Problem #PRU-35457

Problems Calculus Real numbers Integer and fractional parts. Archimedean property Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

13–15 3.0

Problem

Prove that the equation $a_{1} \sin x + b_{1} \cos x + a_{2} \sin 2 x + b_{2} \cos 2 x + \dots + a_{n} \sin n x + b_{n} \cos n x = 0$ has at least one root for any values of $a_{1}, b_{1}, a_{2}, b_{2}, \dots, a_{n}, b_{n}$ .

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