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Shades of Fermat's Last Theorem

The familiar Pythagorean 3-4-5 triple gives one solution to (x-1)^n + x^n = (x+1)^n so what about other solutions for x an integer and n= 2, 3, 4 or 5?

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Two Cubes

Two cubes, each with integral side lengths, have a combined volume equal to the total of the lengths of their edges. How big are the cubes? [If you find a result by 'trial and error' you'll need to prove you have found all possible solutions.]

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Exhaustion

Find the positive integer solutions of the equation (1+1/a)(1+1/b)(1+1/c) = 2

Unit Interval

Stage: 4 and 5 Challenge Level: Challenge Level:1

Given any two numbers between $0$ and $1$ you have to prove that their sum is less than 1 plus their product; that is, given $0 < x < 1$ and $0 < y < 1$, prove that $x + y < 1 + xy$.

Hyeyoun Chung, St Paul's Girls' School, and Andaleeb Ahmed, Woodhouse Sixth Form College, London both produced nice solutions.

Consider $1-x$ and $1-y$. Since $0 < x < 1$ and $0 < y < 1$ it follows that

$\begin{eqnarray} \\ (1 - x)(1 - y) & > & 0 \\ 1 - x - y + xy & > & 0 \\ 1 + xy & > & x + y. \end{eqnarray}$

This is equivalent to $x + y < 1 + xy$.