Problems

This is a famous problem, called Monty Hall problem after a popular TV show in America.
In the problem, you are on a game show, being asked to choose between three doors. Behind each door, there is either a car or a goat. You choose a door. The host, Monty Hall, picks one of the other doors, which he knows has a goat behind it, and opens it, showing you the goat. (You know, by the rules of the game, that Monty will always reveal a goat.) Monty then asks whether you would like to switch your choice of door to the other remaining door. Assuming you prefer having a car more than having a goat, do you choose to switch or not to switch?

Find a representation as a product of \(a^{2n+1} + b^{2n+1}\) for general \(a,b,n\).

Find a representation as a product of \(a^n - b^n\) for general \(a,b,n\).

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 white or black. The numbers \(2016\) and \(2017\) are coloured differently. Prove that there are three identically coloured integers which sum to zero.