Problems

Can three points with integer coordinates be the vertices of an equilateral triangle?

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?

Prove the divisibility rule for $25$: a number is divisible by $25$ if and only if the number made by the last two digits of the original number is divisible by $25$;
Can you come up with a divisibility rule for $125$?

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 fig.)

The rain stopped and all people closed their umbrellas. They now stood keeping a distance of $50$ cm between neighbours. By how many times has the queue length decreased? People can be considered points, and umbrellas are circles with a radius of $50$ cm.

Cut a square into five triangles in such a way that the area of one of these triangles is equal to the sum of the area of other four triangles.