2023 RMO

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Let $n>k>1$ be positive integers. Determine all positive real numbers $a_1, a_2, ..., a_n$ which satisfy $\sum_{i=1}^{n}$ $\sqrt {\frac {ka_{i}^{k}}{k-1a_{i}^{k}+1}}$ $=\sum_{i=1}^{n}$ $a_i$ $=n$ .

Problem 6

Consider a set of $16$ points arranged in a $4\times4$ square grid formation. Prove that if any $7$ of these points are coloured blue, then there exists an isosceles right-angled triangle whose vertices are all blue.

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