Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.7 AIME Find the answer

Let $a$ and $b$ be positive real numbers. Determine the minimum possible value of $\sqrt{a^{2}+b^{2}}+\sqrt{(a-1)^{2}+b^{2}}+\sqrt{a^{2}+(b-1)^{2}}+\sqrt{(a-1)^{2}+(b-1)^{2}}$

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

Let $A B C D$ be a square with $A=(0,0), B=(1,0), C=(1,1), D=(0,1)$ , and $P$ be a point in the same plane as $A B C D$ . Then the desired expression is equivalent to $A P+B P+C P+D P$ . By the triangle inequality, $A P+C P \geq A C$ and $B P+D P \geq B D$ , so the minimum possible value is $A C+B D=2 \sqrt{2}$ . This is achievable when $a=b=\frac{1}{2}$ , so we are done.

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