2018 UMO Problems/Problem 2

Problem 2

Let $P(x)$ be a cubic polynomial $(x-a)(x-b)(x-c)$ , where $a, b,$ and $c$ are positive real numbers. Let Q(x) be the polynomial with $Q(x) = (x-ab)(x-bc)(x-ac)$ . If $P(x) = Q(x)$ for all $x$ , then find the minimum possible value of $a + b + c$ .

Solution 1

Plugging in $x = 0$ , we find that $abc = 1$ . Using AM-GM, we have that $a+b+c \leq 3 ^3\sqrt{abc} = \fbox{3}$

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2018 UMO Problems/Problem 2

Problem 2

Solution 1