MathDB

4

Part of 2009 Saint Petersburg Mathematical Olympiad

Problems(2)

Cells in square

Source: St Petersburg Olympiad 2009, Grade 11, P4

8/30/2017
From 2008×20082008 \times 2008 square we remove one corner cell 1×11 \times 1. Is number of ways to divide this figure to corners from 33 cells odd or even ?
combinatorics
Orthocenter

Source: St Petersburg Olympiad 2009, Grade 9, P4

8/30/2017
Points A1A_1 and C1C_1 are on BCBC and ABAB of acute-angled triangle ABCABC . AA1AA_1 and CC1CC_1 intersect in KK. Circumcircles of AA1B,CC1BAA_1B,CC_1B intersect in PP - incenter of AKCAKC. Prove, that PP - orthocenter of ABCABC
geometrycircumcircleincenter