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Part of the School of Olympiads

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## A picking game (problem on induction)

There is a heap of $n$ matches. 2 players take turns to pick 1 or 2 matches. The winner is the person who picks the last match.

• Who wins for 5 matches (if no player misses a chance to win or draw)?
• Is there a general strategy for any number of matches (to force a win or a tie for either side)?
• What is the strategy if existing?

## A Rootful Question

Prove the following:

• $5<\sqrt{5}+\sqrt[3]{5}+\sqrt[4]{5}$
• $8>\sqrt{8}+\sqrt[3]{8}+\sqrt[4]{8}$
• $9>\sqrt{n}+\sqrt[3]{n}+\sqrt[4]{n},n\geq 9$

## A problem on functions

Find all pairs of real numbers (a,b) such that: f(x)= x2 + ax + b If q is a root of f(x) then q2-2 is also a root