KURSCHAK 1949

Prove that $\sin x + \frac12 \sin 2x + \frac13 \sin 3x > 0$ for $0 < x < 180^o$ .

$P$ is a point on the base of an isosceles triangle. Lines parallel to the sides through $P$ meet the sides at $Q$ and $R$ . Show that the reflection of $P$ in the line $QR$ lies on the circumcircle of the triangle.

Which positive integers cannot be represented as a sum of (two or more) consecutive integers?