Olympiad problems

2022 Tuymaada Olympiad, 8

In an acute triangle $\triangle ABC$ the points $C_m, A_m, B_m$ are the midpoints of $AB, BC, CA$ respectively. Inside the triangle $\triangle ABC$ a point $P$ is chosen so that $\angle PCB = \angle B_mBC$ and $\angle PAB = \angle ABB_m.$ A line passing through $P$ and perpendicular to $AC$ meets the median $BB_m$ at $E.$ Prove that $E$ lies on the circumcircle of the triangle $\triangle A_mB_mC_m.$ [i](K. Ivanov )[/i]

1956 Czech and Slovak Olympiad III A, 1

Tags: algebra , trigonometric equation , system of equations , cosine , sine

Find all $x,y\in\left(0,\frac{\pi}{2}\right)$ such that \begin{align*} \frac{\cos x}{\cos y}&=2\cos^2 y, \\ \frac{\sin x}{\sin y}&=2\sin^2 y. \end{align*}

2004 Bosnia and Herzegovina Team Selection Test, 5

Tags: inequality , cosine , sine , algebra

For $0 \leq x < \frac{\pi}{2} $ prove the inequality: $a^2\tan(x)\cdot(\cos(x))^{\frac{1}{3}}+b^2\sin{x}\geq 2xab$ where $a$ and $b$ are real numbers.