Can you decorate an \(8 \times 8\) cake with chocolate roses in such a way that any \(2 \times 2\) piece would have exactly 2 roses on it, and any \(3 \times 1\) piece would have exactly one rose? Either draw such a cake or explain why this is not possible.
Several films were nominated for the “Best Math Movie“ award. Each of the 10 judges secretly picked the top movie of their choice. It is known that out of any 4 judges, at least 2 voted for the same film. Prove that there exists a film that was picked by at least 4 judges.
Out of \(7\) integer numbers, the sum of any \(6\) is a multiple of \(5\). Show that every one of these numbers is a multiple of \(5\).
An \(8 \times 8\) chessboard has 30 diagonals total (15 in each direction). Is it possible to place several chess pieces on this chessboard in such a way that the total number of pieces on each diagonal would be odd?
Anna’s garden is a grid of \(n \times m\) squares. She wants to have trees in some of these squares, but she wants the total number of trees in each column and in each row to be an odd number (not necessarily the same, they just all need to be odd). Show that it is possible only if \(m\) and \(n\) are both even or both odd and calculate in how many different ways she can place the trees in the grid.
30 pupils in years 7 to 11 took part in the creation of 40 maths problems. Every possible pair of pupils in the same year created the same number of problems. Every possible pair of pupils in different years created a different number of problems. How many pupils created exactly one problem?
Is it possible to fill a \(5 \times 5\) table with numbers so that the sum of the numbers in each row is positive and the sum of the numbers in each column is negative?
One term a school ran 20 sessions of an after-school Astronomy Club. Exactly five pupils attended each session and no two students encountered one another over all of the sessions more than once. Prove that no fewer than 20 pupils attended the Astronomy Club at some point during the term.
The total age of a group of 7 people is 332 years. Prove that it is possible to choose three members of this group so that the sum of their ages is no less than 142 years.
You are given 25 numbers. The sum of any 4 of these numbers is positive. Prove that the sum of all 25 numbers is also positive.