analytic geometry | D | |
1485 | A small square with side length $4 \mathrm{~cm}$ is drawn within a big square with side length $10 \mathrm{~cm}$; their sides are parallel to each other (see diagram). What percentage of the figure is shaded? <image1> | [
"$25 \\%$",
"$30 \\%$",
"$40 \\%$",
"$42 \\%$",
"$45 \\%$",
"I don't know"
] | images/1485.jpg | D | null | 4 | metric geometry - area | D | |
1486 | The big rectangle shown is divided into 30 equally big squares. The perimeter of the area shaded in grey is $240 \mathrm{~cm}$. How big is the area of the big rectangle? <image1> | [
"$480 \\mathrm{~cm}^{2}$",
"$750 \\mathrm{~cm}^{2}$",
"$1080 \\mathrm{~cm}^{2}$",
"$1920 \\mathrm{~cm}^{2}$",
"$2430 \\mathrm{~cm}^{2}$",
"I don't know"
] | images/1486.jpg | D | null | 3 | combinatorial geometry | D | |
1487 | A straight wooden fence is made up of vertical beams stuck in the ground which are each connected to the next beam by 4 horizontal beams. The fence begins and ends with a vertical beam. Out of how many beams could such a fence be made? <image1> | [
"95",
"96",
"97",
"98",
"99",
"I don't know"
] | images/1487.jpg | B | null | 3 | algebra | B | |
1488 | The diagram shows three adjacent squares with side lengths $3 \mathrm{~cm}, 5 \mathrm{~cm}$ and $8 \mathrm{~cm}$. How big is the area of the shaded in trapezium? <image1> | [
"$13 \\mathrm{~cm}^{2}$",
"$\\frac{55}{4} \\mathrm{~cm}^{2}$",
"$\\frac{61}{4} \\mathrm{~cm}^{2}$",
"$\\frac{65}{4} \\mathrm{~cm}^{2}$",
"$\\frac{69}{4} \\mathrm{~cm}^{2}$",
"I don't know"
] | images/1488.jpg | B | null | 4 | metric geometry - area | B | |
1489 | The points $M$ and $N$ are the midpoints of two sides of the big rectangle (see diagram). Which part of the area of the big rectangle is shaded? <image1> | [
"$\\frac{1}{6}$",
"$\\frac{1}{5}$",
"$\\frac{1}{4}$",
"$\\frac{1}{3}$",
"$\\frac{1}{2}$",
"I don't know"
] | images/1489.jpg | C | null | 4 | metric geometry - area | C | |
1490 | The pentagon $A B C D E$ is split into four triangles that all have the same perimeter (see diagram). Triangle $A B C$ is equilateral and the triangles $A E F, D F E$ and $C D F$ are congruent isosceles triangles. How big is the ratio of the perimeter of the pentagon $A B C D E$ to the perimeter of the triangle $A B C$... | [
"2",
"$\\frac{3}{2}$",
"$\\frac{4}{3}$",
"$\\frac{5}{3}$",
"$\\frac{5}{2}$",
"I don't know"
] | images/1490.jpg | D | null | 4 | metric geometry - length | D | |
1497 | Consider the two touching semicircles with radius 1 and their diameters $A B$ and $C D$ respectively that are parallel to each other. The extensions of the two diameters are also tangents to the respective other semicircle (see diagram). How big is the square of the length $A D$ ? <image1> | [
"16",
"$8+4 \\sqrt{3}$",
"12",
"9",
"$5+2 \\sqrt{3}$",
"I don't know"
] | images/1497.jpg | B | null | 4 | metric geometry - length | B | |
1498 | Leon has drawn a closed loop on the surface of a cuboid.
Which net cannot show his loop? <image1> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1498.jpg | C | null | 3 | descriptive geometry | C | |
1500 | This piece of paper was folded in half twice, and then had two equilateral triangles cut out of it. Which diagram shows how the paper will look when it is unfolded again? <image1>
<image2> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1500.jpg | C | The diagram shows how the cut-out triangles form rhombic holes when the paper is unfolded.  | 3 | transformation geometry | C | |
1502 | There used to be 5 parrots in my cage. Their average value was $€ 6000$. One day while I was cleaning out the cage the most beautiful parrot flew away. The average value of the remaining four parrots was $€ 5000$. What was the value of the parrot that escaped? <image1> | [
"$€ 1000$",
"$€ 2000$",
"$€ 5500$",
"$€ 6000$",
"$€ 10000$",
"I don't know"
] | images/1502.jpg | E | The total value of the 5 parrots was $5 \times € 6000=€ 30000$. After one has flown away, the total value is $4 \times € 5000=€ 20000$. So the value of the escaped parrot was $€ 10000$. | 4 | algebra | E | |
1503 | The net on the right can be cut out and folded to make a cube. Which face will then be opposite the face marked $\mathbf{x}$ ? <image1> | [
"a",
"b",
"c",
"d",
"e",
"I don't know"
] | images/1503.jpg | E | A moment's thought will reveal that the faces marked $a, b, c$ and $d$ will all be adjacent to the face marked $\mathbf{x}$. | 4 | descriptive geometry | E | |
1504 | A transparent square sheet of film lies on a table. The letter $\mathbf{Y}$ is drawn (like this) on the sheet. We turn the sheet clockwise through $90^{\circ}$, then turn it over what is now the left edge of the sheet, and then turn it through $180^{\circ}$. Which figure can we now see?
<image1> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1504.jpg | A | After turning clockwise through $90^{\circ}$, the letter will appear as <. Turning it over the left edge of the sheet gives $>$. Then a turn through $180^{\circ}$ gives $<$. | 3 | transformation geometry | A | |
1506 | Two squares of the same size, and with their edges parallel, cover a circle with a radius of $3 \mathrm{~cm}$, as shown. In square centimetres, what is the total shaded area? <image1> | [
"$8(\\pi-1)$",
"$6(2 \\pi-1)$",
"$(9 \\pi-25)$",
"$9(\\pi-2)$",
"$\\frac{6\\pi}{5}$",
"I don't know"
] | images/1506.jpg | D | From the diagram, it can be seen that the area of the central square is half of the dashed square, that is $\frac{1}{2}\times 6\times 6=18$. The shaded area is the area of the circle less the area of the central square, so is $\pi\times 3^2-18=9(\pi -2)$.  each made from 4 little cubes as shown. The shape shaded black is completely visible, but both of the others are only partially visible. Which of the following shapes is the unshaded one? <image1>
<image2> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1507.jpg | A | The unshaded shape is A because the grey shape must continue behind on the bottom row and so the unshaded shape continues with the hidden back corner.  | 4 | combinatorial geometry | A | |
1508 | In a rectangle $A B C D$, the points $P, Q, R$ and $S$ are the midpoints of sides $A B, B C, C D$ and $A D$ respectively, and $T$ is the midpoint of the line $R S$. What fraction of the area of $A B C D$ is the triangle $P Q T$ ? <image1> | [
"$\\frac{5}{16}$",
"$\\frac{1}{4}$",
"$\\frac{1}{5}$",
"$\\frac{1}{6}$",
"$\\frac{3}{8}$",
"I don't know"
] | images/1508.jpg | B | A moment's thought shows that the rhombus $P Q R S$ occupies half the area of the rectangle, and that the triangle $P Q T$ occupies half the area of the rhombus. Note that the exact position of $T$ on $R S$ is irrelevant. [Alternatively: ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-010.jpg?he... | 3 | metric geometry - area | B | |
1510 | In the diagram the large square is divided into 25 smaller squares. Adding up the sizes of the five angles $X P Y, X Q Y, X R Y, X S Y$ and $X T Y$, what total is obtained? <image1> | [
"$30^{\\circ}$",
"$45^{\\circ}$",
"$60^{\\circ}$",
"$75^{\\circ}$",
"$90^{\\circ}$",
"I don't know"
] | images/1510.jpg | B | The diagram shows the angles rearranged to form a $45^{\circ}$ angle.  | 4 | combinatorial geometry | B | |
1512 | In triangle $ABC, AB=AC, AE=AD$ and angle $BAD=30^{\circ}$. What is the size of angle $CDE$ ? <image1> | [
"$10^{\\circ}$",
"$15^{\\circ}$",
"$20^{\\circ}$",
"$25^{\\circ}$",
"$30^{\\circ}$",
"I don't know"
] | images/1512.jpg | B | Let $x^{\circ}=\angle C D E, y^{\circ}=\angle E D A=\angle D E A, z^{\circ}=\angle D A E$. The sum of the angles in triangle $A E D$ shows that $z=180-2 y$. Then, from triangle $A B C, \angle A C B=\frac{1}{2}(180-(30+180-2 y))=y-15$. Now $\angle A E D=x+\angle A C B$, since it is exterior to triangle $D C E$. Hence ... | 5 | metric geometry - angle | B | |
1515 | The diagram shows a net of a cube, with three dotted lines added. If you folded the net into a cube and then cut along the dotted lines you would have a hole in the cube. What would be the shape of the hole? <image1> | [
"an equilateral triangle",
"a rectangle, but not a square",
"a right-angled triangle",
"a square",
"a hexagon",
"I don't know"
] | images/1515.jpg | A | The three dotted lines are equal in length, so the hole has three equal sides. The diagram shows the assembled cube with the hole removed.  | 4 | descriptive geometry | A | |
1518 | Alfonso the Ostrich has been training for the Head in the Sand Competition in the Animolympiad. He buried his head in the sand last week and pulled it out at 8.15 am on Monday to find he had reached a new personal record - he had been underground for 98 hours and 56 minutes. When did Alfonso bury his head in the sand? ... | [
"On Thursday at 5.19 am",
"On Thursday at $5.41 \\mathrm{am}$",
"On Thursday at $11.11 \\mathrm{am}$",
"On Friday at 5.19 am",
"On Friday at 11.11 am",
"I don't know"
] | images/1518.jpg | A | Alfonso's personal record of 98 hours and 56 minutes amounts to 4 days, 2 hours and 56 minutes. Working back from 8.15 am on Monday takes us back to Thursday at $5.19 \mathrm{am}$. | 5 | arithmetic | A | |
1520 | In a square with sides of length $6 \mathrm{~cm}$ the points $A$ and $B$ are on one of the axes of symmetry, as shown. The shaded area is equal to each of the two unshaded areas.
<image1>
What is the length of $A B$ ? | [
"$3.6 \\mathrm{~cm}$",
"$3.8 \\mathrm{~cm}$",
"$4.0 \\mathrm{~cm}$",
"$4.2 \\mathrm{~cm}$",
"$4.4 \\mathrm{~cm}$",
"I don't know"
] | images/1520.jpg | C | The parallelogram has one third of the area of the square, so each of the shaded triangles, with $A B$ as base, has an area of $\frac{1}{6} \times 6 \times 6=6 \mathrm{~cm}^{2}$. The height of each triangle is $3 \mathrm{~cm}$ and so the length $A B=4 \mathrm{~cm}$. | 2 | metric geometry - length | C | |
1521 | Consecutive numbers have been entered diagonally criss-crossing the square on the right. Which of the following numbers could $x$ not be? <image1> | [
"128",
"256",
"81",
"121",
"400",
"I don't know"
] | images/1521.jpg | A | Since $x$ is the largest number entered, it is a square number and 128 is the only option which is not. | 4 | algebra | A | |
1523 | In the diagram, $A B C D$ is a parallelogram. If $A A_{1}=4 \mathrm{~cm}, D D_{1}=5 \mathrm{~cm}$ and $C C_{1}=7 \mathrm{~cm}$, what is the length of $B B_{1}$ ? <image1> | [
"$9 \\mathrm{~cm}$",
"$11 \\mathrm{~cm}$",
"$12 \\mathrm{~cm}$",
"$16 \\mathrm{~cm}$",
"$21 \\mathrm{~cm}$",
"I don't know"
] | images/1523.jpg | D | Construct a rectangle $P Q R S$ around the parallelogram $A B C D$ so that $P S$ is parallel to $A_{1} C_{1}$. Because triangles $A B Q$ and $C D S$ are congruent, $A Q=C S=$ $7+5 \mathrm{~cm}=12 \mathrm{~cm}$ and so $B B_{1}=A Q+A A_{1}=12+4 \mathrm{~cm}$ $=16 \mathrm{~cm}$. ![](https://cdn.mathpix.com/cropped/2023_1... | 2 | metric geometry - length | D | |
1524 | The diagram shows a cube with edges of length $12 \mathrm{~cm}$. An ant crawls from the point $P$ to the point $Q$ along the route shown. What is the length of the ant's path? <image1> | [
"$40 \\mathrm{~cm}$",
"$48 \\mathrm{~cm}$",
"$50 \\mathrm{~cm}$",
"$60 \\mathrm{~cm}$",
"more information is needed",
"I don't know"
] | images/1524.jpg | D | On the bottom edge, wherever the ant turns upward, altogether she still travels the equivalent of five lengths of $12 \mathrm{~cm}$. | 4 | solid geometry | D | |
1525 | The diagram shows the ground plan of a room. Adjoining walls are perpendicular to each other. The letters $a$ and $b$ on the plan show the lengths of some of the walls. What is the area of the room? <image1> | [
"$3 a b+a^{2}$",
"$8 a+2 b$",
"$3 a b-a^{2}$",
"$b^{2}-a^{2}$",
"$3 a b$",
"I don't know"
] | images/1525.jpg | E | The area of the room is the same as that of a rectangular room with dimensions $3 a$ by $b$. | 3 | metric geometry - area | E | |
1526 | Which of the following cubes can be folded from the net on the right?
<image1>
<image2> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1526.jpg | E | When the net is folded up, the two small shaded squares will appear on the same face, and opposite the face consisting of the larger shaded square. | 4 | descriptive geometry | E | |
1528 | The diagram shows a length of string wound over and under $n$ equal circles. The sum of the diameters of the circles is $d \mathrm{~cm}$. What is the length of the string in $\mathrm{cm}$ ?
<image1> | [
"$\\frac{1}{2} \\pi d$",
"$\\pi d n$",
"$2 \\pi d n$",
"$\\pi d$",
"$d n$",
"I don't know"
] | images/1528.jpg | A | Because the diameter of each circle is $\frac{d}{n} \mathrm{~cm}$, the length of each semicircular arc is $\frac{1}{2} \pi \frac{d}{n} \mathrm{~cm}$. For $n$ semicircles, this gives a total length of $\frac{1}{2} \pi \frac{d}{n} \times n \mathrm{~cm}=\frac{\pi}{2} d \mathrm{~cm}$. | 2 | metric geometry - length | A | |
1529 | Two rectangles $A B C D$ and $D B E F$ are shown in the diagram. What is the area of the rectangle $D B E F$ ? <image1> | [
"$10 \\mathrm{~cm}^{2}$",
"$12 \\mathrm{~cm}^{2}$",
"$13 \\mathrm{~cm}^{2}$",
"$14 \\mathrm{~cm}^{2}$",
"$16 \\mathrm{~cm}^{2}$",
"I don't know"
] | images/1529.jpg | B | The area of triangle $B C D$ is equal to half the area of each of the rectangles $A B C D$ and $D B E F$. So the area of $D B E F$ is $3 \mathrm{~cm} \times 4 \mathrm{~cm}=12 \mathrm{~cm}^{2}$. | 3 | metric geometry - area | B | |
1530 | Five straight lines intersect at a common point and five triangles are constructed as shown. What is the total of the 10 angles marked on the diagram? <image1> | [
"$300^{\\circ}$",
"$450^{\\circ}$",
"$360^{\\circ}$",
"$600^{\\circ}$",
"$720^{\\circ}$",
"I don't know"
] | images/1530.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 $900^{\circ}-180^{\circ}=720^{\circ}$... | 5 | metric geometry - angle | E | |
1531 | Gregor's computer is tracing out a path in the first quadrant as shown in the diagram. In the first second the computer draws the line from the origin to $(1,0)$ and after that it continues to follow the directions indicated in the diagram at a speed of 1 unit length per second.
Which point will the traced path reach a... | [
"$(10,0)$",
"$(1,11)$",
"$(10,11)$",
"$(2,10)$",
"$(11,11)$",
"I don't know"
] | images/1531.jpg | A | The computer will draw its path over 2 minutes or 120 seconds. The number of unit lengths successively between the points $(1,0),(0,2),(3,0),(0,4),(5,0)$, and so on, increases by 2 each time. Starting at 1 and adding consecutive odd numbers leads to the sequence of square numbers, and so the path will reach $(n, 0)$, w... | 4 | algebra | A | |
1532 | Let $a$ and $b$ be the lengths of the two shorter sides of the right-angled triangle shown in the diagram. The longest side, $D$, is the diameter of the large circle and $d$ is the diameter of the small circle, which touches all three sides of the triangle.
Which one of the following expressions is equal to $D+d$ ? <im... | [
"$(a+b)$",
"$2(a+b)$",
"$\\frac{1}{2}(a+b)$",
"$\\sqrt{a b}$",
"$\\sqrt{a^{2}+b^{2}}$",
"I don't know"
] | images/1532.jpg | A | The two tangents drawn from a point to a circle are equal in length, so we can mark the lengths of the tangents $r, s$ and $t$ on the diagram. Since the triangle is right-angled, and a tangent is perpendicular to the radius, through the point of contact, the small quadrilateral is a square, with all sides equal to $r$.... | 2 | metric geometry - length | A | |
1533 | Which of the following is a net for the cube with two holes shown alongside?
<image1>
<image2> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1533.jpg | C | The holes are in the middle of opposite edges of the cube; only net $\mathrm{C}$ has this feature. | 4 | solid geometry | C | |
1534 | The solid shown on the right is made from two cubes. The small cube with edges $1 \mathrm{~cm}$ long sits on top of a bigger cube with edges $3 \mathrm{~cm}$ long. What is the surface area of the whole solid? <image1> | [
"$56 \\mathrm{~cm}^{2}$",
"$58 \\mathrm{~cm}^{2}$",
"$60 \\mathrm{~cm}^{2}$",
"$62 \\mathrm{~cm}^{2}$",
"$64 \\mathrm{~cm}^{2}$",
"I don't know"
] | images/1534.jpg | B | The surface areas of the separate cubes are $6 \mathrm{~cm}^{2}$ and $54 \mathrm{~cm}^{2}$. When the smaller one sits on top of the larger, an area of $1 \mathrm{~cm}^{2}$ is lost from each of the two cubes; thus the surface area of the whole solid is $6+54-2=58 \mathrm{~cm}^{2}$. | 4 | solid geometry | B | |
1535 | If all the statements in the box are true, which of $\mathrm{A}, \mathrm{B}, \mathrm{C}$, $\mathrm{D}$ or $\mathrm{E}$ can be deduced? <image1> | [
"It's red",
"It's a blue square",
"It's red and round",
"It's yellow and round",
"It's blue and round",
"I don't know"
] | images/1535.jpg | E | From the third statement, it is either yellow or blue, but from the fourth and second statements, if it is yellow, then it is red, a contradiction. Hence it is blue and, from the first statement, it is also round. | 4 | logic | E | |
1540 | <image1>
In the diagram above there are 11 cards, each printed with two letters. The diagram below shows a rearangement of the cards, but only the top letters are shown.
<image2>
Which one of the following sequences of letters could appear on the bottom row of the second diagram? | [
"ANJAMKILIOR",
"RLIIMKOJNAA",
"JANAMKILIRO",
"RAONJMILIKA",
"ANMAIKOLIRJ",
"I don't know"
] | images/1540.jpg | E | The table below shows all the possible letters from KILIMANJARO underneath the rearranged letters of MISSISSIPPI:  In options A, B, C and D the underlined letters are not possibl ANJAMKILIO... | 4 | algebra | E | |
1541 | The robot in the diagram has been programmed to move in a straight line and, if it meets a wall (shown by bold lines), to turn right by $90^{\circ}$ and then to continue straight on. If it cannot go straight or turn right it will stop. What will happen to this robot? <image1> | [
"It will stop at $\\mathrm{P} 2$.",
"It will stop at P1.",
"It will stop at $\\mathrm{T} 1$.",
"It will stop at $S 1$.",
"It will never stop.",
"I don't know"
] | images/1541.jpg | E | The robot eventually will go round and round the rectangle whose corners are at $\mathrm{T} 4, \mathrm{~T} 1, \mathrm{~S} 1$ and $\mathrm{S} 4$. | 4 | combinatorial geometry | E | |
1546 | In the diagram, $V W X$ and $X Y Z$ are congruent equilateral triangles and angle $V X Y=80^{\circ}$. What is the size of angle $V W Y$ ? <image1> | [
"$25^{\\circ}$",
"$30^{\\circ}$",
"$35^{\\circ}$",
"$40^{\\circ}$",
"$45^{\\circ}$",
"I don't know"
] | images/1546.jpg | D | The angles of an equilateral triangle are $60^{\circ}$ and triangle $W X Y$ is isosceles. $\angle W X Y=80^{\circ}+60^{\circ}=140^{\circ}$, hence $\angle X W Y=\frac{1}{2}\left(180^{\circ}-140^{\circ}\right)=20^{\circ}$ and so $\angle V W Y=60^{\circ}-20^{\circ}=40^{\circ}$. | 5 | metric geometry - angle | D | |
1547 | Each object shown is made up of 7 cubes. Which of $\mathrm{P}, \mathrm{Q}, \mathrm{R}$ and $\mathrm{S}$ can be obtained by rotating $\mathrm{T}$ ?
<image1> | [
"P and R",
"Q and S",
"only R",
"none of them",
"P, Q and R",
"I don't know"
] | images/1547.jpg | A | Stand each object on a table with the central cube uppermost. Then the plan view, from directly above, of $P$, $\mathrm{R}$ and $\mathrm{T}$ is shown in the left diagram, and that of $\mathrm{Q}$ and $\mathrm{S}$ is on the right. In each case the central cube is shown shaded. Mathematicians and scientists say that they... | 4 | descriptive geometry | A | |
1550 | Marta wants to use 16 square tiles like the one shown to form a $4 \times 4$ square design. The tiles may be turned. Each arc bisects the sides it meets and has length $p \mathrm{~cm}$. She is trying to make the arcs connect to make a long path. What is the length, in centimetres, of the longest possible path? <image1> | [
"$15 p$",
"$20 p$",
"$21 p$",
"$22 p$",
"$25 p$",
"I don't know"
] | images/1550.jpg | D | The diagram on the right shows one way to join 22 arcs for a total length of $22 p \mathrm{~cm}$. This is the maximal length as, in order to use as many as possible of the 32 arcs, one cannot use 4 of the corners or more than 2 of the arcs in touching the outside of the square. ![](https://cdn.mathpix.com/cropped/2023... | 2 | metric geometry - length | D | |
1551 | In the diagram, a square with sides of length $4 \mathrm{~cm}$ and a triangle with the same perimeter as the square are joined together to form a pentagon. What is the perimeter of the pentagon? <image1> | [
"$12 \\mathrm{~cm}$",
"$24 \\mathrm{~cm}$",
"$28 \\mathrm{~cm}$",
"$32 \\mathrm{~cm}$",
"It depends on the size of the triangle",
"I don't know"
] | images/1551.jpg | B | The square (and hence the triangle) has perimeter $16 \mathrm{~cm}$. From $2 \times 16 \mathrm{~cm}$, we have to subtract $2 \times 4 \mathrm{~cm}$ for the common side. Thus the perimeter of the pentagon is $24 \mathrm{~cm}$. | 2 | metric geometry - length | B | |
1553 | In the diagram, three lines intersect at one point, forming angles of $108^{\circ}$ and $124^{\circ}$, as shown. What is the size of the angle marked $x^{\circ}$ ? <image1> | [
"$56^{\\circ}$",
"$55^{\\circ}$",
"$54^{\\circ}$",
"$53^{\\circ}$",
"$52^{\\circ}$",
"I don't know"
] | images/1553.jpg | E | One can see that $y+124=180$, so $y=56$. Then $x=108-56=52$.  | 5 | metric geometry - angle | E | |
1558 | The star on the right is formed from 12 identical equilateral triangles. The length of the perimeter of the star is $36 \mathrm{~cm}$. What is the length of the perimeter of the shaded hexagon? <image1> | [
"$12 \\mathrm{~cm}$",
"$18 \\mathrm{~cm}$",
"$24 \\mathrm{~cm}$",
"$30 \\mathrm{~cm}$",
"$36 \\mathrm{~cm}$",
"I don't know"
] | images/1558.jpg | B | The perimeter of the star is formed from 12 sides of the equilateral triangles and that of the hexagon from 6 sides. So the perimeter of the hexagon is $\frac{1}{2} \times 36=18 \mathrm{~cm}$. | 2 | metric geometry - length | B | |
1559 | In the diagram on the right, $Q S R$ is a straight line, $\angle Q P S=12^{\circ}$ and $P Q=P S=R S$.
What is the size of $\angle Q P R$ ? <image1> | [
"$36^{\\circ}$",
"$42^{\\circ}$",
"$54^{\\circ}$",
"$60^{\\circ}$",
"$84^{\\circ}$",
"I don't know"
] | images/1559.jpg | C | Observing that triangle $P Q S$ is isosceles, we have $\angle P S Q=\frac{1}{2}\left(180^{\circ}-12^{\circ}\right)=84^{\circ}$ and hence $\angle P S R=180^{\circ}-84^{\circ}=96^{\circ}$. Since triangle $P R S$ is also isosceles, we have $\angle S P R=\frac{1}{2}\left(180^{\circ}-96^{\circ}\right)=42^{\circ}$. Hence $... | 5 | metric geometry - angle | C | |
1560 | Which of the following knots consist of more than one loop of rope?
<image1> | [
"$P, R$ and $T$",
"$R, S$ and $T$",
"$P, R, S$ and $T$",
"$$ all of $P, Q, R, S$ and $T$",
"$$ none of $\\mathrm{A}, \\mathrm{B}, \\mathrm{C}$ or $\\mathrm{D}$",
"I don't know"
] | images/1560.jpg | A | The diagrams below show that only $\mathrm{P}, \mathrm{R}$ and $\mathrm{T}$ are made from more than one loop.  | 3 | topology | A | |
1562 | The diagram shows four circles each of which touches the largest square and two adjacent circles. A second square has its vertices at the midpoints of the sides of the largest square and the central square has its vertices at the centres of the circles.
<image1>
What is the ratio of the total shaded area to the area of... | [
"$\\pi: 12$",
"$1: 4$",
"$(\\pi+2): 16$",
"$1: 3$",
"$\\pi: 4$",
"I don't know"
] | images/1562.jpg | B | First, note that the middle-sized square passes through the centres of the four circles. Each side of the middle-sized square together with the edges of the outer square creates a right-angled isosceles triangle with angles of $45^{\circ}$. Thus the angles these sides make with the inner square are also $45^{\circ}$. E... | 3 | metric geometry - area | B | |
1564 | In each of the squares in the grid, one of the letters $P, Q, R$ and $S$ must be entered in such a way that adjacent squares (whether connected by an edge or just a corner) do not contain the same letter. Some of the letters have already been entered as shown.
What are the possibilities for the letter in the shaded squ... | [
"only $Q$",
"only $R$",
"only $S$",
"either $R$ or $S$, but no others",
"it is impossible to complete the grid",
"I don't know"
] | images/1564.jpg | D | It is clear that there is a unique way to complete the top three rows, as shown on the right (start in the second square of the third row). Thereafter it is possible to complete the fourth row with $R$ and $S$ alternating and the fifth row $Q P Q P Q$.  with two of the sides extended to meet at the point $X$. What is the size of the acute angle at $X$ ? <image1> | [
"$40^{\\circ}$",
"$45^{\\circ}$",
"$50^{\\circ}$",
"$55^{\\circ}$",
"$60^{\\circ}$",
"I don't know"
] | images/1565.jpg | E | The exterior angles of a regular nonagon are $360^{\circ} \div 9=40^{\circ}$, whence the interior angles are $180^{\circ}-40^{\circ}=140^{\circ}$. In the arrowhead quadrilateral whose rightmost vertex is $X$, three of the angles are $40^{\circ}, 40^{\circ}$ and $360^{\circ}-140^{\circ}=220^{\circ}$ and these add up to ... | 5 | metric geometry - angle | E | |
1568 | The fractions $\frac{1}{3}$ and $\frac{1}{5}$ have been placed on the
<image1>
number-line shown on the right. At which position should the fraction $\frac{1}{4}$ be placed? | [
"$a$",
"$b$",
"$C$",
"$d$",
"$e$",
"I don't know"
] | images/1568.jpg | A | The difference between $\frac{1}{3}$ and $\frac{1}{5}$ is $\frac{1}{3}-\frac{1}{5}=\frac{2}{15}$. This section of the number line is divided into 16 intervals, each of length $\frac{2}{15} \div 16=\frac{1}{120}$. The difference between $\frac{1}{4}$ and $\frac{1}{5}$ is $\frac{1}{4}-\frac{1}{5}=\frac{1}{20}=\frac{6}{12... | 4 | algebra | A | |
1569 | Three cuts are made through a large cube to make eight smaller cuboids, as shown in the diagram on the right. What is the ratio of the total surface area of these eight cuboids to the total surface area of the original cube?
<image1> | [
"$1: 1$",
"$4: 3$",
"$3: 2$",
"$2: 1$",
"$4: 1$",
"I don't know"
] | images/1569.jpg | D | After the cuts, eight smaller cuboids are formed and so we can conclude that the cuts are parallel to the faces of the large cube. Each of the smaller cuboids has three matching pairs of faces, one on the outside of the large cube and one inside. So the total surface area of the smaller cuboids is twice the surface are... | 4 | solid geometry | D | |
1570 | The diagram shows the plan of a room. Adjoining walls are perpendicular to each other and the lengths of some of the walls are shown. What is the length of the perimeter of the room? <image1> | [
"$3 a+4 b$",
"$3 a+8 b$",
"$6 a+4 b$",
"$6 a+6 b$",
"$6 a+8 b$",
"I don't know"
] | images/1570.jpg | E | One long wall has length $b+2 b+b=4 b$ and the perpendicular long wall has length $a+a+a=3 a$. So the length of the perimeter is $6 a+8 b$. | 2 | metric geometry - length | E | |
1573 | The diagram shows a quadrilateral $A B C D$, in which $A D=B C$, $\angle C A D=50^{\circ}, \angle A C D=65^{\circ}$ and $\angle A C B=70^{\circ}$.
<image1>
What is the size of $\angle A B C$ ? | [
"$50^{\\circ}$",
"$55^{\\circ}$",
"$60^{\\circ}$",
"$65^{\\circ}$",
"Impossible to determine",
"I don't know"
] | images/1573.jpg | B | From the angle sum of a triangle, $\angle A D C=65^{\circ}$. Since $\angle A D C=\angle A C D$, triangle $A C D$ is isosceles and so $A C=A D=B C$. Triangle $A B C$ is therefore isosceles and from the angle sum of a triangle, $\angle B A C=\angle A B C=55^{\circ}$. | 5 | metric geometry - angle | B | |
1574 | Andrea has wound some rope around a piece of wood, as shown in the diagram on the right. She rotates the wood $180^{\circ}$ as shown by the arrow in the diagram. What does she see after the rotation?
<image1>
<image2> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1574.jpg | D | When Andrea started, the rope passed through each notch at the top of the piece of wood. When the wood is rotated through $180^{\circ}$, the rope must now pass through each notch at the bottom of the piece of wood. This means, of the options available, she must see D. However, there is an alternative view that she cou... | 3 | transformation geometry | D | |
1576 | The diagram shows a logo made entirely from semicircular arcs, each with a radius of $2 \mathrm{~cm}, 4 \mathrm{~cm}$ or $8 \mathrm{~cm}$. What fraction of the logo is shaded? <image1> | [
"$\\frac{1}{3}$",
"$\\frac{1}{4}$",
"$\\frac{1}{5}$",
"$\\frac{2}{3}$",
"$\\frac{3}{4}$",
"I don't know"
] | images/1576.jpg | B | The shaded shape and the whole logo are in proportion and the ratio of corresponding lengths is $1: 2$. Therefore the ratio of their areas is $1: 4$ and the shaded area is $\frac{1}{4}$ of the logo. Alternatively: The shaded area can be rearranged into a semicircle of radius $4 \mathrm{~cm}$ which has an area of $\fra... | 3 | metric geometry - area | B | |
1581 | The diagram shows three squares. The medium square is formed by joining the midpoints of the sides of the large square. The small square is formed by joining the midpoints of the sides of the medium square. The area of the small square is $6 \mathrm{~cm}^{2}$. What is the difference between the area of the medium squar... | [
"$3 \\mathrm{~cm}^{2}$",
"$6 \\mathrm{~cm}^{2}$",
"$9 \\mathrm{~cm}^{2}$",
"$12 \\mathrm{~cm}^{2}$",
"$15 \\mathrm{~cm}^{2}$",
"I don't know"
] | images/1581.jpg | D | The diagram on the right shows how the shape can be dissected into sixteen congruent triangles. The small square has been dissected into four triangles, each of area $6 \div 4=1.5 \mathrm{~cm}^{2}$. The difference in area between the medium and the large square is eight of these triangles, that is $8 \times 1.5=12 \mat... | 3 | metric geometry - area | D | |
1582 | Each region in the figure is to be coloured with one of four colours: red $(\mathrm{R})$, green $(\mathrm{G})$, orange $(\mathrm{O})$ or yellow $(\mathrm{Y})$. The colours of only three regions are shown. Any two regions that touch must have different colours. <image1> The colour of the region $\mathrm{X}$ is: | [
"red",
"orange",
"green",
"yellow",
"impossible to determine",
"I don't know"
] | images/1582.jpg | A | Label the regions 1 to 5 as shown in the diagram. Region 1 must be coloured yellow as it touches a red, a green and an orange region. Then region 2 must be coloured red as it touches an orange, a yellow and a green region. Now region 3 must be coloured green as it touches an orange, a yellow and a red region. Then regi... | 4 | logic | A | |
1583 | A square piece of paper is cut into six rectangular pieces as shown in the diagram. When the lengths of the perimeters of the six rectangular pieces are added together, the result is $120 \mathrm{~cm}$. What is the area of the square piece of paper? <image1> | [
"$48 \\mathrm{~cm}^{2}$",
"$64 \\mathrm{~cm}^{2}$",
"$110.25 \\mathrm{~cm}^{2}$",
"$144 \\mathrm{~cm}^{2}$",
"$256 \\mathrm{~cm}^{2}$",
"I don't know"
] | images/1583.jpg | D | Let the square have side $x \mathrm{~cm}$. Label each rectangle and its sides as shown where the units are $\mathrm{cm}$. The perimeters are: \begin{tabular}{lll} $A$ & $B$ & $C$ \\ $2(a+b)$ & $2(a+c)$ & $2(a+d)$ \\ $D$ & $E$ & $F$ \\ $2(e+h)$ & $2(f+h)$ & $2(g+h)$ \end{tabular} Thus the total of the perimeters is: $$ ... | 3 | metric geometry - area | D | |
1584 | Lina has placed two shapes on a $5 \times 5$ board, as shown in the picture on the right. Which of the following five shapes should she place on the empty part of the board so that none of the remaining four shapes will fit in the empty space that is left? (The shapes may be rotated or turned over, but can only be plac... | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1584.jpg | D | By inspection, it is possible to spot the answer is shape $\mathrm{D}$. We can justify this as follows: When Lina places any of the other shapes on the empty part of the board, the shape must cover at least one square on the bottom row so that Lina cannot place shape $\mathrm{C}$ on the board. ![](https://cdn.mathpix.... | 4 | combinatorial geometry | D |
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