TR-MO 1994

For $n\in\mathbb{N}$ , let $a_{n}$ denote the closest integer to $\sqrt{n}$ . Evaluate $$\sum_{n=1}^\infty{\frac{1}{a_{n}^{3}}}.$$

Let $ABCD$ be a cyclic quadrilateral $\angle{BAD}< 90^\circ$ and $\angle BCA = \angle DCA$ . Point $E$ is taken on segment $DA$ such that $BD=2DE$ . The line through $E$ parallel to $CD$ intersects the diagonal $AC$ at $F$ . Prove that $$\frac{AC\cdot BD}{AB\cdot FC}=2.$$

Let $n$ blue lines, no two of which are parallel and no three concurrent, be drawn on a plane. An intersection of two blue lines is called a blue point. Through any two blue points that have not already been joined by a blue line, a red line is drawn. An intersection of two red lines is called a red point, and an intersection of red line and a blue line is called a purple point. What is the maximum possible number of purple points?

Let $f: \mathbb{R}^{+}\rightarrow \mathbb{R}+$ be an increasing function. For each $u\in\mathbb{R}^{+}$ , we denote $g(u)=\inf\{ f(t)+u/t \mid t>0\}$ . Prove that: $(a)$ If $x\leq g(xy)$ , then $x\leq 2f(2y)$ ; $(b)$ If $x\leq f(y)$ , then $x\leq 2g(xy)$ .

Find the set of all ordered pairs $(s,t)$ of positive integers such that $$t^{2}+1=s(s+1).$$

The incircle of triangle $ABC$ touches $BC$ at $D$ and $AC$ at $E$ . Let $K$ be the point on $CB$ with $CK=BD$ , and $L$ be the point on $CA$ with $AE=CL$ . Lines $AK$ and $BL$ meet at $P$ . If $Q$ is the midpoint of $BC$ , $I$ the incenter, and $G$ the centroid of $\triangle ABC$ , show that: $(a)$ $IQ$ and $AK$ are parallel, $(b)$ the triangles $AIG$ and $QPG$ have equal area.