MathDB

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Part of 1988 Greece National Olympiad

Problems(3)

\sqrt{1987+a}+\sqrt{1987+b}=2\sqrt{1987+c} 1988 Greece MO Grade X p1

Source:

9/7/2024
Let a>0,b>0,c>0a>0,b>0,c>0 and 1987+a+1987+b=21987+c\sqrt{1987+a}+\sqrt{1987+b}=2\sqrt{1987+c}. Prove that 12(a+b)≥c\frac{1}{2} (a+b )\ge c .
algebrainequalities
min x^2+y^2, x+y=2a-4, xy=a^2-3a+5 - 1988 Greece MO Grade XI p1

Source:

9/7/2024
Given x,y,a∈Rx,y,a\in \mathbb{R} , x+y=2a−4x+y=2a-4 and xy=a2−3a+5xy=a^2-3a+5. What is the minimum value of x2+y2x^2+y^2?
algebrainequalities
2f(x+y+xy)= a f(x)+ bf(y)+f(xy)

Source: 1988 Greece MO Grade X p1

9/6/2024
Find all functions f:R→Rf: \mathbb{R}\to\mathbb{R} that satidfy : 2f(x+y+xy)=af(x)+bf(y)+f(xy)2f(x+y+xy)= a f(x)+ bf(y)+f(xy) for any x,y∈Rx,y \in\mathbb{R} όπου a,b∈Ra,b\in\mathbb{R} with a2−a≠b2−ba^2-a\ne b^2-b
functionalgebrafunctional equation