Difference between revisions of "2023 RMO"

Revision as of 12:30, 9 December 2024

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

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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 ==Problem 5==
-Let <math>n>k>1</math> be positive integers. Determine all positive real numbers <math>a_1, a_2, ..., a_n</math> which satisfy <math>\sum_{i=1}^{n}</math> <math>\sqrt {\frac {ka_{i}^{k}}{k-1a_{i}^{k}+1}}</math> <math>=\sum_{i=1}^{n}</math> <math>a_i</math> <math>=n</math>.
 ==Problem 6==
 Consider a set of <math>16</math> points arranged in a <math>4\times4</math> square grid formation. Prove that if any <math>7</math> of these points are coloured blue, then there exists an isosceles right-angled triangle whose vertices are all blue.

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