Let \(a,b,c,d\) be positive real numbers. Prove that \((a+b)\times(c+d) = ac+ad+bc+bd\). Find both algebraic solution and geometric interpretation.
Let \(a,b,c,d\) be positive real numbers such that \(a\geq b\) and \(c\geq d\). Prove that \((a-b)\times(c-d) = ac-ad-bc+bd\). Find both algebraic solution and geometric interpretation.
Using the area of a rectangle prove that \(a\times b=b\times a\).
Jason has \(20\) red balls and \(14\) bags to store them. Prove that there is a bag which contains at least two balls.
One of the most useful tools for proving mathematical statements is the Pigeonhole principle. Here is one example: suppose that a flock of \(10\) pigeons flies into a set of \(9\) pigeonholes to roost. Prove that at least one of these \(9\) pigeonholes must have at least two pigeons in it.
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 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 \(k\).
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 such that \(n^2\) is divisible by \(2\). Prove that \(n\) is divisible by \(2\).