MathDB

3

Part of 2001 Cono Sur Olympiad

Problems(2)

Three acute inscribed triangles determine a nonobtuse one

Source: XII Cono Sur Mathematical Olympiad (2001)

7/28/2011
Three acute triangles are inscribed in the same circle with their vertices being nine distinct points. Show that one can choose a vertex from each triangle so that the three chosen points determine a triangle each of whose angles is at most 90∘90^\circ.
geometryrectanglecombinatorics unsolvedcombinatorics
How long before a function with g(n+1)=g(n)+-1 reaches 2001

Source: XII Cono Sur Mathematical Olympiad (2001)

7/28/2011
A function gg defined for all positive integers nn satisfies [*]g(1)=1g(1) = 1; [*]for all n≥1n\ge 1, either g(n+1)=g(n)+1g(n+1)=g(n)+1 or g(n+1)=g(n)−1g(n+1)=g(n)-1; [*]for all n≥1n\ge 1, g(3n)=g(n)g(3n) = g(n); and [*]g(k)=2001g(k)=2001 for some positive integer kk. Find, with proof, the smallest possible value of kk.
functioninductionalgebra unsolvedalgebra