MathDB

1

Part of 2008 Brazil National Olympiad

Problems(2)

Multiples beginning with 2008

Source: Brazilian Math Olympiad 2008, Problem 1

10/28/2008
A positive integer is dapper if at least one of its multiples begins with 2008 2008. For example, 7 7 is dapper because 200858 200858 is a multiple of 7 7 and begins with 2008 2008. Observe that 200858 \equal{} 28694\times 7. Prove that every positive integer is dapper.
pigeonhole principlenumber theory unsolvednumber theory
Cyclic quadrilateral and reflections wrt bisectors

Source: Brazilian Math Olympiad 2008, Problem 4

10/28/2008
Let ABCD ABCD be a cyclic quadrilateral and r r and s s the lines obtained reflecting AB AB with respect to the internal bisectors of ∠CAD \angle CAD and ∠CBD \angle CBD, respectively. If P P is the intersection of r r and s s and O O is the center of the circumscribed circle of ABCD ABCD, prove that OP OP is perpendicular to CD CD.
geometrygeometric transformationreflectioncircumcircleincenterhomothetycyclic quadrilateral