MathDB

6

Part of 2005 IMO Shortlist

Problems(2)

A=b

Source: Taiwan 1st TST 2006, 1st day, problem 3

3/29/2006
Let aa, bb be positive integers such that bn+nb^n+n is a multiple of an+na^n+n for all positive integers nn. Prove that a=ba=b.
Proposed by Mohsen Jamali, Iran
modular arithmeticnumber theoryDivisibilityIMO ShortlistChinese Remainder TheoremHi
Isotomic conjugates [intersections of median & incircle]

Source: belgian IMO preparation; IMO Shortlist 2005 geometry problem G6

3/25/2006
Let ABCABC be a triangle, and MM the midpoint of its side BCBC. Let γ\gamma be the incircle of triangle ABCABC. The median AMAM of triangle ABCABC intersects the incircle γ\gamma at two points KK and LL. Let the lines passing through KK and LL, parallel to BCBC, intersect the incircle γ\gamma again in two points XX and YY. Let the lines AXAX and AYAY intersect BCBC again at the points PP and QQ. Prove that BP=CQBP = CQ.
geometryhomothetyreflectiontrapezoidIMO Shortlistprojective geometryPolars