MathDB

3

Part of 2011 Belarus Team Selection Test

Problems(3)

max value of summands each natural >=3 may be represented as sum

Source: 2011 Belarus TST 1.3

11/7/2020
Any natural number n,n≥3n, n\ge 3 can be presented in different ways as a sum several summands (not necessarily different). Find the greatest possible value of these summands.
Folklore
algebranumber theorySum
f(f(x+y))=xf(y)+g(x)

Source: 2011 Belarus TST 2.3

11/7/2020
Find all functions f:R→R,g:R→Rf: R \to R ,g: R \to R satisfying the following equality f(f(x+y))=xf(y)+g(x)f(f(x+y))=xf(y)+g(x) for all real xx and yy.
I. Gorodnin
algebrafunctional equationfunctional
f(x-f(x/y))=xf(1-f(1/y))

Source: 2011 Belarus TST 7.3

11/7/2020
Find all functions f:R→Rf:R\to R such that for all real x,yx,y with y≠0y\ne 0 f(x−f(x/y))=xf(1−f(1/y))f(x-f(x/y))=xf(1-f(1/y)) and a) f(1−f(1))≠0f(1-f(1))\ne 0 b) f(1−f(1))=0 f(1-f(1))= 0
S. Kuzmich, I.Voronovich
algebrafunctionalfunctional equation