MathDB

3

Part of 2024 Mongolian Mathematical Olympiad

Problems(2)

Unexpectedly easy functional equation

Source: Mongolian Mathematical Olympiad 2024 P3

4/12/2024
Let R+\mathbb{R}^+ denote the set of positive real numbers. Determine all functions f:R+→R+f: \mathbb{R}^+ \to \mathbb{R}^+ such that for all positive real numbers xx and yy : f(x)f(y+f(x))=f(1+xy)f(x)f(y+f(x))=f(1+xy)
Proposed by Otgonbayar Uuye.
functional equationfunctionalgebra
Subsets of a set form a complete residue system mod 2^n

Source: Mongolia national olympiad 2024

4/10/2024
A set XX consisting of nn positive integers is called <spanclass=′latex−italic′>good</span><span class='latex-italic'>good</span> if the following condition holds: For any two different subsets of XX, say AA and BB, the number s(A)−s(B)s(A) - s(B) is not divisible by 2n2^n. (Here, for a set AA, s(A)s(A) denotes the sum of the elements of AA) Given nn, find the number of good sets of size nn, all of whose elements is strictly less than 2n2^n.
combinatoricsnumber theorycomplete residue system