MathDB

4

Part of 2020 European Mathematical Cup

Problems(2)

An inequality with square roots and minima

Source: European Mathematical Cup 2020, Problem J4

12/22/2020
Let a,b,ca,b,c be positive real numbers such that ab+bc+ac=a+b+cab+bc+ac = a+b+c. Prove the following inequality: a+bc+b+ca+c+ab≤2⋅min⁡{ab+bc+ca, ba+cb+ac}.\sqrt{a+\frac{b}{c}} + \sqrt{b+\frac{c}{a}} + \sqrt{c+\frac{a}{b}} \leq \sqrt{2} \cdot \min \left\{ \frac{a}{b}+\frac{b}{c}+\frac{c}{a},\ \frac{b}{a}+\frac{c}{b}+\frac{a}{c} \right\}. \\ \\ Proposed by Dorlir Ahmeti.
inequalitiesemc
interesting function

Source: 9th EMC, 12th December 2020 - 20th December 2020, SENIOR league, P4.

12/22/2020
Let R+\mathbb{R^+} denote the set of all positive real numbers. Find all functions f:R+→R+f: \mathbb{R^+}\rightarrow \mathbb{R^+} such that xf(x+y)+f(xf(y)+1)=f(xf(x))xf(x + y) + f(xf(y) + 1) = f(xf(x)) for all x,y∈R+.x, y \in\mathbb{R^+}.
Proposed by Amadej Kristjan Kocbek, Jakob Jurij Snoj
algebrafunctional equationfunction