Olympiad problems

1978 Romania Team Selection Test, 4

Solve the equation $ \sin x\sin 2x\cdots\sin nx+\cos x\cos 2x\cdots\cos nx =1, $ for $ n\in\mathbb{N} $ and $ x\in\mathbb{R} . $

2009 Moldova National Olympiad, 12.3

Tags: trigonometry , trigonometric equation , equation , algebra

Find all pairs $(a,b)$ of real numbers, so that $\sin(2009x)+\sin(ax)+\sin(bx)=0$ holds for any $x\in \mathbf {R}$.

1967 IMO Longlists, 42

Tags: trigonometry , factorization , trigonometric equation , algebra

Decompose the expression into real factors: \[E = 1 - \sin^5(x) - \cos^5(x).\]

1967 IMO Shortlist, 4

Tags: trigonometry , geometry , algebra , trigonometric equation

[b](i)[/b] Solve the equation: \[ \sin^3(x) + \sin^3\left( \frac{2 \pi}{3} + x\right) + \sin^3\left( \frac{4 \pi}{3} + x\right) + \frac{3}{4} \cos {2x} = 0.\] [b](ii)[/b] Supposing the solutions are in the form of arcs $AB$ with one end at the point $A$, the beginning of the arcs of the trigonometric circle, and $P$ a regular polygon inscribed in the circle with one vertex in $A$, find: 1) The subsets of arcs having the other end in $B$ in one of the vertices of the regular dodecagon. 2) Prove that no solution can have the end $B$ in one of the vertices of polygon $P$ whose number of sides is prime or having factors other than 2 or 3.

1967 IMO Longlists, 46

Tags: algebra , trigonometric equation , trigonometry , system of equations

If $x,y,z$ are real numbers satisfying relations \[x+y+z = 1 \quad \textrm{and} \quad \arctan x + \arctan y + \arctan z = \frac{\pi}{4},\] prove that $x^{2n+1} + y^{2n+1} + z^{2n+1} = 1$ holds for all positive integers $n$.