Problems

Let \(n\) be an integer. Prove that if \(n^3\) is divisible by \(3\), then \(n\) is divisible by \(3\).

The numbers \(x\) and \(y\) satisfy \(x+3 = y+5\). Prove that \(x>y\).

The numbers \(x\) and \(y\) satisfy \(x+7 \geq y+8\). Prove that \(x>y\).

Prove that there are infinitely many natural numbers \(\{1,2,3,4,...\}\).

Prove that there are infinitely many prime numbers \(\{2,3,5,7,11,13...\}\).

Is it possible to colour the cells of a \(3\times 3\) board red and yellow such that there are the same number of red cells and yellow cells?

A coin is tossed six times. How many different sequences of heads and tails can you get?

Each cell of a \(3 \times 3\) square can be painted either black, or white, or grey. How many different ways are there to colour in this table?

Consider a set of numbers \(\{1,2,3,4,...n\}\) for natural \(n\). Find the number of permutations of this set, namely the number of possible sequences \((a_1,a_2,...a_n)\) where \(a_i\) are different numbers from \(1\) to \(n\).

Eleven people were waiting in line in the rain, each holding an umbrella. They stood closely together, so that the umbrellas of the neighbouring people were touching (see the figure)

The rain stopped and everyone closed their umbrellas. They then shuffled closer, keeping a distance of \(50\) cm between neighbours. What is the ratio of the old queue length to the new queue length? People can be considered points, and umbrellas are circles with a radius of \(50\) cm.