International Mathematics Competition
for University Students
2024

IMC2024: Day 1, Problem 1

Problem 1. Determine all pairs \(\displaystyle (a,b) \in \mathbb{C} \times \mathbb{C}\) satisfying

\(\displaystyle |a| = |b| = 1 \quad \text{and} \quad a + b + a\overline{b} \in \mathbb{R}. \)

Mike Daas, Universiteit Leiden

Hint: Write \(\displaystyle a = e^{ix}\) and \(\displaystyle b = e^{iy}\), and transform the RHS to a product.

  Solution