Yuting6/geoqa-r1v-8k-rotated huggingface dataset:
Problem: <image>Question: The diagram shows a square with sides of length $a$. The shaded part of the square is bounded by a semicircle and two quarter-circle arcs. What is the shaded area? <image1>
Choices:
A. $\frac{\pi a^{2}}{8}$
B. $\frac{a^{2}}{2}$
C. $\frac{\pi a^{2}}{2}$
D. $\frac{a^{2}}{4}$
E. $\frac{\pi a^{2}}{4}$
Answer: B
Problem: <image>Question: Mr Hyde can't remember exactly where he has hidden his treasure. He knows it is at least $5 \mathrm{~m}$ from his hedge, and at most $5 \mathrm{~m}$ from his tree. Which of the following shaded areas could represent the largest region where his treasure could lie?
<image1>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: A
Problem: <image>Question: The picture shows the same die in three different positions. When the die is rolled, what is the probability of rolling a 'YES'?
<image1>
Choices:
A. $\frac{1}{3}$
B. $\frac{1}{2}$
C. $\frac{5}{9}$
D. $\frac{2}{3}$
E. $\frac{5}{6}$
Answer: B
Problem: <image>Question: In the grid, each small square has side of length 1 . What is the minimum distance from 'Start' to 'Finish' travelling only on edges or diagonals of the squares? <image1>
Choices:
A. $2 \sqrt{2}$
B. $\sqrt{10}+\sqrt{2}$
C. $2+2 \sqrt{2}$
D. $4 \sqrt{2}$
E. $6$
Answer: C
Problem: <image>Question: The square $F G H I$ has area 80 . Points $J, K, L, M$ are marked on the sides of the square so that $F K=G L=H M=I J$ and $F K=3 K G$. What is the area of the shaded region? <image1>
Answer: 25
Problem: <image>Question: A large set of weighing scales has two identical sets of scales placed on it, one on each pan. Four weights $W, X, Y, Z$ are placed on the weighing scales as shown in the left diagram.
<image1>
Then two of these weights are swapped, and the pans now appear as shown in the diagram on the right. Which two weights were swapped?
Choices:
A. $W$ and $Z$
B. $W$ and $Y$
C. $W$ and $X$
D. $X$ and $Z$
E. $X$ and $Y$
Answer: A
Problem: <image>Question: The figure shows seven regions enclosed by three circles. We call two regions neighbouring if their boundaries have more than one common point. In each region a number is written. The number in any region is equal to the sum of the numbers of its neighbouring regions. Two of the numbers are shown. What number is written in the central region? <image1>
Answer: 0
Problem: <image>Question: In the triangle $F G H$, we can draw a line parallel to its base $F G$, through point $X$ or $Y$. The areas of the shaded regions are the same. The ratio $H X: X F=4: 1$. What is the ratio $H Y: Y F$ ? <image1>
Choices:
A. $1: 1$
B. $2: 1$
C. $3: 1$
D. $3: 2$
E. $4: 3$
Answer: D
Problem: <image>Question: A square is split into nine identical squares, each with sides of length one unit. Circles are inscribed in two of these squares, as shown. What is the shortest distance between the two circles? <image1>
Choices:
A. $2 \sqrt{2}-1$
B. $\sqrt{2}+1$
C. $2 \sqrt{2}$
D. 2
E. 3
Answer: A
Problem: <image>Question: The large triangle shown has sides of length 5 units. What percentage of the area of the triangle is shaded? <image1>
Choices:
A. $80 \%$
B. $85 \%$
C. $88 \%$
D. $90 \%$
E. impossible to determine
Answer: C
Problem: <image>Question: Sepideh is making a magic multiplication square using the numbers 1, 2, 4, 5, 10, 20, 25, 50 and 100 . The products of the numbers in each row, in each column and in the two diagonals should all be the same. In the figure you can see how she has started. Which number should Sepideh place in the cell with the question mark? <image1>
Answer: 4
Problem: <image>Question: Peter wants to colour the cells of a $3 \times 3$ square in such a way that each of the rows, each of the columns and both diagonals have cells of three different colours. What is the least number of colours Peter could use? <image1>
Answer: 5
Problem: <image>Question: The picture shows a cube with four marked angles, $\angle W X Y$, $\angle X Y Z, \angle Y Z W$ and $\angle Z W X$. What is the sum of these angles? <image1>
Choices:
A. $315^{\circ}$
B. $330^{\circ}$
C. $345^{\circ}$
D. $360^{\circ}$
E. $375^{\circ}$
Answer: B
Problem: <image>Question: 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 number of moves required in order to obtain the chess-like colouring shown on the right?
Answer: 12
Problem: <image>Question: In the number pyramid shown each number is the sum of the two numbers immediately below. What number should appear in the lefthand cell of the bottom row?
<image1>
Answer: 16
Problem: <image>Question: Which of the following diagrams shows the locus of the midpoint of the wheel when the wheel rolls along the zig-zag curve shown?
<image1>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: E
Problem: <image>Question: A circle of radius 1 rolls along a straight line from the point $K$ to the point $L$, where $K L=11 \pi$. Which of the following pictures shows the correct appearance of the circle when it reaches $L$ ?
<image1>
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: D
Problem: <image>Question: 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>
Answer: 35
Problem: <image>Question: A belt drive system consists of the wheels $K, L$ and $M$, which rotate without any slippage. The wheel $L$ makes 4 full turns when $K$ makes 5 full turns; also $L$ makes 6 full turns when $M$ makes 7 full turns.
<image1>
The perimeter of wheel $M$ is $30 \mathrm{~cm}$. What is the perimeter of wheel $K$ ?
Choices:
A. $27 \mathrm{~cm}$
B. $28 \mathrm{~cm}$
C. $29 \mathrm{~cm}$
D. $30 \mathrm{~cm}$
E. $31 \mathrm{~cm}$
Answer: B
Problem: <image>Question: Maja decided to enter numbers into the cells of a $3 \times 3$ grid. She wanted to do this in such a way that the numbers in each of the four $2 \times 2$ grids that form part of the $3 \times 3$ grid have the same totals. She has already written numbers in three of the corner cells, as shown in the diagram. Which number does she need to write in the bottom right corner?
<image1>
Answer: 0
Problem: <image>Question: Carlos wants to put numbers in the number pyramid shown in such a way that each number above the bottom row is the sum of the two numbers immediately below it. What is the largest number of odd numbers that Carlos could put in the pyramid?
<image1>
Answer: 14
Problem: <image>Question: On a balance scale, three different masses were put at random on each pan and the result is shown in the picture. The masses are of 101, 102, 103, 104, 105 and 106 grams. What is the probability that the 106 gram mass stands on the heavier pan?
<image1>
Choices:
A. $75 \%$
B. $80 \%$
C. $90 \%$
D. $95 \%$
E. $100 \%$
Answer: B
Problem: <image>Question: The points $G$ and $I$ are on the circle with centre $H$, and $F I$ is tangent to the circle at $I$. The distances $F G$ and $H I$ are integers, and $F I=F G+6$. The point $G$ lies on the straight line through $F$ and $H$. How many possible values are there for $H I$ ? <image1>
Answer: 6
Problem: <image>Question: The diagram shows a triangle $F H I$, and a point $G$ on $F H$ such that $G H=F I$. The points $M$ and $N$ are the midpoints of $F G$ and $H I$ respectively. Angle $N M H=\alpha^{\circ}$. Which of the following gives an expression for $\angle I F H$ ?
<image1>
Choices:
A. $2 \alpha^{\circ}$
B. $(90-\alpha)^{\circ}$
C. $45+\alpha^{\circ}$
D. $\left(90-\frac{1}{2} \alpha\right)^{\circ}$
E. $60^{\circ}$
Answer: A
Problem: <image>Question: The distance from the top of the can on the floor to the top of the bottle on the table is $150 \mathrm{~cm}$. The distance from the top of the bottle on the floor to the top of the can on the table is $110 \mathrm{~cm}$. What is the height of the table?
<image1>
Choices:
A. $110 \mathrm{~cm}$
B. $120 \mathrm{~cm}$
C. $130 \mathrm{~cm}$
D. $140 \mathrm{~cm}$
E. $150 \mathrm{~cm}$
Answer: C
Problem: <image>Question: The diagram shows three congruent regular hexagons. Some diagonals have been drawn, and some regions then shaded. The total shaded areas of the hexagons are $X, Y, Z$ as shown. Which of the following statements is true? <image1>
Choices:
A. $X, Y$ and $Z$ are all the same
B. $\quad Y$ and $Z$ are equal, but $X$ is different
C. $X$ and $Z$ are equal, but $Y$ is different
D. $X$ and $Y$ are equal, but $Z$ is different
E. $X, Y, Z$ are all different
Answer: A
Problem: <image>Question: Some of the digits in the following correct addition have been replaced by the letters $P, Q, R$ and  $S$ , as shown. What is the value of $P+Q+R+S$ ?
<image1>
Answer: 15
Problem: <image>Question: Monika plans to travel across the network in the diagram from point $P$ to point $Q$, travelling only in the direction of the arrows. How many different routes are possible? <image1>
Answer: 16
Problem: <image>Question: Eight congruent semicircles are drawn inside a square of side-length 4 . Each semicircle begins at a vertex of the square and ends at a midpoint of an edge of the square. What is the area of the non-shaded part of the square? <image1>
Answer: 8
Problem: <image>Question: An annulus is a shape made from two concentric circles. The diagram shows an annulus consisting of two concentric circles of radii 2 and 9. Inside this annulus two circles are drawn without overlapping, each being tangent to both of the concentric circles that make the annulus. In a different annulus made by concentric circles of radii 1 and 9 , what would be the largest possible number of non-overlapping circles that could be drawn in this way?
<image1>
Answer: 3
Problem: <image>Question: Diana drew a rectangular grid of 12 squares on squared paper. Some of the squares were then painted black. In each white square she wrote the number of black squares that shared an edge with it (a whole edge, not just a vertex). The figure shows the result. Then she did the same with a rectangular grid of 2 by 1009 squares. What is the maximum value that she could obtain as the result of the sum of all the numbers in this grid? <image1>
Answer: 3025
Problem: <image>Question: At each vertex of the 18 -gon in the picture a number should be written which is equal to the sum of the numbers at the two adjacent vertices. Two of the numbers are given. What number should be written at the vertex $P$ ? <image1>
Answer: 38
Problem: <image>Question: Two chords $P Q$ and $P R$ are drawn in a circle with diameter $P S$. The point $T$ lies on $P R$ and $Q T$ is perpendicular to $P R$. The angle $Q P R=60^{\circ}, P Q=24 \mathrm{~cm}, R T=3 \mathrm{~cm}$. What is the length of the chord $Q S$ in $\mathrm{cm}$ ? <image1>
Choices:
A. $\sqrt{3}$
B. 2
C. 3
D. $2 \sqrt{3}$
E. $3 \sqrt{2}$
Answer: D
Problem: <image>Question: Two angles are marked on the $3 \times 3$ grid of squares.
<image1>
Which of the following statements about the angles is correct?
Choices:
A. $\alpha=\beta$
B. $2 \alpha+\beta=90$
C. $\alpha+\beta=60$
D. $2 \beta+\alpha=90$
E. $\alpha+\beta=45$
Answer: B
Problem: <image>Question: Inside each unit square a certain part has been shaded. In which square is the total shaded area the largest?
<image1>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: A
Problem: <image>Question: On each of three pieces of paper a five-digit number is written as shown. Three of the digits are covered. The sum of the three numbers is 57263 . What are the covered digits? <image1>
Choices:
A. 0,2 and 2
B. 1,2 and 9
C. 2, 4 and 9
D. 2,7 and 8
E. 5,7 and 8
Answer: B
Problem: <image>Question: The flag of Kangaria is a rectangle with side-lengths in the ratio $3: 5$. The flag is divided into four rectangles of equal area as shown. What is the ratio of the length of the shorter sides of the white rectangle to the length of its longer sides? <image1>
Choices:
A. $1: 3$
B. $1: 4$
C. $2: 7$
D. $3: 10$
E. $4: 15$
Answer: E
Problem: <image>Question: The diagram shows a shape made of arcs of three circles, each with radius $R$. The centres of the circles lie on the same straight line, and the middle circle passes through the centres of the other two circles. What is the perimeter of the shape? <image1>
Choices:
A. $\frac{2 \pi R \sqrt{3}}{3}$
B. $\frac{5 \pi R}{3}$
C. $\frac{10 \pi R}{3}$
D. $2 \pi R \sqrt{3}$
E. $4 \pi R$
Answer: C
Problem: <image>Question: The diagram shows a net of an octahedron. When this is folded to form the octahedron, which of the labelled line segments will coincide with the line segment labelled $x$ ? <image1>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: E
Problem: <image>Question: A square has two of its vertices on a semicircle and the other two on the diameter of the semicircle as shown. The radius of the circle is 1 . What is the area of the square? <image1>
Choices:
A. $\frac{4}{5}$
B. $\frac{\pi}{4}$
C. 1
D. $\frac{4}{3}$
E. $\frac{2}{\sqrt{3}}$
Answer: A
Problem: <image>Question: A network consists of 16 vertices and 24 edges that connect them, as shown. An ant begins at the vertex labelled Start. Every minute, it walks from one vertex to a neighbouring vertex, crawling along a connecting edge. At which of the vertices labelled $P, Q, R, S, T$ can the ant be after 2019 minutes? <image1>
Choices:
A. only $P, R$ or $S$,
B. not $Q$
C. only $Q$
D. only $T$
E. all of the vertices are possible
Answer: C
Problem: <image>Question: The diagram shows triangle $J K L$ of area $S$. The point $M$ is the midpoint of $K L$. The points $P, Q, R$ lie on the extended lines $L J, M J, K J$, respectively, such that $J P=2 \times J L, J Q=3 \times J M$ and $J R=4 \times J K$.
What is the area of triangle $P Q R$ ? <image1>
Choices:
A. $S$
B. $2 S$
C. $3 S$
D. $\frac{1}{2} S$
E. $\frac{1}{3} S$
Answer: A
Problem: <image>Question: The diagram shows a shape made from ten squares of side-length $1 \mathrm{~cm}$, joined edge to edge. What is the length of its perimeter, in centimetres?
<image1>
Answer: 18
Problem: <image>Question: In the calculations shown, each letter stands for a digit. They are used to make some two-digit numbers. The two numbers on the left have a total of 79. What is the total of the four numbers on the right? <image1>
Answer: 158
Problem: <image>Question: The shortest path from Atown to Cetown runs through Betown. Two of the signposts that can be seen on this path are shown, but one of them is broken and a number missing. What distance was written on the broken sign? <image1>
Choices:
A. $2 \mathrm{~km}$
B. $3 \mathrm{~km}$
C. $4 \mathrm{~km}$
D. $5 \mathrm{~km}$
E. $6 \mathrm{~km}$
Answer: A
Problem: <image>Question: Freda wants to write a number in each of the nine cells of this figure so that the sum of the three numbers on each diameter is 13 and the sum of the eight numbers on the circumference is 40. What number must be written in the central cell? <image1>
Answer: 3
Problem: <image>Question: Two squares of different sizes are drawn inside an equilateral triangle. One side of one of these squares lies on one of the sides of the triangle as shown. What is the size of the angle marked by the question mark? <image1>
Choices:
A. $25^{\circ}$
B. $30^{\circ}$
C. $35^{\circ}$
D. $45^{\circ}$
E. $50^{\circ}$
Answer: E
Problem: <image>Question: A vertical stained glass square window of area $81 \mathrm{~cm}^{2}$ is made out of six triangles of equal area (see figure). A fly is sitting on the exact spot where the six triangles meet. How far from the bottom of the window is the fly sitting? <image1>
Choices:
A. $3 \mathrm{~cm}$
B. $5 \mathrm{~cm}$
C. $5.5 \mathrm{~cm}$
D. $6 \mathrm{~cm}$
E. $7.5 \mathrm{~cm}$
Answer: D
Problem: <image>Question: Two identical rectangles with sides of length $3 \mathrm{~cm}$ and $9 \mathrm{~cm}$ are overlapping as in the diagram. What is the area of the overlap of the two rectangles? <image1>
Choices:
A. $12 \mathrm{~cm}^{2}$
B. $13.5 \mathrm{~cm}^{2}$
C. $14 \mathrm{~cm}^{2}$
D. $15 \mathrm{~cm}^{2}$
E. $16 \mathrm{~cm}^{2}$
Answer: D
Problem: <image>Question: In each of the cells, a number is to be written so that the sum of the 4 numbers in each row and in each column are the same.
<image1>
What number must be written in the shaded cell?
Answer: 7
Problem: <image>Question: A zig-zag line starts at the point $P$, at one end of the diameter $P Q$ of a circle. Each of the angles between the zig-zag line and the diameter $P Q$ is equal to $\alpha$ as shown. After four peaks, the zig-zag line ends at the point $Q$. What is the size of angle $\alpha$ ? <image1>
Choices:
A. $60^{\circ}$
B. $72^{\circ}$
C. $75^{\circ}$
D. $80^{\circ}$
E. $86^{\circ}$
Answer: B
Problem: <image>Question: A rectangle with perimeter $30 \mathrm{~cm}$ is divided by two lines, forming a square of area $9 \mathrm{~cm}^{2}$, as shown in the figure.
<image1>
What is the perimeter of the shaded rectangle?
Choices:
A. $14 \mathrm{~cm}$
B. $16 \mathrm{~cm}$
C. $18 \mathrm{~cm}$
D. $21 \mathrm{~cm}$
E. $24 \mathrm{~cm}$
Answer: C
Problem: <image>Question: Six rectangles are arranged as shown. The number inside each rectangle gives the area, in $\mathrm{cm}^{2}$, of that rectangle. The rectangle on the top left has height $6 \mathrm{~cm}$.
<image1>
What is the height of the bottom right rectangle?
Choices:
A. $4 \mathrm{~cm}$
B. $5 \mathrm{~cm}$
C. $6 \mathrm{~cm}$
D. $7.5 \mathrm{~cm}$
E. $10 \mathrm{~cm}$
Answer: B
Problem: <image>Question: Five line segments are drawn inside a rectangle as shown.
<image1>
What is the sum of the six marked angles?
Choices:
A. $360^{\circ}$
B. $720^{\circ}$
C. $900^{\circ}$
D. $1080^{\circ}$
E. $1120^{\circ}$
Answer: D
Problem: <image>Question: The numbers from 1 to 6 are to be placed at the intersections of three circles, one number in each of the six squares. The number 6 is already placed. Which number must replace $x$, so that the sum of the four numbers on each circle is the same? <image1>
Answer: 1
Problem: <image>Question: The diagram shows a semicircle with centre $O$. Two of the angles are given. What is the value of $x$ ? <image1>
Answer: 9
Problem: <image>Question: Each box in the strip shown is to contain one number. The first box and the eighth box each contain 2021. Numbers in adjacent boxes have $\operatorname{sum} T$ or $T+1$ as shown. What is the value of $T$ ?
<image1>
Answer: 4045
Problem: <image>Question: In the $4 \times 4$ grid some cells must be painted black. The numbers to the right of the grid and those below the grid show how many cells in that row or column must be black.
<image1>
In how many ways can this grid be painted?
Answer: 5
Problem: <image>Question: Tatiana's teacher drew a $3 \times 3$ grid on the board, with zero in each cell. The students then took turns to pick a $2 \times 2$ square of four adjacent cells, and to add 1 to each of the numbers in the four cells. After a while, the grid looked like the diagram on the right (some of the numbers in the cells have been rubbed out.)
<image1>
What number should be in the cell with the question mark?
Answer: 16
Problem: <image>Question: The smaller square in the picture has area 16 and the grey triangle has area 1. What is the area of the larger square? <image1>
Answer: 18
Problem: <image>Question: A caterpillar crawled up a smooth slope from $A$ to $B$, and crept down the stairs from $B$ to $C$. What is the ratio of the distance the caterpillar travelled from $B$ to $C$ to the distance it travelled from $A$ to $B$ ? <image1>
Choices:
A. $1: 1$
B. 2:1
C. 3:1
D. $\sqrt{2}: 1$
E. $\sqrt{3}: 1$
Answer: E
Problem: <image>Question: Carolina has a box of 30 matches. She begins to make the number 2022 using matchsticks. The diagram shows the first two digits.
<image1>
How many matchsticks will be left in the box when she has finished?
Answer: 9
Problem: <image>Question: Some shapes are drawn on a piece of paper. The teacher folds the left-hand side of the paper over the central bold line. How many of the shapes on the left-hand side will fit exactly on top of a shape on the right-hand side? <image1>
Answer: 3
Problem: <image>Question: Katrin arranges tables measuring $2 \mathrm{~m}$ by $1 \mathrm{~m}$ according to the number of participants in a meeting. The diagrams show the plan view for a small, a medium and a large meeting. How many tables are needed for a large meeting? <image1>
Answer: 12
Problem: <image>Question: On Nadya's smartphone, the diagram shows how much time she spent last week on four of her apps. This week she halved the time spent on two of these apps, but spent the same amount of time as the previous week on the other two apps.
<image1>
Which of the following could be the diagram for this week?
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: E
Problem: <image>Question: There were five candidates in the school election. After $90 \%$ of the votes had been counted, the preliminary results were as shown on the right. How many students still had a chance of winning the election?
<image1>
Answer: 3
Problem: <image>Question: Five squares and two right-angled triangles are positioned as shown. The areas of three squares are $3 \mathrm{~m}^{2}, 7 \mathrm{~m}^{2}$ and $22 \mathrm{~m}^{2}$ as shown. What is the area, in $\mathrm{m}^{2}$, of the square with the question mark? <image1>
Answer: 18
Problem: <image>Question: A ladybird aims to travel from hexagon $\mathrm{X}$ to hexagon $\mathrm{Y}$, passing through each of the seven unshaded hexagons once and only once. She can move from one hexagon to another only through a common edge. How many different routes could she take? <image1>
Answer: 5
Problem: <image>Question: The centres of the seven circles shown all lie on the same line. The four smaller circles have radius $1 \mathrm{~cm}$. The circles touch, as shown.
<image1>
What is the total area of the shaded regions?
Choices:
A. $\pi \mathrm{cm}^{2}$
B. $2 \pi \mathrm{cm}^{2}$
C. $3 \pi \mathrm{cm}^{2}$
D. $4 \pi \mathrm{cm}^{2}$
E. $5 \pi \mathrm{cm}^{2}$
Answer: B
Problem: <image>Question: Twelve congruent rectangles are placed together to make a rectangle $P Q R S$ as shown. What is the ratio $P Q: Q R$ ? <image1>
Choices:
A. $2: 3$
B. $3: 4$
C. $5: 6$
D. $7: 8$
E. $8: 9$
Answer: E
Problem: <image>Question: The diagram shows a square $P Q R S$ of side-length $1 . W$ is the centre of the square and $U$ is the midpoint of $R S$. Line segments $T W, U W$ and $V W$ split the square into three regions of equal area. What is the length of $S V$ ? <image1>
Choices:
A. $\frac{1}{2}$
B. $\frac{2}{3}$
C. $\frac{3}{4}$
D. $\frac{4}{5}$
E. $\frac{5}{6}$
Answer: E
Problem: <image>Question: Two congruent isosceles right-angled triangles each have squares inscribed in them as shown. The square $\mathrm{P}$ has an area of $45 \mathrm{~cm}^{2}$.
What is the area, in $\mathrm{cm}^{2}$, of the square $\mathrm{R}$ ? <image1>
Answer: 40
Problem: <image>Question: Cuthbert is going to make a cube with each face divided into four squares. Each square must have one shape drawn on it; either a cross, a triangle or a circle. Squares that share an edge must have different shapes on them. One possible cube is shown in the diagram. Which of the following combinations of crosses and triangles is possible on such a cube (with the other shapes being circles)?
<image1>
Choices:
A. 6 crosses, 8 triangles
B. 7 crosses, 8 triangles
C. 5 crosses, 8 triangles
D. 7 crosses, 7 triangles
E. none of these are possible
Answer: E
Problem: <image>Question: The bases of the two touching squares shown lie on the same straight line. The lengths of the diagonals of the larger square and the smaller square are $10 \mathrm{~cm}$ and $8 \mathrm{~cm}$ respectively. $P$ is the centre of the smaller square. What is the area, in $\mathrm{cm}^{2}$, of the shaded triangle $P Q R$ ? <image1>
Answer: 20
Problem: <image>Question: The cube shown has sides of length 2 units. Holes in the shape of a hemisphere are carved into each face of the cube. The six hemispheres are identical and their centres are at the centres of the faces of the cube. The holes are just large enough to touch the hole on each neighbouring face. What is the diameter of each hole? <image1>
Choices:
A. 1
B. $\sqrt{2}$
C. $2-\sqrt{2}$
D. $3-\sqrt{2}$
E. $3-\sqrt{3}$
Answer: B