Difference between revisions of "2023 RMO"

Latest revision as of 12:30, 9 December 2024

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 ==Problem 4==
-Let <math>\Omega_1,\Omega_2</math> be two intersecting circles with centres <math>O_1,O_2</math> respectively. Let <math>l</math> be a line that intersects <math>\Omega_1</math> at points <math>A,C</math> and <math>\Omega_2</math> at points <math>B,D</math> such that <math>A, B, C, D</math> are collinear in that order. Let the perpendicular bisector of segment <math>AB</math> intersect <math>\Omega_1</math> at points <math>P,Q</math>; and the perpendicular bisector of segment <math>CD</math> intersect <math>\Omega_1</math> at points <math>R,S</math> such that <math>P,R</math> are on the same side of <math>l</math>. Prove that the midpoints of <math>PR, QS</math> and <math>\Omega_{1} \Omega_{2}</math> are collinear.
 ==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.