JMO 1996

A plane is partitioned into triangles. Let $\mathcal{T}_0$ denote the set of vertices of triangles in the partition. Let $ABC$ be a triangle with $A,B,C\in\mathcal{T}_0$ and $\theta$ be its smallest angle. Assume that no point of $\mathcal{T}_0$ lies inside the circumcircle of $\triangle ABC$. Prove that there exists a triangle $\sigma$ in the partition such that its intersection with $\triangle ABC$ is nonempty and whose every angle is greater than $\theta$.

Let $m,n$ be positive integers with $(m,n)=1$. Find $(5^m+7^m,5^n+7^n)$.

Let $x>1$ be a real number which is not an integer. For each $n\in\mathbb{N}$, let $a_n=\lfloor x^{n+1}\rfloor - x\lfloor x^n\rfloor$. Prove that the sequence $(a_n)$ is not periodic.

Let $\theta$ be the maximum of the six angles between six edges of a regular tetrahedron in space and a fixed plane. When the tetrahedron is rotated in space, find the maximum of $\theta$.

Let $q$ be a real number with $\frac{1+\sqrt{5}}{2}<q<2$. If a positive integer $n$ is represented in binary system as $2^k+a_{k-1}2^{k-1}+\cdots +2a_1+a_0$, where $a_i\in\{0,1\}$, define $$p_n=q^k+a_{k-1}q^{k-1}+\cdots +qa_1+a_0.$$

Prove that there exist infinitely many positive integers $k$ with the property that there is no $l\in\mathbb{N}$ such that $p_{2k}<p_l< p_{2k+1}$.