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Berkeley Math Circle

Algebra Difficulty 4.7 AIME Prove it United States

Problem:
Suppose $x$ and $y$ are real numbers satisfying $x + y = 5$ . What is the largest possible value of $x^{2} + 2 x y$ ?

Solution

Solution:
The quantity in question is $(x + y)^{2} - y^{2} \leq (x + y)^{2} = 25$ . Equality occurs when $x = 5$ and $y = 0$ , hence the maximum possible value is $25$ .

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