MathDB

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Part of 2019 Middle European Mathematical Olympiad

Problems(2)

a FE on MEMO

Source: 2019 MEMO Problem I-1

8/28/2019
Find all functions f:R→Rf:\mathbb{R} \to \mathbb{R} such that for any two real numbers x,yx,y holds f(xf(y)+2y)=f(xy)+xf(y)+f(f(y)).f(xf(y)+2y)=f(xy)+xf(y)+f(f(y)).
Proposed by Patrik Bak, Slovakia
algebrafunctional equationmemoMEMO 2019injective function
Minimum and maximum

Source: 2019 MEMO Problem T-1

8/30/2019
Determine the smallest and the greatest possible values of the expression (1a2+1+1b2+1+1c2+1)(a2a2+1+b2b2+1+c2c2+1)\left( \frac{1}{a^2+1}+\frac{1}{b^2+1}+\frac{1}{c^2+1}\right)\left( \frac{a^2}{a^2+1}+\frac{b^2}{b^2+1}+\frac{c^2}{c^2+1}\right) provided a,ba,b and cc are non-negative real numbers satisfying ab+bc+ca=1ab+bc+ca=1.
Proposed by Walther Janous, Austria
algebrainequalitiesmemoMEMO 2019