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Estonian Math Competitions

Algebra Difficulty 4.8 AIME Prove it Estonia

Do there exist numbers $a$ , $b$ , $c$ that satisfy the equation
$2a(c-a) - b(2a+b) + c(2b-c) = 2020?$

Solution

Transforming the l.h.s. of the equation gives
$\begin{aligned} 2a(c-a) - b(2a+b) + c(2b-c) &= 2ac - 2a^2 - 2ab - b^2 + 2bc - c^2 \\ &= -a^2 - (a+b-c)^2. \end{aligned}$
The equality $-a^2 - (a+b-c)^2 = 2020$ cannot be valid since all terms of its l.h.s. are non-positive whereas the r.h.s. is positive.

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