NL-NWO 1986

$f(x) = \frac{12x+9}{19x+86}, \, x \ne -\frac{86}{19}$ Prove that $\exists ! \,\, {x_o \in R} \,\, \forall h_1,h_2 \in R [f(x_0+h_1)f(x_0-h_1)=f(x_0+h_2)f(x_0-h_2)]$ , and calculate $x_0$ .

Prove that for all positive integers $n$ holds that $$\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+...+\frac{1}{(2n-1) \cdot 2n}=\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{2n}$$

The following apply: $a,b,c,d \ge 0$ and $abcd=1$ Prove that $$a^2+b^2+c^2+d^2+ab+ac+ad+bc+bd+cd \ge 10$$

The lines $a$ and $b$ are parallel and the point $A$ lies on $a$ . One chooses one circle $\gamma$ through A tangent to $b$ at $B$ . $a$ intersects $\gamma$ for the second time at $T$ . The tangent line at $T$ of $\gamma$ is called $t$ . Prove that independently of the choice of $\gamma$ , there is a fixed point $P$ such that $BT$ passes through $P$ . Prove that independently of the choice of $\gamma$ , there is a fixed circle $\delta$ such that $t$ is tangent to $\delta$ .