We are given a \(100\times 100\) square grid and \(N\) counters. All of the possible arrangements of the counters on the grid which follow the following rule are considered: no two counters lie in adjacent squares.
What is the largest value of \(N\) for which, in every single possible arrangement of counters following this rule, it is possible to find at least one counter such that moving it to an adjacent square does not break the rule. Squares are considered adjacent if they share a side.
Sarah believes that two watermelons are heavier than three melons, Anna believes that three watermelons are heavier than four melons. It is known that one of the girls is right, and the other is mistaken. Is it true that 12 watermelons are heavier than 18 melons? (It is believed that all watermelons weigh the same and all melons weigh the same.)
What is the largest number of counters that can be put on the cells of a chessboard so that on each horizontal, vertical and diagonal (not only on the main ones) there is an even number of counters?
On a particular day it turned out that every person living in a particular city made no more than one phone call. Prove that it is possible to divide the population of this city into no more than three groups, so that within each group no person spoke to any other by telephone.
We are given a convex 200-sided polygon in which no three diagonals intersect at the same point. Each of the diagonals is coloured in one of 999 colours. Prove that there is some triangle inside the polygon whose sides lie some of the diagonals, so that all 3 sides are the same colour. The vertices of the triangle do not necessarily have to be the vertices of the polygon.
Is it possible to arrange natural numbers from 1 to \(2002^2\) in the cells of a \(2002\times2002\) table so that for each cell of this table one could choose a triplet of numbers, from a row or column, where one of the numbers is equal to the product of the other two?
All of the points with whole number co-ordinates in a plane are plotted in one of three colours; all three colours are present. Prove that there will always be possible to form a right-angle triangle from these points so that its vertices are of three different colours.
A regular hexagon with sides of length \(5\) is divided by straight lines, that are parallel to its sides, to form regular triangles with sides of length 1. We call the vertices of all such triangles nodes. It is known that more than half of the nodes are marked. Prove that there are five marked nodes lying on one circle.
Prove that in any set of 117 unique three-digit numbers it is possible to pick 4 non-overlapping subsets, so that the sum of the numbers in each subset is the same.
A target consists of a triangle divided by three families of parallel lines into 100 equilateral unit triangles. A sniper shoots at the target. He aims at a particular equilateral triangle and either hits it or hits one of the adjacent triangles that share a side with the one he was aiming for. He can see the results of his shots and can choose when to stop shooting. What is the largest number of triangles that the sniper can guarantee he can hit exactly 5 times?