Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.0 AIME Find the answer

Given points $a$ and $b$ in the plane, let $a \oplus b$ be the unique point $c$ such that $a b c$ is an equilateral triangle with $a, b, c$ in the clockwise orientation. Solve $(x \oplus(0,0)) \oplus(1,1)=(1,-1)$ for $x$ .

Your answer

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

Solution

It is clear from the definition of $\oplus$ that $b \oplus(a \oplus b)=a$ and if $a \oplus b=c$ then $b \oplus c=a$ and $c \oplus a=b$ . Therefore $x \oplus(0,0)=(1,1) \oplus(1,-1)=(1-\sqrt{3}, 0)$ . Now this means $x=(0,0) \oplus(1-\sqrt{3}, 0)=\left(\frac{1-\sqrt{3}}{2}, \frac{3-\sqrt{3}}{2}\right)$ .

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