MathDB

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Part of 2022 Middle European Mathematical Olympiad

Problems(2)

A functional equation from MEMO

Source: Middle European Mathematical Olympiad 2022, problem I-1

9/1/2022
Find all functions f:R→Rf: \mathbb R \to \mathbb R such that f(x+f(x+y))=x+f(f(x)+y)f(x+f(x+y))=x+f(f(x)+y) holds for all real numbers xx and yy.
Two sum-product sequences

Source: MEMO 2022 T1

9/2/2022
Given a pair (a0,b0)(a_0, b_0) of real numbers, we define two sequences a0,a1,a2,...a_0, a_1, a_2,... and b0,b1,b2,...b_0, b_1, b_2, ... of real numbers by an+1=an+bna_{n+1}= a_n + b_n and bn+1=anbnb_{n+1}=a_nb_n for all n=0,1,2,...n = 0, 1, 2,.... Find all pairs (a0,b0)(a_0, b_0) of real numbers such that a2022=a0a_{2022}= a_0 and b2022=b0b_{2022}= b_0.
algebra