2013 Singapore Math Olympiad | IMO Challenge

#brainexercise #imo #matholympiad

2013 Singapore Math Olympiad | IMO Challenge

[NadiehFan]

This can be done much easier. You should have started by observing that

(x − 1)(x² + x + 1) = x³ − 1

Consequently, the roots of

x² + x + 1 = 0

are the two complex third roots of unity, which implies that x³ = 1 and therefore also x³ⁿ = 1 for any integer n. So, we can immediately conclude that

x⁴⁹ + x⁵⁰ + x⁵¹ + x⁵² + x⁵³ = x + x² + 1 + x + x² = x + x² = −1