MathDB

2

Part of 2010 Spain Mathematical Olympiad

Problems(2)

Spanish functional equation, find n for f(n)=2010

Source: Spain MO 2010

2/11/2011
Let N0\mathbb{N}_0 and Z\mathbb{Z} be the set of all non-negative integers and the set of all integers, respectively. Let f:N0→Zf:\mathbb{N}_0\rightarrow\mathbb{Z} be a function defined as f(n)=−f(⌊n3⌋)−3{n3}f(n)=-f\left(\left\lfloor\frac{n}{3}\right\rfloor \right)-3\left\{\frac{n}{3}\right\} where ⌊x⌋\lfloor x \rfloor is the greatest integer smaller than or equal to xx and {x}=x−⌊x⌋\{ x\}=x-\lfloor x \rfloor. Find the smallest integer nn such that f(n)=2010f(n)=2010.
functionfloor functioninductionalgebra proposedalgebra
PA' and B'C' intersect on A-median

Source: Spain MO 2010

2/11/2011
In a triangle ABCABC, let PP be a point on the bisector of ∠BAC\angle BAC and let A′,B′A',B' and C′C' be points on lines BC,CABC,CA and ABAB respectively such that PA′PA' is perpendicular to BC,PB′⊥ACBC,PB'\perp AC, and PC′⊥ABPC'\perp AB. Prove that PA′PA' and B′C′B'C' intersect on the median AMAM, where MM is the midpoint of BCBC.
geometryincentergeometry proposed