Canadian Mathematical Olympiad 1984
PROBLEM 1
Prove that the sum of the squares of 1984 consecutive positive integers cannot be the square of an integer.
Problem 2
Alice and Bob are in a hardware store. The store sells coloured sleeves that fit over keys to distinguish them. The following conversation takes place:
Alice: Are you going to cover your keys? Bob: I would like to, but there are only 7 colours and I have 8 keys. Alice: Yes, but you could always distinguish a key by noticing that the red key next to the green key was different from the red key next to the blue key. Bob: You must be careful what you mean by "next to" or "three keys over from" since you can turn the key ring over and the keys are arranged in a circle. Alice: Even so, you don't need 8 colours. Problem: What is the smallest number of colours needed to distinguish $n$ keys if all the keys are to be covered.
Problem 3
An integer is digitally divisible if (a) none of its digits is zero; (b) it is divisible by the sum of its digits (e.g., 322 is digitally divisible).
Show that there are infinitely many digitally divisible integers. Problem 4 An acute-angled triangle has unit area. Show that there is a point inside the triangle whose distance from each of the vertices is at least $\frac{2}{\sqrt[4]{27}}$.
Problem 5 Given any 7 real numbers, prove that there are two of them, say $x$ and $y$, such that