How Can The Triangle Help Us Solve This British Mathematical Olympiad Question?

Let a, b and c be the lengths of the sides of a triangle. Suppose that ab + bc + ca = 1.

Show that (a + 1)(b + 1)(c + 1) < 4.

This maths challenge comes from Round 1 of the British Olympiad in 2010.

I believe the ingenuity involved is worth learning for anyone who wants to hone their competition skills!

So please give this a go :)

Once again, this is a good moment to pause the article and give the problem a go yourself.

When you’re ready, keep reading for the solution.

And if you come up with your own approach, feel free to share it in the comments — I’d love to see how you tackled it.

Hey, I’m Barry, your friendly neighbourhood mathematician. My mission is to create accessible maths that inspires students and teachers around the world.

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