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Problem 8

Part of 1999 Ukraine National Mathematical Olympiad

Problems(2)

f(f(n))+f(n)=2n

Source:

9/14/2014
find all function such that f:N−>Nf:N->N for every n∈N n\in\mathbb{N} , f(f(n))+f(n)=2nf(f(n))+f(n)=2n
PLZ->(solution not to use characterastic equation)<-
functioninductionstrong inductionalgebra unsolvedalgebra
concurrency of lines formed by perpendiculars

Source: Ukraine 1999 Grade 10 P8

5/11/2021
Let AA1,BB1,CC1AA_1,BB_1,CC_1 be the altitudes of an acute-angled triangle ABCABC, and let OO be an arbitrary interior point. Let M,N,P,Q,R,SM,N,P,Q,R,S be the feet of the perpendiculars from OO to the lines AA1,BC,BB1,CA,CC1,ABAA_1,BC,BB_1,CA,CC_1,AB, respectively. Prove that the lines MN,PQ,RSMN,PQ,RS are concurrent.
geometryTriangles