MathDB

6

Part of 2023 Kazakhstan National Olympiad

Problems(2)

Beautiful geometry problem, HM line

Source: Kazakhstan (Junior) National Olympiad 2023, 9th grade, P6

3/22/2023
The altitudes of an acute triangle ABCABC intersect at HH. The tangent line at HH to the circumcircle of triangle BHCBHC intersects the lines ABAB and ACAC at points QQ and PP respectively. The circumcircles of triangles ABCABC and APQAPQ intersect at point KK (K≠AK\neq A). The tangent lines at the points AA and KK to the circumcircle of triangle APQAPQ intersect at TT. Prove that THTH passes through the midpoint of segment BCBC.
geometry
1.061, intersection of two rhombuses

Source: Kazakhstan National Olympiad 2023, 10-11 grade, P6

3/22/2023
Inside an equilateral triangle with side 33 there are two rhombuses with sides 1,0611,061 and acute angles 60∘60^\circ. Prove that these two rhombuses intersect each other. (The vertices of the rhombus are strictly inside the triangle.)
geometry