Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.6 AIME Prove it United States

Problem:

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 $x$ .

Solution

Solution:

Answer: $\left(\frac{1-\sqrt{3}}{2}, \frac{3-\sqrt{3}}{2}\right)$

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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