Question bank IMAS 2023

Exam papers

120 questions

Question 1

If the square root of a number is between 10 and 11, then the cube root of this number is between which numbers?

3 and 4
4 and 5
5 and 6
6 and 7
7 and 8

Ann bought two types of rice: the first type costs 4 dollars per kg and the second type costs 5 dollars per kg. She mixed the two types of rice in some ratio, then she sold the mixture at 6 dollars per kg, making a 30% profit. What was the ratio (type 1 : type 2) of the mixture?

$6:11$
$3:5$
$5:8$
$3:4$
$8:5$

Question 3

A grid has been divided into several non-overlapping rectangles along its grid lines. Exactly one square in each rectangle has been labelled with the area of the rectangle. For example, if the given grid is

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then the dividing method is as follows:

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Now, consider the following grid:

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What is the area of the rectangle containing the ★?

Question 4

Two towns A and B are at a distance of $n$ km along a river bank. A boat, starting from A and traveling upstream, reaches B in 8 hours. Another boat, traveling the same route, with speed 3 km/h greater than the speed of the first one, covers the distance in 5 hours. What is the value of $n$?

$40$
$60$
$80$
$100$
$120$

Question 5

The greatest common divisor of the positive integers $3m+4$ and $5m+2$ is equal to $d$. What is the sum of all possible values of $d$?

$24$
$25$
$26$
$27$
$28$

Question 6

What is the remainder when the sum $1\times2+2\times3+3\times4+\cdots+2022\times2023$ is divided by 11?

$10$
$8$
$4$
$2$
$0$

Question 7

In the diagram below, two circles, each of radius 6 cm, are there such that their centres lie on the circumference of the other. What is the area, in $\mathrm{cm}^2$, of the region common to both circles?

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$12\pi-9\sqrt3$
$24\pi-18\sqrt3$
$24\pi-9\sqrt3$
$4\pi-\sqrt3$
$24\pi-3\sqrt3$

Question 8

Eight different digits are written in each cell of a table with 2 rows and 4 columns. Each cell is written exactly one digit. The sum of the digits in any of the two rows is divisible by 7, while the sum of the digits in any of the four columns is an even number. If $a$ and $b$ are the only two digits that are not written in the table, what is the number of possible values for $|a-b|$?

Question 9

Several boxes were delivered to a pastry shop. Each box either contain 8 cakes and 6 pastries or 4 cakes and $n$ pastries. The owner of the shop ate 1 cake and 1 pastry and exactly re-distributed the remaining items in boxes of 3 cakes and 2 pastries each. What is the value of $n$?

Question 10

Betty paints a $7\times7$ grid of white squares. At the start, she colours the square in the centre in black. Then, she picks any white square that shares a side with the one and only one other black square and colours it black, repeating this until either she wants to stop or there are no suitable white square exists. Which of the following configurations is a possible diagram that Betty could have coloured?

Question 11

There are knights, which are numbered from 1 to 10, inclusively, are to be seated at a evenly spaced round table. The numbers of any two neighbouring knights differ by at most two. What is the number of the knight sitting directly opposite (marked by “?”) to knight number 6?

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Question 12

There are 2 types of games at a school party, denoted by X and Y. A student gets 7 points for game X and get 11 points for game Y. The students can play $x$ times of game X and $y$ times of game Y. If a student gets exactly 472 points in total, then he can get the special prize. What is the minimum possible value of $x+y$ to get the special prize?

$44$
$48$
$52$
$60$
$64$

Question 13

Let $PQR$ be an acute triangle with $PQ=6$ cm and $PR=10$ cm. Let $PX$ and $PY$ be the altitude and the angle bisector from $P$. If $XY:XR=1:4$, then what is the length, in cm, of $QR$?

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$4\sqrt3$
$4\sqrt6$
$6\sqrt6$
$8\sqrt3$
$8\sqrt6$

Question 14

If $x,y,z$ are positive integers such that $x+2xy+3xyz=70$, how many possible values of $xyz$ are there?

Question 15

As shown in the diagram below, when the angle of the sunbeam changes from $45^\circ$ to $30^\circ$, the shadow of a leaning tower doubles in length. What is the measure of the angle $\alpha$, in degrees, that the leaning tower makes with the horizontal line?

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$60^\circ$
$65^\circ$
$70^\circ$
$75^\circ$
$80^\circ$

Question 16

The positive real numbers $a,b,c$ satisfy $2a+3b+4c=9$. What is the maximum possible value of $a^4b^3c^2$?

Question 17

Let $a=\frac12(\sqrt6+\sqrt8)$, $b=\sqrt7$ and $c=\sqrt[4]{48}$. Which of the inequalities is correct?

$a\lt b\lt c$
$a\lt c\lt b$
$b\lt c\lt a$
$c\lt b\lt a$
$c\lt a\lt b$

Question 18

A ball is shot from corner $A$ of a $2\times1$ rectangular billiard table. The ball bounces three times off the walls and then falls into one of the corners of the table. The dashed line is one possible path. What is the longest path it could have possibly travelled?

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$\sqrt{17}$
$2\sqrt5$
$2\sqrt{10}$
$\sqrt{65}$
$9$

Question 19

From a group of 5 girls and 4 boys, a team is chosen so that it contains at least two girls. In how many ways can this team be chosen?

$208$
$248$
$416$
$432$
$512$

Question 20

Let $ABC$ be an acute triangle with circumcentre $O$ and $\angle ACB=60^\circ$. If $AO$ intersects $BC$ at point $X$, $BO$ intersects $AC$ at point $Y$ and $XY=7$ cm, what is the length, in cm, of $AY+BX$?

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$10$
$11$
$12$
$13$
$14$