HMMT 2014 #0jli

HMMT 2014Hard

From module Geometry with Complex Numbers

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Understand the scale

Scale 0.5: Easiest math competition problems, often solvable even for early middle school students without experience. 1: Problems strictly for beginners, often at a moderate middle school level. 1.5: Problems for stronger beginner students, on the level of the middling problems in most middle school contests. 2: For motivated beginners, harder questions from the previous categories. 2.5: More advanced beginner problems, hardest questions from previous categories. 3: Early intermediate problems that require more creative thinking. 3.5: Tougher early intermediate problems that consistently stretch into higher-level creative thinking or conceptual knowledge. 4: Intermediate-level problems. 4.5: Upper intermediate problems approaching the upper bound for non-invitational sprint math competitions. 5: More difficult AIME problems or simple proof-based Olympiad-style problems. 6: High-level AIME-styled questions or introductory Olympiad-level questions. 7: Tougher Olympiad-level questions, may require more technical knowledge. 8: High-level Olympiad-level questions. 9: Expert Olympiad-level questions. 9.5: The hardest problems appearing on Olympiads which the strongest students could reasonably solve. 10: Historically notorious problems, generally unsuitable for even very hard competitions because they are exceedingly tedious, long, and difficult.

Complex numbersComplex numbers in geometryConstructions and loci

Problem

Problem:

Prove that there exists a nonzero complex number $c$ and a real number $d$ such that

\left|\left|\frac{1}{1+z+z^{2}}\right|-\left|\frac{1}{1+z+z^{2}}-c\right| \right|=d

for all $z$ with $|z|=1$ and $1+z+z^{2} \neq 0$ . (Here, $|z|$ denotes the absolute value of the complex number $z$ , so that $|a+b i|=\sqrt{a^{2}+b^{2}}$ for real numbers $a, b$ .)

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