TW-TST 1993

A sequence $(a_{n})$ of positive integers is given by $a_{n}=[n+\sqrt{n}+\frac{1}{2}]$ . Find all of positive integers which belong to the sequence.

Let $E$ and $F$ are distinct points on the diagonal $AC$ of a parallelogram $ABCD$ . Prove that , if there exists a cricle through $E,F$ tangent to rays $BA,BC$ then there also exists a cricle through $E,F$ tangent to rays $DA,DC$ .

Find all $x,y,z\in\mathbb{N}_{0}$ such that $7^{x} + 1 = 3^{y} + 5^{z}$ .

Alternative formulation: Solve the equation $1+7^{x}=3^{y}+5^{z}$ in nonnegative integers $x$ , $y$ , $z$ .

In the Cartesian plane, let $C$ be a unit circle with center at origin $O$ . For any point $Q$ in the plane distinct from $O$ , define $Q'$ to be the intersection of the ray $OQ$ and the circle $C$ . Prove that for any $P\in C$ and any $k\in\mathbb{N}$ there exists a lattice point $Q(x,y)$ with $|x|=k$ or $|y|=k$ such that $PQ'<\frac{1}{2k}$ .

Assume $A=\{a_{1},a_{2},...,a_{12}\}$ is a set of positive integers such that for each positive integer $n \leq 2500$ there is a subset $S$ of $A$ whose sum of elements is $n$ . If $a_{1}<a_{2}<...<a_{12}$ , what is the smallest possible value of $a_{1}$ ?

Let $m$ be equal to $1$ or $2$ and $n<10799$ be a positive integer. Determine all such $n$ for which $\sum_{k=1}^{n}\frac{1}{\sin{k}\sin{(k+1)}}=m\frac{\sin{n}}{\sin^{2}{1}}$ .