An algebra problem by Aly Ahmed

Algebra Level 3

Let x x and y y positive integers whose sum is 100. Find the minimum value of x 2 + y 2 x^2 + y^2 .


The answer is 5000.

Aly Ahmed by Aly Ahmed , 34, Egypt

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1 solution

Elijah L
Jun 19, 2020

Using RMS-AM:

x 2 + y 2 2 ≥ x + y 2 \displaystyle \sqrt{\frac{x^2 + y^2}{2}} \ge \frac{x+y}{2}

x 2 + y 2 ≥ 2 ( x + y 2 ) 2 \displaystyle x^2 + y^2 \ge 2\left(\frac{x+y}{2}\right)^2

x 2 + y 2 ≥ 5000 x^2 + y^2 \ge \boxed{5000}

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