CN-SMO 2007

Determine the number of real number $a$ , such that for every $a$ , equation $x^3=ax+a+1$ has a root $x_0$ satisfying following conditions: (a) $x_0$ is an even integer; (b) $|x_0|<1000$ .

$AB$ is the diameter of semicircle $O$ . $C$ , $D$ are two arbitrary points on semicircle $O$ . Point $P$ lies on line $CD$ such that line $PB$ is tangent to semicircle $O$ at $B$ . Line $PO$ intersects line $CA$ , $AD$ at point $E$ , $F$ respectively. Prove that $OE$ = $OF$ .

Let $a_i=min\{ k+\dfrac{i}{k}|k \in N^*\}$ , determine the value of $S_{n^2}=[a_1]+[a_2]+\cdots +[a_{n^2}]$ , where $n\ge 2$ . ( $[x]$ denotes the greatest integer not exceeding x)

Let $f(x)$ be a function satisfying $f(x+1)-f(x)=2x+1 (x \in \mathbb{R})$ .In addition, $|f(x)|\le 1$ holds for $x\in [0,1]$ . Prove that $|f(x)|\le 2+x^2$ holds for $x \in \mathbb{R}$ .

In right-angle triangle $ABC$ , $\angle C=90$ °, Point $D$ is the midpoint of side $AB$ . Points $M$ and $C$ lie on the same side of $AB$ such that $MB\bot AB$ , line $MD$ intersects side $AC$ at $N$ , line $MC$ intersects side $AB$ at $E$ . Show that $\angle DBN=\angle BCE$ .

Find all triples $(a,b,c)$ satisfying the following conditions: (i) $a,b,c$ are prime numbers, where $a<b<c<100$ . (ii) $a+1,b+1,c+1$ form a geometric sequence.