MathDB

2

Part of 2012 Spain Mathematical Olympiad

Problems(2)

Spanish functional equation

Source: Spanish MO 2012 Q2

6/7/2012
Find all functions f:R→Rf:\mathbb{R}\to\mathbb{R} such that (x−2)f(y)+f(y+2f(x))=f(x+yf(x))(x-2)f(y)+f(y+2f(x))=f(x+yf(x)) for all x,y∈Rx,y\in\mathbb{R}.
functionalgebra proposedalgebra
Show every term in sequence is an integer

Source: Spanish MO 2012 Q5

6/7/2012
A sequence (an)n≥1(a_n)_{n\ge 1} of integers is defined by the recurrence a1=1, a2=5, an=an−12+4an−2 for n≥2.a_1=1,\ a_2=5,\ a_n=\frac{a_{n-1}^2+4}{a_{n-2}}\ \text{for}\ n\ge 2. Prove that all terms of the sequence are integers and find an explicit formula for ana_n.
algebra unsolvedalgebra