MathDB

2

Part of 2010 Saint Petersburg Mathematical Olympiad

Problems(2)

Geometry with midpoints

Source: St Petersburg Olympiad 2010, Grade 11, P2

9/14/2017
ABCABC is triangle with AB=BCAB=BC. X,YX,Y are midpoints of ACAC and ABAB. ZZ is base of perpendicular from BB to CYCY. Prove, that circumcenter of XYZXYZ lies on ACAC
geometrycircumcircle
Divisors of big numbers.

Source: St Petersburg Olympiad 2010, Grade 10, P2

9/14/2017
There are 1010 consecutive 30-digit numbers. We write the biggest divisor for every number ( divisor is not equal number). Prove that some written numbers ends with same digit.
number theory