IMSO 2021 Mathematics - International Essay

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?