MathDB

4

Part of 2014 European Mathematical Cup

Problems(2)

An function form N to N

Source: EMC 2014 P4

12/14/2014
Find all functions ff from positive integers to themselves such that: 1)f(mn)=f(m)f(n)f(mn)=f(m)f(n) for all positive integers m,nm, n 2){1,2,...,n}={f(1),f(2),...f(n)}\{1, 2, ..., n\}=\{f(1), f(2), ... f(n)\} is true for infinitely many positive integers nn.
functioninductionnumber theory proposednumber theory
Weird looking function

Source: European Mathematical Cup 2014, Senior Division, P4

12/14/2014
Find all functions f:R→R f:\mathbb{R}\rightarrow\mathbb{R} such that for all x,y∈Rx,y\in{{\mathbb{R}}} holds f(x2)+f(2y2)=(f(x+y)+f(y))(f(x−y)+f(y))f(x^2)+f(2y^2)=(f(x+y)+f(y))(f(x-y)+f(y))
Proposed by Matija Bucić
functionalgebra