Show the following: Pigeonhole principle strong form: Let \(q_1, \,q_2,\, . . . ,\, q_n\) be positive integers. If \(q_1+ q_2+ . . . + q_n - n + 1\) objects are put into \(n\) boxes, then either the \(1\)st box contains at least \(q_1\) objects, or the \(2\)nd box contains at least \(q_2\) objects, . . ., or the \(n\)th box contains at least \(q_n\) objects.
How can you deduce the usual Pigeonhole principle from this statement?
Each integer point on the numerical axis is colored either white or black and the numbers \(2016\) and \(2017\) are colored differently. Prove that there are three identically colored integers which sum up to zero.
Each integer on the number line is coloured either yellow or blue. Prove that there is a colour with the following property: For every natural number \(k\), there are infinitely many numbers of this colour divisible by \(k\).
There are \(100\) non-zero numbers written in a circle. Between every two adjacent numbers, their product was written, and the previous numbers were erased. It turned out that the number of positive numbers after the operation coincides with the amount of positive numbers before. What is the minimum amount of positive numbers that could have been written initially?
Let \(r\) be a rational number and \(x\) be an irrational number(i.e. not a rational one). Prove that the number \(r+x\) is irrational.
If \(r\) and \(s\) are both irrational, then must \(r+s\) be irrational as well?
Definition: We call a number \(x\) rational if there exist two integers \(p\) and \(q\) such that \(x=\frac{p}{q}\). We assume that \(p\) and \(q\) are coprime.
Prove that \(\sqrt{2}\) is not rational.
Let \(n\) be an integer. Prove that if \(n^2\) is divisible by \(2\), then \(n\) is divisible by \(2\).
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 \(x>y\).