CMO 1984

Prove that the sum of the squares of $1984$ consecutive positive integers cannot be the square of an integer.

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.

An integer is digitally divisible if both of the following conditions are fulfilled: $(a)$ 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.

An acute triangle has unit area. Show that there is a point inside the triangle whose distance from each of the vertices is at least $\frac{2}{\sqrt[4]{27}}$ .

Given any $7$ real numbers, prove that there are two of them $x,y$ such that $0\le\frac{x-y}{1+xy}\le\frac{1}{\sqrt{3}}$ .