MathDB

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Part of 2011 Polish MO Finals

Problems(2)

Sum of the numbers is divisible by k

Source: LXII Polish Olympiad 2011, Problem 1

5/24/2011
Find all integers n≥1n\geq 1 such that there exists a permutation (a1,a2,...,an)(a_1,a_2,...,a_n) of (1,2,...,n)(1,2,...,n) such that a1+a2+...+aka_1+a_2+...+a_k is divisible by kk for k=1,2,...,nk=1,2,...,n
number theory unsolvednumber theory
All functions with f(x)f(y)=g(x)g(y)+g(x)+g(y)

Source: LXII Polish Olympiad 2011, Problem 4

5/24/2011
Determine all pairs of functions f,g:R→Rf,g:\mathbb{R}\rightarrow\mathbb{R} such that for any x,y∈Rx,y\in \mathbb{R}, f(x)f(y)=g(x)g(y)+g(x)+g(y).f(x)f(y)=g(x)g(y)+g(x)+g(y).
functionquadraticsalgebra unsolvedalgebra