Olympiad problems

2024 Thailand TSTST, 9

Let triangle $ ABC $ be an acute-angled triangle. Square $ AEFB $ and $ ADGC $ lie outside triangle $ ABC $. $ BD $ intersects $ CE $ at point $ H $, and $ BG $ intersects $ CF $ at point $ I $. The circumcircle of triangle $ BFI $ intersects the circumcircle of triangle $ CGI $ again at point $ K $. Prove that line segment $ HK $ bisects $ BC $.

2024 Thailand October Camp, 3

Tags: cartesian coordinates , spiral similarity , complex bash , inversion , geometry

Let triangle $ ABC $ be an acute-angled triangle. Square $ AEFB $ and $ ADGC $ lie outside triangle $ ABC $. $ BD $ intersects $ CE $ at point $ H $, and $ BG $ intersects $ CF $ at point $ I $. The circumcircle of triangle $ BFI $ intersects the circumcircle of triangle $ CGI $ again at point $ K $. Prove that line segment $ HK $ bisects $ BC $.

2024 Bangladesh Mathematical Olympiad, P7

Tags: perpendicular lines , cartesian coordinates , geometry

Let $ABCD$ be a square. $E$ and $F$ lie on sides $AB$ and $BC$, respectively, such that $BE = BF$. The line perpendicular to $CE$, which passes through $B$, intersects $CE$ and $AD$ at points $G$ and $H$, respectively. The lines $FH$ and $CE$ intersect at point $P$ and the lines $GF$ and $CD$ intersect at point $Q$. Prove that the line $DP$ is perpendicular to the line $BQ$.

2008 Peru MO (ONEM), 4

Tags: combinatorics , geometry , lattice points , existence , cartesian coordinates

All points in the plane that have both integer coordinates are painted, using the colors red, green, and yellow. If the points are painted so that there is at least one point of each color. Prove that there are always three points $X$, $Y$ and $Z$ of different colors, such that $\angle XYZ = 45^{\circ} $