MathDB

5

Part of 1999 Irish Math Olympiad

Problems(2)

find a term of a sequence

Source: Ireland 1999

7/4/2009
The sequence un u_n, n\equal{}0,1,2,... is defined by u_0\equal{}0, u_1\equal{}1 and for each n≥1 n \ge 1, u_{n\plus{}1} is the smallest positive integer greater than un u_n such that \{ u_0,u_1,...,u_{n\plus{}1} \} contains no three elements in arithmetic progression. Find u100 u_{100}.
inductionnumber theory proposednumber theory
concurrent lines

Source: Ireland 1999

7/4/2009
A convex hexagon ABCDEF ABCDEF satisfies AB\equal{}BC, CD\equal{}DE, EF\equal{}FA and: \angle ABC\plus{}\angle CDE\plus{}\angle EFA \equal{} 360^{\circ}. Prove that the perpendiculars from A,C A,C and E E to FB,BD FB,BD and DF DF respectively are concurrent.
geometry proposedgeometry