In the last issue of Parabola George Szekeres (see the end of his article on Carmichael numbers) stated that one of the most famous problems in mathematics, Fermat's last theorem, appeared to have been solved.
In my first year at university, our lecturer offered a $\$100$ prize to anyone who could work out $$ \int e^{-x^2} dx$$
Before reading the rest of this article try to prove the following statements.
- If a drawer contains a large number of socks of the same colour but two different sizes, and I take out three socks, then two of these will make a pair.
Consider a fixed reflector $R$ on the rim of a bicycle wheel of radius $r$.
One of the questions common to the Senior and Junior divisions of this year's School Mathematics Competition was the following.
Q.903 Find all positive integers $n$ such that $1+2+3+\cdots + n$ is a factor of $1\times 2\times 3 \times \cdots \times n$.
Q.893 Find all positive integer solutions of $$x^2 - 84 = 6y +3x - 2xy.$$