Question bank IMSO Maths 2021

Mali tries to insert some number of “+” signs between the digits of $777777777777$ (twelve copies of 7) so that the resulting expression is a multiple of 30. How many ways can she do it?

Place each of the numbers 2, 3, 4, 5, 6, 7 and 8 exactly once in the eight empty circles (the position of 1 is already fixed) such that the sum of each pair of adjacent numbers is a prime number. Diagrams are considered the same if one can be obtained from the other by reflection or rotation. List all possible diagrams.



Source wording note: “eight empty circles” is printed in the original. The diagram actually has eight circles in total, with seven empty circles and a fixed 1. The task is interpreted as filling those seven empty circles. The original blank answer diagrams are retained below.

Four runners (A, B, C and D) participated in a race. Before the event, all four of them (I, II, III and IV) anonymously predicted the final rankings as shown in the table below.



I and III each got two positions correct, while II and IV each got one position correct. Determine the final placings of the four participants.

As shown in the diagram below, five persons (A, B, C, D and E) live in one village. The table shows the sum of the distances travelled by the other four people when going to one person's house. If a party is held at E's house, what is the total distance travelled, in meters, by A, B, C and D?

Three motorcyclists started their rides at the same time. During the whole journey, they travelled for some time, took a rest for some time and then travelled to return to their homes at the same time. The first motorcyclist travelled twice as much as the second motorcyclist rested, the second motorcyclist travelled three times as much as the third motorcyclist rested, and the third motorcyclist travelled four times as much as the first motorcyclist rested. If the sum of the speeds of the first and third is 48 km/h, find the speed of the second one.

Note: Each motorcyclist travelled at the same speed for the whole journey.

A pair of circles are called “neighbors” if they touch each other. How many pairs of “neighbors” are there in the diagram below?

In the diagram below, the square has side length 20 cm, and the area of the “middle pentagon” inside the square is 21 cm². What is the total area, in cm², of all the shaded regions?

In a chess tournament, each participant plays every other participant exactly once. A participant gets 1 point for a win, 0.5 point for a draw and 0 points for a loss. At most how many of the 40 participants can score 24 points or more?

There are eight consecutive non-zero positive integers such that their sum can be expressed as the sum of seven consecutive positive integers and cannot be expressed as the sum of three consecutive positive integers. What is the minimum possible value of the largest of the eight integers?

Dissect the diagram below into two congruent parts, which may be rotated or reflected. Find the perimeter, in cm, of each part if the area of the original diagram is 72 cm².

Five teams took part in a soccer tournament, where each team plays every other team exactly once. Each team gets 3 points for a win, 1 point for a draw and 0 points for a loss. After all matches have been played, is it possible for the total points of the five teams to be five consecutive positive integers? If yes, provide an example.

In each cell of the $5\times5$ grid shown below, Peter may place several stones or leave it empty so that the total numbers of stones in all five rows and all five columns are all different. What is the minimum number of stones Peter must use?

In the diagram below, the area of triangle EFG equals the area of triangle CDG, the ratio of the areas of triangles EBC and BCD is $3:2$, and $AD:DC=2:1$. If triangle BDF has area 29 cm², what is the area, in cm², of triangle ABC?

Suppose that we are living in a city where the streets are laid out on a square grid as shown below. In this city, you are only allowed to move along vertical lines (streets running north–south) or horizontal lines (streets running east–west). Suppose your home is at point $A$ and you want to go to your friend's house at point $B$. The distance travelled is not the straight-line distance measured with a ruler, but the distance walked along the streets. We define the TCG distance between $A$ and $B$ to be the length of a shortest such route. For example, from $A$ to $B$ you may move 3 right and 3 down, or 3 down and 3 right, so the TCG distance in the diagram is 6.

(a) How many grid points have TCG distance 6 from both $A$ and $B$? Draw the diagram. (2 marks)

(b) How many grid points have TCG distance 12 from both $A$ and $B$? Draw the diagram. (4 marks)



Answer-sheet instruction: no partial score for each subpart. The original answer sheet supplies a larger blank grid for drawing.

Divide the diagram below, cutting only along the drawn lines, into 7 identical pieces. Reflected pieces are allowed.



Answer-sheet instruction: no partial score.

The diagram shows a triangular chessboard. A chess piece called a castle, indicated by $C$, may attack along any of the three directions highlighted below.



(a) Place as many castles as possible on the board so that no two castles can attack each other. (3 marks)



(b) A chess piece called a bishop, indicated by $B$, may attack along the direction of any of its angles. The two diagrams show examples of its attack ranges; the broken lines indicate the directions of the angles.





Place as many bishops as possible on the board so that no bishop attacks another. (3 marks)



The answer sheet asks for the number of pieces and a placement diagram for each part, with no partial score within each part.

The diagram shows a map of 23 towns connected by roads, with a video arcade in each town. Andy, who loves playing video games, works in an office in town $O$. Every day after work, he visits all the towns exactly once to play before reaching his home in town $H$. He spends much of his free time at the arcades and is ignoring his girlfriend. One day she decides to meet him on his way home, but he varies his routes all the time. She has discovered that there is one road he must pass along in every possible route, so she plans to wait for him on it. Which road is it?



The answer sheet also asks you to draw one possible route and name the two towns at the ends of the road. No partial score.

Anna and Boris play a game using the integers $1,2,3,4,5,6,7,8$. Anna first selects a positive integer $k\le8$. Boris then chooses $k$ of these eight numbers, and Anna chooses three of the eight numbers. Anna wins if the sum of every pair of her chosen numbers is among the numbers chosen by Boris.

(a) What is the minimum value of $k$ for which Anna has a winning strategy? (2 marks)

(b) If Anna has a winning strategy for $k$, give a counterexample showing that for $k-1$ she cannot guarantee a win. (4 marks)

Answer-sheet instruction: no partial score for each subpart.

There are 100 positive integers whose sum is equal to their product. Find all possible products.

Answer-sheet scoring note: the number of correct answers minus the number of incorrect answers is the net number of correct answers. The paper assigns this question 6 marks, but does not specify how that net count is converted into 6 marks.

In year 2021, Ada's little brother is 10 years old. If Ada's age in 2021 is the sum of all the digits of the year she was born in, how old is she in 2021?