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 into 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 \(125\)?
Which of the following numbers are divisible by \(11\) and which are not? \[121,\, 143,\, 286, 235, \, 473,\, 798, \, 693,\, 576, \,748\] Can you write down and prove a divisibility rule which helps to determine if a three digit number is divisible by \(11\)?
In how many ways can eight rooks be arranged on the chessboard in such a way that none of them can take any other. The color of the rooks does not matter, it’s everyone against everyone.
How many five-digit numbers are there which are written in the same from left to right and from right to left? For example the numbers \(54345\) and \(12321\) satisfy the condition, but the numbers \(23423\) and \(56789\) do not.
Definition A set is a collection of elements, containing only one copy of each element. The elements are not ordered, nor they are governed by any rule. We consider an empty set as a set too.
There is a set \(C\) consisting of \(n\) elements. How many sets can be constructed using the elements of \(C\)?
There are six letters in the alphabet of the Bim-Bam tribe. A word is any sequence of six letters that has at least two identical letters. How many words are there in the language of the Bim-Bam tribe?