id stringlengths 1 4 | question stringlengths 38 1.5k | options listlengths 0 5 | image stringlengths 12 15 | decoded_image imagewidth (px) 96 2.52k | answer stringlengths 1 24 | solution stringlengths 0 2.39k ⌀ | level int64 1 5 | subject stringclasses 15
values |
|---|---|---|---|---|---|---|---|---|
2175 | Triangle $ ABC$ has a right angle at $ B$, $ AB = 1$, and $ BC = 2$. The bisector of $ \angle BAC$ meets $ \overline{BC}$ at $ D$. What is $ BD$?
<image1> | [
"$\\frac{\\sqrt{3} - 1}{2}$",
"$\\frac{\\sqrt{5} - 1}{2}$",
"$\\frac{\\sqrt{5} + 1}{2}$",
"$\\frac{\\sqrt{6} + \\sqrt{2}}{2}$",
"$2\\sqrt{3} - 1$"
] | images/2175.jpg | B | null | 4 | metric geometry - length | |
2580 | In the figure, $\angle A$, $\angle B$, and $\angle C$ are right angles. If $\angle AEB = 40^\circ $ and $\angle BED = \angle BDE$, then $\angle CDE = $
<image1> | [
"$75^\\circ$",
"$80^\\circ$",
"$85^\\circ$",
"$90^\\circ$",
"$95^\\circ$"
] | images/2580.jpg | E | null | 5 | metric geometry - angle | |
2693 | Jerry cuts a wedge from a $6$-cm cylinder of bologna as shown by the dashed curve. Which answer choice is closest to the volume of his wedge in cubic centimeters?
<image1> | [
"48",
"75",
"151",
"192",
"603"
] | images/2693.jpg | C | null | 4 | solid geometry | |
1726 | <image1>
The diagram above shows the front and right-hand views of a solid made up of cubes of side $3 \mathrm{~cm}$. The maximum volume that the solid could have is $\mathrm{V} \mathrm{cm}^{3}$. What is the value of $\mathrm{V}$ ? | [] | images/1726.jpg | 540 | Each cube has volume $3^{3} \mathrm{~cm}^{3}=27 \mathrm{~cm}^{3}$. There are four cubes visible in the base layer from both the front and the side so the maximum number of cubes in the base layer is $4 \times 4=16$. Similarly, the maximum number of cubes in the second layer is $2 \times 2=4$. Hence the maximum number o... | 5 | descriptive geometry | |
2736 | In the given figure hexagon $ABCDEF$ is equiangular, $ABJI$ and $FEHG$ are squares with areas $18$ and $32$ respectively, $\triangle JBK$ is equilateral and $FE=BC$. What is the area of $\triangle KBC$?
<image1> | [
"$6\\sqrt{2}$",
"$9$",
"$12$",
"$9\\sqrt{2}$",
"$32$"
] | images/2736.jpg | C | null | 3 | metric geometry - area | |
1471 | Five squares and two right-angled triangles are placed as shown in the diagram. The numbers 3, 8 and 22 in the squares state the size of the area in $\mathrm{m}^{2}$. How big is the area (in $\mathrm{m}^{2}$ ) of the square with the question mark?
<image1> | [] | images/1471.jpg | 17 | null | 4 | metric geometry - area | |
1046 | Four identical dice are arranged in a row (see the fig.).
<image1>
Each dice has faces with 1, 2, 3, 4, 5 and 6 points, but the dice are not standard, i.e., the sum of the points on the opposite faces of the dice is not necessarily equal to 7. What is the total sum of the points in all the 6 touching faces of the dice? | [] | images/1046.jpg | 20 | null | 5 | arithmetic | |
2823 | The figure below depicts two congruent triangles with angle measures $40^\circ$, $50^\circ$, and $90^\circ$. What is the measure of the obtuse angle $\alpha$ formed by the hypotenuses of these two triangles?\n<image1> | [] | images/2823.jpg | 170 | The intersection of the two triangles is a convex quadrilateral with angle measures $90^{\circ}, 50^{\circ}, 50^{\circ}$, and $\alpha$, whence $\alpha=360^{\circ}-90^{\circ}-2 \cdot 50^{\circ}=170^{\circ}$. | 5 | metric geometry - length | |
2385 | A particle moves through the first quadrant as follows. During the first minute it moves from the origin to $(1,0)$. Thereafter, it continues to follow the directions indicated in the figure, going back and forth between the positive $x$ and $y$ axes, moving one unit of distance parallel to an axis in each minute. At w... | [
"(35,44)",
"(36,45)",
"(37,45)",
"(44,35)",
"(45,36)"
] | images/2385.jpg | D | null | 5 | algebra | |
1655 | Valeriu draws a zig-zag line inside a rectangle, creating angles of $10^{\circ}, 14^{\circ}, 33^{\circ}$ and $26^{\circ}$ as shown. What is the size of the angle marked $\theta$ ? <image1> | [
"$11^{\\circ}$",
"$12^{\\circ}$",
"$16^{\\circ}$",
"$17^{\\circ}$",
"$33^{\\circ}$"
] | images/1655.jpg | A | Add to the diagram three lines parallel to two of the sides of the rectangle, creating angles $a, b, c$, $d, e$ and $f$ as shown. Since alternate angles formed by parallel lines are equal, we have $a=26^{\circ}$ and $f=10^{\circ}$. Since $a+b=33^{\circ}$ and $e+f=14^{\circ}$, we have $b=7^{\circ}$ and $e=4^{\circ}$. S... | 5 | metric geometry - angle | |
620 | A mushroom grows up every day. For five days Maria took a picture of this mushroom, but she wrongly ordered the photos beside. What is the sequence of photos that correctly shows the mushroom growth, from left to right?
<image1> | [
"2-5-3-1-4",
"2-3-4-5-1",
"5-4-3-2-1",
"1-2-3-4-5",
"2-3-5-1-4"
] | images/620.jpg | A | null | 3 | combinatorics | |
1079 | On the inside of a square with side length $7 \mathrm{~cm}$ another square is drawn with side length $3 \mathrm{~cm}$. Then a third square with side length $5 \mathrm{~cm}$ is drawn so that it cuts the first two as shown in the picture on the right. How big is the difference between the black area and the grey area?
<i... | [
"$0 \\mathrm{~cm}^{2}$",
"$10 \\mathrm{~cm}^{2}$",
"$11 \\mathrm{~cm}^{2}$",
"$15 \\mathrm{~cm}^{2}$",
"It can not be decided from the information given."
] | images/1079.jpg | D | null | 3 | metric geometry - area | |
2641 | The area of triangle $ XYZ$ is 8 square inches. Points $ A$ and $ B$ are midpoints of congruent segments $ \overline{XY}$ and $ \overline{XZ}$. Altitude $ \overline{XC}$ bisects $ \overline{YZ}$. What is the area (in square inches) of the shaded region?
<image1> | [
"$1\\frac{1}{2}$",
"$2$",
"$2\\frac{1}{2}$",
"$3$",
"$3\\frac{1}{2}$"
] | images/2641.jpg | D | null | 3 | metric geometry - area | |
353 | After the storm last night, the flagpole on our school building is leaning over. Looking from northwest, its tip is to the right of its bottom point. Looking from the east, its tip is also to the right of its bottom point. In which direction could the flagpole be leaning over?
<image1> | [
"A",
"B",
"C",
"D",
"E"
] | images/353.jpg | A | null | 1 | descriptive geometry | |
1849 | An $n$-pyramid is defined to be a stack of $n$ layers of balls, with each layer forming a triangular array. The layers of a 3-pyramid are shown in the diagram.
An 8-pyramid is now formed where all the balls on the outside of the 8 -pyramid are black (including the base layer) and the balls on the inside are all white. ... | [] | images/1849.jpg | 4 | To form the white pyramid, we must remove all the black balls. The entire base layer is black. The top three layers of an 8-pyramid are the same as the 3-pyramid shown in the question and all the balls will be coloured black. The highest white ball is in the centre of the fourth layer down, and the lowest white balls a... | 2 | solid geometry | |
608 | Linda fixes 3 photos on a pin board next to each other. She uses 8 pins to do so. Peter wants to fix 7 photos in the same way. How many pins does he need for that?
<image1> | [] | images/608.jpg | 16 | null | 3 | arithmetic | |
410 | A cube (on the right) is colored in three colors so that each face has exactly one color and the opposite face has the same color. Which of the following developments is the development of this cube?
<image1>
<image2> | [
"A",
"B",
"C",
"D",
"E"
] | images/410.jpg | E | null | 5 | solid geometry | |
1319 | A 3-pyramid is a stack of the following 3 layers of balls. In the same way we have a 4-pyramid, a 5-pyramid, etc. All the outside balls of an 8-pyramid are removed. What kind of figure form the rest balls?
<image1> | [
"3-pyramid",
"4-pyramid",
"5-pyramid",
"6-pyramid",
"7-pyramid"
] | images/1319.jpg | B | null | 2 | solid geometry | |
1060 | How many lines of symmetry does this figure have?
<image1> | [] | images/1060.jpg | 2 | null | 3 | transformation geometry | |
2438 | A $ 9\times9\times9$ cube is composed of twenty-seven $ 3\times3\times3$ cubes. The big cube is 'tunneled' as follows: First, the six $ 3\times3\times3$ cubes which make up the center of each face as well as the center of $ 3\times3\times3$ cube are removed. Second, each of the twenty remaining $ 3\times3\times3$ cubes... | [] | images/2438.jpg | 1056 | null | 3 | solid geometry | |
423 | The figure shows a rectangular garden with dimensions $16 \mathrm{~m}$ and $20 \mathrm{~m}$. The gardener has planted six identical flowerbeds (they are gray in the diagram). What is the perimeter (in metres) of each of the flowerbeds?
<image1> | [] | images/423.jpg | 24 | null | 3 | metric geometry - length | |
2303 | <image1>
Chords $AB$ and $CD$ in the circle above intersect at $E$ and are perpendicular to each other. If segments $AE$, $EB$, and $ED$ have measures $2$, $3$, and $6$ respectively, then the length of the diameter of the circle is | [
"$4\\sqrt{5}$",
"$\\sqrt{65}$",
"$2\\sqrt{17}$",
"$3\\sqrt{7}$",
"$6\\sqrt{2}$"
] | images/2303.jpg | B | null | 5 | metric geometry - length | |
2619 | Triangles $ABC$, $ADE$, and $EFG$ are all equilateral. Points $D$ and $G$ are midpoints of $\overline{AC}$ and $\overline{AE}$, respectively. If $AB = 4$, what is the perimeter of figure $ABCDEFG$?
<image1> | [] | images/2619.jpg | 15 | null | 2 | metric geometry - length | |
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"
] | 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 | |
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> | [
"$20^{\\circ}$",
"$25^{\\circ}$",
"$30^{\\circ}$",
"$35^{\\circ}$",
"$40^{\\circ}$"
] | 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 | |
1375 | Tarzan wanted to draw a rhombus made up of two equilateral triangles. He drew the line segments inaccurately. When Jane checked the measurements of the four angles shown, she sees that they are not equally big (see diagram). Which of the five line segments in this diagram is the longest?
<image1> | [
"AD",
"AC",
"AB",
"BC",
"BD"
] | images/1375.jpg | A | null | 4 | metric geometry - length | |
552 | Which of the following sentences fits to the picture?
<image1> | [
"There are equally many circles as squares.",
"There are fewer circles than triangles.",
"There are twice as many circles as triangles.",
"There are more squares than triangles.",
"There are two more triangles than circles."
] | images/552.jpg | C | null | 2 | counting | |
1677 | The shortest path from Atown to Cetown runs through Betown. The two signposts shown are set up at different places along this path. What distance is written on the broken sign? <image1> | [
"$1 \\mathrm{~km}$",
"$3 \\mathrm{~km}$",
"$4 \\mathrm{~km}$",
"$5 \\mathrm{~km}$",
"$9 \\mathrm{~km}$"
] | images/1677.jpg | A | The information on the signs pointing to Atown and the signs pointing to Cetown both tell us that the distance between the signs is $(7-2) \mathrm{km}=(9-4) \mathrm{km}=5 \mathrm{~km}$. Therefore the distance which is written on the broken sign is $(5-4) \mathrm{km}=1 \mathrm{~km}$. | 2 | metric geometry - length | |
703 | The picture shows the clown Dave dancing on top of two balls and one cubic box. The radius of the lower ball is $6 \mathrm{dm}$, the radius of the upper ball is three times less. The side of the cubic box is $4 \mathrm{dm}$ longer than the radius of the upper ball. At what height (in $\mathrm{dm}$ ) above the ground is... | [] | images/703.jpg | 22 | null | 2 | algebra | |
1777 | The diagram shows the eight vertices of an octagon connected by line segments. Jodhvir wants to write one of the integers 1,2,3 or 4 at each of the vertices so that the two integers at the ends of every line segment are different. He has already written three integers as shown.
<image1>
How many times will the integer ... | [] | images/1777.jpg | 4 | Let the integers at the vertices of Jodhvir's completed diagram be as shown. The vertices where $a, b, c$ and $e$ are written are all joined by line segments to vertices labelled 1,2 and 3. Therefore, since the two integers at the ends of any line segment are different, each of $a, b, c$ and $e$ is equal to 4 . The ve... | 2 | graph theory | |
995 | Martin has three cards that are labelled on both sides with a number. Martin places the three cards on the table without paying attention to back or front. He adds the three numbers that he can then see. How many different sums can Martin get that way?
<image1> | [
"3",
"5",
"6",
"9",
"A different amount."
] | images/995.jpg | E | null | 3 | combinatorics | |
114 | Two equal trains, each with 31 numbered wagons, travel in opposite directions. When the wagon number 7 of a train is side by side with the wagon number 12 of the other train, which wagon is side by side with the wagon number 11 ?
<image1> | [] | images/114.jpg | 8 | null | 1 | algebra | |
2761 | In triangle $ABC$, point $D$ divides side $\overline{AC}$ so that $AD:DC=1:2$. Let $E$ be the midpoint of $\overline{BD}$ and let $F$ be the point of intersection of line $BC$ and line $AE$. Given that the area of $\triangle ABC$ is $360$, what is the area of $\triangle EBF$?
<image1> | [] | images/2761.jpg | 30 | null | 3 | metric geometry - area | |
18 | A fifteen-meter log has to be sawn into three-meter pieces. How many cuts are needed for that?
<image1> | [] | images/18.jpg | 4 | null | 2 | arithmetic | |
1457 | The numbers from 1 to 6 are placed in the circles at the intersections of 3 rings. The position of number 6 is shown. The sums of the numbers on each ring are the same. What number is placed in the circle with the question mark?
<image1> | [] | images/1457.jpg | 1 | null | 5 | logic | |
1306 | A coin with diameter $1 \mathrm{~cm}$ rolls around the contour outside of a regular hexagon with sides $1 \mathrm{~cm}$ long, as shown. How long is the path traced by the centre of the coin (in $\mathrm{cm}$ )?
<image1> | [
"$6+\\frac{\\pi}{2}$",
"$6+\\pi$",
"$12+\\pi$",
"$6+2 \\pi$",
"$12+2 \\pi$"
] | images/1306.jpg | B | null | 4 | transformation geometry | |
2412 | In the $xy$-plane, consider the L-shaped region bounded by horizontal and vertical segments with vertices at $(0,0), (0,3), (3,3), (3,1), (5,1)$ and $(5,0)$. The slope of the line through the origin that divides the area of this region exactly in half is
<image1> | [
"$\\frac{2}{7}$",
"$\\frac{1}{3}$",
"$\\frac{2}{3}$",
"$\\frac{3}{4}$",
"$\\frac{7}{9}$"
] | images/2412.jpg | E | null | 4 | analytic geometry | |
1339 | In front of a supermarket there are two rows of interconnected trolleys.The first one is $2.9 \mathrm{~m}$ long and consists of 10 trolleys. The second one is $4.9 \mathrm{~m}$ long and consists of twenty trolleys. How long is one trolley?
<image1> | [
"$0.8 \\mathrm{~m}$",
"$1 \\mathrm{~m}$",
"$1.1 \\mathrm{~m}$",
"$1.2 \\mathrm{~m}$",
"$1.4 \\mathrm{~m}$"
] | images/1339.jpg | C | null | 3 | algebra | |
1922 | As shown in the diagram, $F G H I$ is a trapezium with side $G F$ parallel to $H I$. The lengths of $F G$ and $H I$ are 50 and 20 respectively. The point $J$ is on the side $F G$ such that the segment $I J$ divides the trapezium into two parts of equal area. What is the length of $F J$ ? <image1> | [] | images/1922.jpg | 35 | Let $x$ be the length $F J$, and $h$ be the height of the trapezium. Then the area of triangle $F J I$ is $\frac{1}{2} x h$ and the area of trapezium FGHI is $\frac{1}{2} h(20+50)=35 h$. The triangle is half the area of the trapezium, so $\frac{1}{2} x h=\frac{1}{2} \times 35 h$, so $x=35$. | 4 | metric geometry - length | |
1614 | The diagram shows the triangle $P Q R$ in which $R H$ is a perpendicular height and $P S$ is the angle bisector at $P$. The obtuse angle between $R H$ and $P S$ is four times angle $S P Q$. What is angle $R P Q$ ? <image1> | [
"$30^{\\circ}$",
"$45^{\\circ}$",
"$60^{\\circ}$",
"$75^{\\circ}$",
"$90^{\\circ}$"
] | images/1614.jpg | C | Let $X$ be the point where $R H$ meets $P S$. In $\triangle H X P$, $\alpha+90^{\circ}+\angle H X P=180^{\circ}$. This gives $\angle H X P=90^{\circ}-\alpha$. Angles on a straight line add to $180^{\circ}$ so $4 \alpha+90^{\circ}-\alpha=180^{\circ}$ with solution $\alpha=30^{\circ}$. Hence the size of $\angle R P Q$ is... | 5 | metric geometry - angle | |
283 | A scatter diagram on the $x y$-plane gives the picture of a kangaroo as shown on the right. Now the $x$- and the $y$-coordinate are swapped around for every point. What does the resulting picture look like?
<image1>
<image2> | [
"A",
"B",
"C",
"D",
"E"
] | images/283.jpg | A | null | 4 | analytic geometry | |
2667 | How many distinct triangles can be drawn using three of the dots below as vertices?
<image1> | [] | images/2667.jpg | 18 | null | 4 | combinatorial geometry | |
1868 | An oval is constructed from four arcs of circles. Arc $P Q$ is the same as arc $R S$, and has radius $1 \mathrm{~cm}$. Arc $Q R$ is the same as arc PS. At the points $P, Q, R, S$ where the arcs touch, they have a common tangent. The oval touches the midpoints of the sides of a rectangle with dimensions $8 \mathrm{~cm}$... | [] | images/1868.jpg | 6 | Let $C$ be the centre of the $\operatorname{arc} P S$ with radius $r$, and let $T$ be the centre of the arc $P Q$. The tangent at $P$ is common to both arcs so the perpendicular at $P$ to this tangent passes through both centres $T$ and $C$. Let $M$ be the midpoint of the top of the rectangle. The rectangle is tangent ... | 4 | metric geometry - length | |
243 | An archer tries his art on the target shown below on the right. With each of his three arrows he always hits the target. How many different scores could he total with three arrows?
<image1> | [] | images/243.jpg | 19 | null | 5 | combinatorics | |
2032 | The smallest four two-digit primes are written in different squares of a $2 \times 2$ table.
The sums of the numbers in each row and column are calculated.
Two of these sums are 24 and 28.
The other two sums are $c$ and $d$, where $c<d$.
What is the value of $5 c+7 d$ ?
<image1> | [] | images/2032.jpg | 412 | The only way of obtaining sums of 24 and 28 are using $11+13=24$ and $11+17=28$. One possible configuration is shown in the diagram, with other two sums $c=13+19=32$ and $d=17+19=36$. The value of $5 c+7 d$ is $5 \times 32+7 \times 36=412$. \begin{tabular}{|l|l|} \hline 11 & 13 \\ \hline 17 & 19 \\ \hline \end{tabular... | 5 | algebra | |
520 | How many dots are in the picture?
<image1> | [] | images/520.jpg | 181 | null | 2 | counting | |
2634 | The students in Mrs. Sawyer's class were asked to do a taste test of five kinds of candy. Each student chose one kind of candy. A bar graph of their preferences is shown. What percent of her class chose candy E?
<image1> | [] | images/2634.jpg | 20 | null | 2 | statistics | |
2307 | In the adjoining figure $ABCD$ is a square and $CMN$ is an equilateral triangle. If the area of $ABCD$ is one square inch, then the area of $CMN$ in square inches is
<image1> | [
"$2\\sqrt{3}-3$",
"$1-\\frac{\\sqrt{3}}{3}$",
"$\\frac{\\sqrt{3}}{4}$",
"$\\frac{\\sqrt{2}}{3}$",
"$4-2\\sqrt{3}$"
] | images/2307.jpg | A | null | 5 | metric geometry - area | |
2994 | What is the angle of rotation in degrees about point $C$ that maps the darker figure to the lighter image? <image1> | [] | images/2994.jpg | 180 | By looking at the diagram provided, we can see that the line containing the point of rotation lands on top of itself, but the arrow is facing the opposite direction. This tells us that 1/2 of a full $360^{\circ}$ rotation was completed; therefore, the image rotated $360^{\circ}/2 = \boxed{180^{\circ}}$ about point $C$. | 2 | transformation geometry | |
2723 | Toothpicks are used to make a grid that is 60 toothpicks long and 32 toothpicks wide. How many toothpicks are used altogether?
<image1> | [] | images/2723.jpg | 3932 | null | 4 | algebra | |
1398 | Jack wants to keep six tubes each of diameter $2 \mathrm{~cm}$ together using a rubber band. He chooses between the two possible variations shown. How are the lengths of the rubber bands related to each other?
<image1> | [
"In the left picture the band is $\\pi \\mathrm{cm}$ shorter.",
"In the left picture the band is $4 \\mathrm{~cm}$ shorter.",
"In the right picture the band is $\\pi \\mathrm{cm}$ shorter.",
"In the right picture the band is $4 \\mathrm{~cm}$ shorter.",
"Both bands are equally long."
] | images/1398.jpg | E | null | 4 | metric geometry - length | |
2817 | Suppose that in a group of $6$ people, if $A$ is friends with $B$, then $B$ is friends with $A$. If each of the $6$ people draws a graph of the friendships between the other $5$ people, we get these $6$ graphs, where edges represent\nfriendships and points represent people.\n\nIf Sue drew the first graph, how many frie... | [] | images/2817.jpg | 4 | We see that each friendship is shown in all but two of these graphs, so the total number of edges (friendships) is $\frac{1}{4}$ times the sum of edges in all graphs. This indicates there are 11 friendships within the group. The number of friendships Sue has is the number missing in the first graph; this gives $11-7=4$... | 5 | graph theory | |
1074 | Fridolin the hamster runs through the maze shown on the right. On the path there are 16 pumpkin seeds. He is only allowed to cross each junction once. What is the maximum number of pumpkin seeds that he can collect?
<image1> | [] | images/1074.jpg | 13 | null | 3 | graph theory | |
842 | Each square in the shape has an area of $4 \mathrm{~cm}^{2}$. How long is the thick line?
<image1> | [
"$16 \\mathrm{~cm}$",
"$18 \\mathrm{~cm}$",
"$20 \\mathrm{~cm}$",
"$21 \\mathrm{~cm}$",
"$23 \\mathrm{~cm}$"
] | images/842.jpg | B | null | 2 | combinatorial geometry | |
1823 | What is the sum of the 10 angles marked on the diagram on the right? <image1> | [
"$300^{\\circ}$",
"$450^{\\circ}$",
"$360^{\\circ}$",
"$600^{\\circ}$",
"$720^{\\circ}$"
] | 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 | |
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"
] | 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 | |
498 | How many more bricks does the right hand pyramid have than the left hand pyramid?
<image1> | [] | images/498.jpg | 5 | null | 2 | counting | |
2586 | Six different digits from the set
\[\{ 1,2,3,4,5,6,7,8,9\}\]
are placed in the squares in the figure shown so that the sum of the entries in the vertical column is 23 and the sum of the entries in the horizontal row is 12.
The sum of the six digits used is
<image1> | [] | images/2586.jpg | 29 | null | 2 | combinatorics | |
1586 | The first diagram on the right shows a shape constructed from two rectangles. The lengths of two sides are marked: 11 and 13. The shape is cut into three parts and the parts are rearranged, as shown in the second diagram on the right. What is the length marked $x$ ?
<image1> | [] | images/1586.jpg | 37 | The original shape constructed from two rectangles has base of length $11+13=24$. By considering the rearrangement, the lengths 11,13,24 and $x$ can be identified as shown in the diagram. Hence $x=13+24=37$. ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981bb911326g-132.jpg?height=302&width=382&top_left_y=16... | 2 | metric geometry - length | |
1672 | Prab painted each of the eight circles in the diagram red, yellow or blue such that no two circles that are joined directly were painted the same colour. Which two circles must have been painted the same colour? <image1> | [
"5 and 8",
"1 and 6",
"2 and 7",
"4 and 5",
"3 and 6"
] | images/1672.jpg | A | Since no two circles that are joined directly are painted the same colour and circles 2,5 and 6 are joined to each other, they are all painted different colours. Similarly circles 2, 6 and 8 join to each other and hence are painted different colours. Therefore circles 5 and 8 must have been painted the same colour. It ... | 3 | graph theory | |
1491 | A tower consists of blocks that are labelled from bottom to top with the numbers from 1 to 90 . Bob uses these blocks to build a new tower. For each step he takes the top three blocks from the old tower and places them on the new tower without changing their order (see diagram). How many blocks are there in the new tow... | [] | images/1491.jpg | 4 | null | 3 | algebra | |
1237 | The numbers 1 to 8 are written into the circles shown so that there is one number in each circle. Along each of the five straight arrows the three numbers in the circles are multiplied. Their product is written next to the tip of the arrow. How big is the sum of the numbers in the three circles on the lowest row of the... | [] | images/1237.jpg | 17 | null | 4 | algebra | |
286 | The diagram shows a circle with centre $O$ as well as a tangent that touches the circle in point $P$. The arc $A P$ has length 20, the arc $B P$ has length 16. What is the size of the angle $\angle A X P$?
<image1> | [
"$30^{\\circ}$",
"$24^{\\circ}$",
"$18^{\\circ}$",
"$15^{\\circ}$",
"$10^{\\circ}$"
] | images/286.jpg | E | null | 4 | metric geometry - angle | |
2960 | Let $ABCD$ be a rectangle. Let $E$ and $F$ be points on $BC$ and $CD$, respectively, so that the areas of triangles $ABE$, $ADF$, and $CEF$ are 8, 5, and 9, respectively. Find the area of rectangle $ABCD$.
<image1> | [] | images/2960.jpg | 40 | Let $u = BE$, $v = CE$, $x = CF$, and $y = DF$. [asy]
unitsize(1.5 cm);
pair A, B, C, D, E, F;
A = (0,2);
B = (0,0);
C = (3,0);
D = (3,2);
E = (3*B + 2*C)/5;
F = (2*D + C)/3;
draw(A--B--C--D--cycle);
draw(A--E--F--cycle);
label("$A$", A, NW);
label("$B$", B, SW);
label("$C$", C, SE);
label("$D$", D, NE);
label("$E$... | 4 | metric geometry - area | |
3003 | All of the triangles in the figure and the central hexagon are equilateral. Given that $\overline{AC}$ is 3 units long, how many square units, expressed in simplest radical form, are in the area of the entire star? <image1> | [] | images/3003.jpg | 3\sqrt{3} | We divide the hexagon into six equilateral triangles, which are congruent by symmetry. The star is made up of 12 of these triangles. [asy]
pair A,B,C,D,E,F;
real x=sqrt(3);
F=(0,0);
E=(x,1);
D=(x,3);
C=(0,4);
A=(-x,1);
B=(-x,3);
draw(A--C--E--cycle); draw(B--D--F--cycle);
label("$D$",D,NE); label("$C$",C,N); label(... | 4 | metric geometry - area | |
1999 | Using this picture we can observe that
$1+3+5+7=4 \times 4$.
What is the value of
$1+3+5+7+9+11+13+15+17+19+21$ ?
<image1> | [] | images/1999.jpg | 121 | The sum $1+3+5+7+9+11+13+15+17+19+21$ has eleven terms. Therefore the value of the required sum is $11 \times 11=121$. | 5 | algebra | |
1705 | An isosceles triangle $P Q R$, in which $P Q=P R$, is split into three separate isosceles triangles, as shown, so that $P S=S Q, R T=R S$ and $Q T=R T$.
What is the size, in degrees, of angle $Q P R$ ? <image1> | [] | images/1705.jpg | 36 | Let the size, in degrees, of angle $Q P R$ be $x$. Since triangle $P S Q$ is isosceles, angle $P Q S=x$ and, using the external angle theorem, angle $R S T=2 x$. Since triangle $S T R$ is isosceles, angle $S T R=2 x$ and, since angles on a straight line add to $180^{\circ}$, angle $Q T R=180-2 x$. Since triangle $Q T R... | 5 | metric geometry - angle | |
2072 | A triangular array of numbers has a first row consisting of the odd integers $ 1,3,5,\ldots,99$ in increasing order. Each row below the first has one fewer entry than the row above it, and the bottom row has a single entry. Each entry in any row after the top row equals the sum of the two entries diagonally above it ... | [] | images/2072.jpg | 17 | null | 5 | algebra | |
2679 | In trapezoid $ABCD$, $AD$ is perpendicular to $DC$, $AD=AB=3$, and $DC=6$. In addition, E is on $DC$, and $BE$ is parallel to $AD$. Find the area of $\Delta BEC$.
<image1> | [] | images/2679.jpg | 4.5 | null | 3 | metric geometry - area | |
2206 | The centers of the faces of the right rectangular prism shown below are joined to create an octahedron, What is the volume of the octahedron?
<image1> | [
"$\\frac{75}{12}$",
"$10$",
"$12$",
"$10\\sqrt{2}$",
"$15$"
] | images/2206.jpg | B | null | 2 | solid geometry | |
19 | Eve has taken 2 bananas to school. At first she changed each of them into 4 apples, later on she exchanged each apple into 3 mandarins. How many mandarins has Eve got? <image1> | [
"$2+4+3$",
"$2 \\cdot 4+3$",
"$2+4 \\cdot 3$",
"$2 \\cdot 4 \\cdot 3$",
"$2+4-3$"
] | images/19.jpg | D | null | 2 | arithmetic | |
1267 | The diagram shows four semicircles with radius 1. The centres of the semicircles are at the mid-points of the sides of a square. What is the radius of the circle which touches all four semicircles?
<image1> | [
"$\\sqrt{2}-1$",
"$\\frac{\\pi}{2}-1$",
"$\\sqrt{3}-1$",
"$\\sqrt{5}-2$",
"$\\sqrt{7}-2$"
] | images/1267.jpg | A | null | 4 | metric geometry - length | |
299 | Which quadrant contains no points of the graph of the linear function $f(x)=-3.5 x+7$?
<image1> | [
"I",
"II",
"III",
"IV",
"Every quadrant contains at least one point of the graph."
] | images/299.jpg | C | null | 4 | analytic geometry | |
224 | In a rectangle JKLM the angle bisector in $\mathrm{J}$ intersects the diagonal KM in $\mathrm{N}$. The distance of $\mathrm{N}$ to $\mathrm{LM}$ is 1 and the distance of $\mathrm{N}$ to $\mathrm{KL}$ is 8. How long is LM?
<image1> | [
"$8+2 \\sqrt{2}$",
"$11-\\sqrt{2}$",
"10",
"$8+3 \\sqrt{2}$",
"$11+\\frac{\\sqrt{2}}{2}$"
] | images/224.jpg | A | null | 5 | metric geometry - length | |
1711 | In the diagram shown, sides $P Q$ and $P R$ are equal. Also $\angle Q P R=40^{\circ}$ and $\angle T Q P=\angle S R Q$. What is the size of $\angle T U R$ ? <image1> | [
"$55^{\\circ}$",
"$60^{\\circ}$",
"$65^{\\circ}$",
"$70^{\\circ}$",
"$75^{\\circ}$"
] | images/1711.jpg | D | Let $\angle S R Q$ be $x^{\circ}$. Since sides $P R$ and $P Q$ are equal, triangle $P Q R$ is isosceles and hence $\angle P R Q=\angle P Q R=\left(180^{\circ}-40^{\circ}\right) / 2=70^{\circ}$. Therefore, since we are given that $\angle T Q P=\angle S R Q$, we have $\angle U Q R$ is $70^{\circ}-x^{\circ}$. Hence, since... | 5 | metric geometry - angle | |
978 | Matchsticks are arranged to form numbers as shown. To form the number 15 one needs 7 matchsticks. To form the number 8 one needs the same amount. What is the biggest number that one can build using 7 matchsticks? <image1> | [] | images/978.jpg | 711 | null | 2 | algebra | |
2 | Which bike is most expensive?
<image1> | [
"A",
"B",
"C",
"D",
"E"
] | images/2.jpg | A | null | 2 | arithmetic | |
405 | In the picture below you can see a road from town $M$ to town $N$ (a solid line) and a detour (a dashed line) of segment $K L$, which is under repair. How many more kilometers does one have to travel from $M$ to $N$ using the detour?
<image1> | [] | images/405.jpg | 6 | null | 3 | metric geometry - length | |
515 | The solid in the diagram is made out of 8 identical cubes. What does the solid look like when viewed from above?
<image1>
<image2> | [
"A",
"B",
"C",
"D",
"E"
] | images/515.jpg | C | null | 5 | descriptive geometry | |
1255 | An ant walks along the sides of an equilateral triangle (see diagram). Its velocity is $5 \mathrm{~cm} / \mathrm{min}$ along the first side, $15 \mathrm{~cm} / \mathrm{min}$ along the second and $20 \mathrm{~cm} / \mathrm{min}$ along the third. With which average velocity in $\mathrm{cm} / \mathrm{min}$ does the ant wa... | [
"$10$",
"$\\frac{80}{11}$",
"$\\frac{180}{19}$",
"$15$",
"$\\frac{40}{3}$"
] | images/1255.jpg | C | null | 4 | algebra | |
2049 | Two skaters, Allie and Billie, are at points $A$ and $B$, respectively, on a flat, frozen lake. The distance between $A$ and $B$ is $100$ meters. Allie leaves $A$ and skates at a speed of $8$ meters per second on a straight line that makes a $60^\circ$ angle with $AB$. At the same time Allie leaves $A$, Billie leaves $... | [] | images/2049.jpg | 160 | null | 5 | algebra | |
1791 | Nico is learning to drive. He knows how to turn right but has not yet learned how to turn left. What is the smallest number of right turns he could make to travel from $\mathrm{P}$ to $\mathrm{Q}$, moving first in the direction shown? <image1> | [] | images/1791.jpg | 4 | Since Nico can only turn right, he cannot approach Q from the right as to do so would require a left turn. Therefore he must approach Q from below on the diagram. Hence he will be facing in the same direction as he originally faced. As he can only turn right, he must make a minimum of four right turns to end up facing ... | 2 | graph theory | |
1918 | A $5 \times 5$ square is divided into 25 cells. Initially all its cells are white, as shown. Neighbouring cells are those that share a common edge. On each move two neighbouring cells have their colours changed to the opposite colour (white cells become black and black ones become white). <image1>
What is the minimum n... | [] | images/1918.jpg | 12 | Note that, for each black square that we wish to produce, there will need to be a move which makes it black. This move will not change the colour of any of the other squares which we wish to make black (since the desired black cells are not neighbouring). Since there are 12 such squares, we must necessarily make at lea... | 4 | combinatorics | |
626 | Gaspar has these seven different pieces, formed by equal little squares.
<image1>
He uses all these pieces to assemble rectangles with different perimeters, that is, with different shapes. How many different perimeters can he find? | [] | images/626.jpg | 3 | null | 2 | combinatorial geometry | |
480 | In each square of the maze there is a piece of cheese. Ronnie the mouse wants to enter and leave the maze as shown in the picture. He doesn't want to visit a square more than once, but would like to eat as much cheese as possible. What is the maximum number of pieces of cheese that he can eat?
<image1> | [] | images/480.jpg | 37 | null | 2 | graph theory | |
366 | Martin's smartphone displays the diagram on the right. It shows how long he has worked with four different apps in the previous week. The apps are sorted from top to bottom according to the amount of time they have been used. This week he has spent only half the amount of time using two of the apps and the same amount ... | [
"A",
"B",
"C",
"D",
"E"
] | images/366.jpg | E | null | 3 | statistics | |
2025 | How many of the figures shown can be drawn with one continuous line without drawing a segment twice?
<image1> | [] | images/2025.jpg | 3 | All are possible except for the second figure. Each of the small lines must either start or end the continuous line. But then only one of the semicircles in the second figure can be included. | 5 | graph theory | |
2738 | The following bar graph represents the length (in letters) of the names of 19 people. What is the median length of these names?
<image1> | [] | images/2738.jpg | 4 | null | 2 | statistics | |
2893 | The measure of one of the smaller base angles of an isosceles trapezoid is $60^\circ$. The shorter base is 5 inches long and the altitude is $2 \sqrt{3}$ inches long. What is the number of inches in the perimeter of the trapezoid? <image1> | [] | images/2893.jpg | 22 | The triangle in the diagram is a $30-60-90$ triangle and has long leg $2\sqrt{3}$. Therefore the short leg has length 2 and hypotenuse has length 4. The bases of the trapezoid are then 5 and $2+5+2=9$ and the legs are both length 4. The entire trapezoid has perimeter $9+4+5+4=\boxed{22}$. | 1 | metric geometry - length | |
1084 | One vertex of the triangle on the left is connected to one vertex of the triangle on the right using a straight line so that no connecting line segment dissects either of the two triangles into two parts. In how many ways is this possible?
<image1> | [] | images/1084.jpg | 4 | null | 4 | combinatorial geometry | |
2861 | Concave pentagon $ABCDE$ has a reflex angle at $D$, with $m\angle EDC = 255^o$. We are also told that $BC = DE$, $m\angle BCD = 45^o$, $CD = 13$, $AB + AE = 29$, and $m\angle BAE = 60^o$. The area of $ABCDE$ can be expressed in simplest radical form as $a\sqrt{b}$. Compute $a + b$.\n<image1> | [] | images/2861.jpg | 59 | Observe the pentagon as it fits in the following figure. The area we are looking for is one-third of the difference in areas of the equilateral triangles, which is $\frac{29^2 \sqrt{3} / 4-13^2 \sqrt{3} / 4}{3}=56 \sqrt{3}$. The answer is $56+3=59$. | 5 | metric geometry - area | |
416 | At noon the minute hand of a clock is in the position shown on the left and after the quarter of an hour -- in the position shown on the right. Which position the minute hand will take after seventeen quarters from the noon?
<image1>
<image2> | [
"A",
"B",
"C",
"D",
"E"
] | images/416.jpg | A | null | 3 | arithmetic | |
693 | A building block is made up of five identical rectangles: <image1>
How many of the patterns shown below can be made with two such building blocks without overlap?
<image2>
<image3>
<image4>
<image5>
<image6> | [] | images/693.jpg | 4 | null | 2 | combinatorial geometry | |
2034 | Each cell in this cross-number can be filled with a non-zero digit so that all of the conditions in the clues are satisfied. The digits used are not necessarily distinct.
<image1>
\section*{ACROSS}
1. Four less than a factor of 105.
3. One more than a palindrome.
5. The square-root of the answer to this Kangaroo questi... | [] | images/2034.jpg | 841 | The two-digit factors of 105 are 15,21 and 35 , so 1 ACROSS is one of 11,17 or 31. However, as 2 DOWN is 400 less than a cube it cannot start with a 7, so 1 ACROSS is either 11 or 31 . If 1 ACROSS is 11 then 1 DOWN is 14, 2 DOWN is 112 and 3 ACROSS is 415 . This means 4 DOWN is at least 50 which is too large to satisfy... | 1 | logic | |
2323 | <image1>
If $\triangle A_1A_2A_3$ is equilateral and $A_{n+3}$ is the midpoint of line segment $A_nA_{n+1}$ for all positive integers $n$, then the measure of $\measuredangle A_{44}A_{45}A_{43}$ equals | [
"$30^\\circ$",
"$45^\\circ$",
"$60^\\circ$",
"$90^\\circ$",
"$120^\\circ$"
] | images/2323.jpg | E | null | 4 | metric geometry - angle | |
2373 | An $8'\text{ X }10'$ table sits in the corner of a square room, as in Figure 1 below. The owners desire to move the table to the position shown in Figure 2. The side of the room is $S$ feet. What is the smallest integer value of $S$ for which the table can be moved as desired without tilting it or taking it apart?
... | [] | images/2373.jpg | 13 | null | 5 | metric geometry - length | |
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> | [
"$30^{\\circ}$",
"$45^{\\circ}$",
"$60^{\\circ}$",
"$75^{\\circ}$",
"$90^{\\circ}$"
] | images/1510.jpg | B | The diagram shows the angles rearranged to form a $45^{\circ}$ angle.  | 4 | combinatorial geometry | |
1813 | Barney has 16 cards: 4 blue $(B), 4$ red $(R), 4$ green $(G)$ and 4 yellow (Y). He wants to place them in the square shown so that every row and every column has exactly one of each card. The diagram shows how he started. How many different ways can he finish?
<image1> | [] | images/1813.jpg | 4 | The left-hand two columns can only be completed in one way as shown. \begin{tabular}{|cc|c|} \hline$B$ & $R$ & 1 \\ $R$ & $B$ & 1 \\ \hline$Y$ & $G$ & 2 \\ $G$ & $Y$ & \\ \hline \end{tabular} Then the square labelled 1 can only be $\begin{array}{llll}Y & G & G & Y \\ G & Y\end{array}$ or $\begin{array}{ll}Y & G\end{ar... | 4 | combinatorics | |
2039 | A point $P$ is chosen in the interior of $\triangle ABC$ so that when lines are drawn through $P$ parallel to the sides of $\triangle ABC$, the resulting smaller triangles, $t_1$, $t_2$, and $t_3$ in the figure, have areas 4, 9, and 49, respectively. Find the area of $\triangle ABC$.
<image1> | [] | images/2039.jpg | 144 | null | 5 | metric geometry - area | |
1341 | In the figure $\alpha=7^{\circ}$. All lines $\mathrm{OA}_{1}, \mathrm{~A}_{1} \mathrm{~A}_{2}, \mathrm{~A}_{2} \mathrm{~A}_{3}, \ldots$ are equally long. What is the maximum number of lines that can be drawn in this way if no two lines are allowed to intersect each other?
<image1> | [] | images/1341.jpg | 13 | null | 4 | metric geometry - length |
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