Olympiad problems

1952 Putnam, B7

Given any real number $N_0,$ if $N_{j+1}= \cos N_j ,$ prove that $\lim_{j\to \infty} N_j$ exists and is independent of $N_0.$

2017 Bosnia And Herzegovina - Regional Olympiad, 2

Let $ABC$ be an isosceles triangle such that $AB=AC$. Find angles of triangle $ABC$ if $\frac{AB}{BC}=1+2\cos{\frac{2\pi}{7}}$

geometry isosceles angle cosine

2004 Bosnia and Herzegovina Team Selection Test, 5

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.

sine inequality cosine algebra AM-GM inequality inequalities

1957 Putnam, A3

Let $a,b$ be real numbers and $k$ a positive integer. Show that $$ \left| \frac{ \cos kb \cos a - \cos ka \cos b}{\cos b -\cos a} \right|<k^2 -1$$ whenever the left side is defined.

trigonometry inequalities cosine algebra number theory

1974 Putnam, B3

Prove that if $a$ is a real number such that $$\cos \pi a= \frac{1}{3},$$ then $a$ is irrational.

cosine irrational algebra

1956 Czech and Slovak Olympiad III A, 1

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*}

system of equations sine trigonometric equation cosine algebra systems of equations trigonometric equations