MathDB

1

Part of 2018 Middle European Mathematical Olympiad

Problems(2)

Simple inequality

Source: MEMO 2018 T1

9/2/2018
Let a,ba,b and cc be positive real numbers satisfying abc=1.abc=1. Prove thata2−b2a+bc+b2−c2b+ca+c2−a2c+ab≤a+b+c−3.\frac{a^2-b^2}{a+bc}+\frac{b^2-c^2}{b+ca}+\frac{c^2-a^2}{c+ab}\leq a+b+c-3.
inequalitiesam-gm inequalityeasy problem
Determine all functions

Source: MEMO 2018 I1

9/2/2018
Let Q+Q^+ denote the set of all positive rational number and let α∈Q+.\alpha\in Q^+. Determine all functions f:Q+→(α,+∞)f:Q^+ \to (\alpha,+\infty ) satisfying f(x+yα)=f(x)+f(y)αf(\frac{ x+y}{\alpha}) =\frac{ f(x)+f(y)}{\alpha} for all x,y∈Q+.x,y\in Q^+ .
algebrafunctionfunctional equation