MathDB

1

Part of 2023 Middle European Mathematical Olympiad

Problems(2)

Functional inequality

Source: MEMO 2023 I1

8/24/2023
For each pair (α,β)(\alpha, \beta) of non-negative reals with α+β≥2\alpha+\beta \geq 2, determine all functions f:R→Rf:\mathbb{R} \rightarrow \mathbb{R}, such that f(x)f(y)≤f(xy)+αx+βyf(x)f(y) \leq f(xy)+\alpha x+\beta y for all reals x,yx, y.
inequalitiesMEMO2023
Maximal number of distinct values

Source: MEMO 2023 T1

8/25/2023
(a) A function f:Z→Zf:\mathbb{Z} \rightarrow \mathbb{Z} is called Z\mathbb{Z}-good if f(a2+b)=f(b2+a)f(a^2+b)=f(b^2+a) for all a,b∈Za, b \in \mathbb{Z}. What is the largest possible number of distinct values that can occur among f(1),…,f(2023)f(1), \ldots, f(2023), where ff is a Z\mathbb{Z}-good function?
(b) A function f:N→Nf:\mathbb{N} \rightarrow \mathbb{N} is called N\mathbb{N}-good if f(a2+b)=f(b2+a)f(a^2+b)=f(b^2+a) for all a,b∈Na, b \in \mathbb{N}. What is the largest possible number of distinct values that can occur among f(1),…,f(2023)f(1), \ldots, f(2023), where ff is a N\mathbb{N}-good function?
algebra