Let R+ denote the set of positive real numbers. Find all functions f:R+→R and g:R+→R such that for all x,y∈R+, g(x)−g(y)=(x−y)f(xy). Linus Tang
Problem
Source: ELMO Shortlist 2024/A6
Tags: Elmo, algebra
23.06.2024 02:10
Rewrite as ∑cyca(b−c)f(a)=0. If f(a)=m+na and f(b)=m+nb, then we get f(c)=m+nc, which gives g(x)=mx−nx+C.
23.06.2024 11:35
Basically the same idea as in ISL 2018 A5.
23.06.2024 20:28
Trivial . ig. Let S be the set of all g for such f exists. Claim: For any two elements x,y∈S and real nos a,b ax+by∈S. Proof: trivial Claim: For an element in g∈S which is not injective , then g is a linear combination of x and 1x . and a constant. Proof: Say g(a)=g(b) , a≠b observe g(x)=g(abx)∀x∈R . Observe f(xy)(x−y)=(g(x)−g(y)) and f(dxy)(x−dy)=g(x)−g(y) divide the two [ u can check divisible So f(xy)f(dxy)=x−yx−dy∀xy≠d from here u can get an explicit formula for f and g Now observe any constant function is in S and of form cx and cx , say we have another solution g′ which is not a linear combination of these . construct a linear combination of ocnstant x and 1x and g′ which is not injective and use last claim