MathDB

6

Part of 2019 Caucasus Mathematical Olympiad

Problems(2)

Three congruent lines connected with right-angled triangle and its bisector

Source: IV Caucasus Mathematic Olympiad

4/7/2019
In a triangle ABCABC with ∠BAC=90∘\angle BAC = 90^{\circ} let BLBL be the bisector, L∈ACL\in AC. Let DD be a point symmetrical to AA with respect to BLBL. Let MM be the circumcenter of ADCADC. Prove that CMCM, DLDL, and ABAB are concurrent.
geometry
copy of existing problem

Source: IV Caucasus Mathematic Olympiad

4/7/2019
15 boxes are given. They all are initially empty. By one move it is allowed to choose some boxes and to put in them numbers of apricots which are pairwise distinct powers of 2. Find the least positive integer kk such that it is possible to have equal numbers of apricots in all the boxes after kk moves.
combinatorics