Browse · MathNet Print → Ukrainian Mathematical Olympiad Ukraine geometry Problem Prove that for any real numbers x and y the inequality ∣cosx∣+∣cosy∣+∣cos(x+y)∣≥1 holds. Solution — click to reveal ∣cosx∣+∣cosy∣+∣cos(x+y)∣≥∣cosxsiny∣+∣cosysinx∣+cos2(x+y)≥∣sin(x+y)∣+cos2(x+y)≥sin2(x+y)+cos2(x+y)=1. Techniques Trigonometry ← Previous problem Next problem →