MathDB

3

Part of 2022 Moscow Mathematical Olympiad

Problems(2)

Geometry with circumcircles

Source: Moscow mathematic olympiad 2022, P11.3

9/29/2022
Bisector ALAL is drawn in an acute triangle ABCABC. On the line LALA beyond the point AA, the point K is chosen with AK=ALAK = AL. Circumcirles of triangles BLKBLK and CLKCLK intersect segments ACAC and ABAB at points PP and QQ respectively. Prove that lines PQPQ and BCBC are parallel.
geometrycircumcircle
Area of 12-gon

Source: Moscow mathematic olympiad 2022, Day 2 P11.3

9/29/2022
In a convex 1212-gon, all angles are equal. It is known that the lengths of some 1010 of its sides are equal to 11, and the length of one more equals 22. What can be the area of ​​this 1212-gon?
geometry