NL-NWO 1991

Prove that for any three positive real numbers $a,b,c, \frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a} \ge \frac{9}{2} \cdot \frac{1}{a+b+c}$ .

An angle with vertex $A$ and measure $\alpha$ and a point $P_0$ on one of its rays are given so that $AP_0=2$ . Point $P_1$ is chose on the other ray. The sequence of points $P_1,P_2,P_3,...$ is defined so that $P_n$ lies on the segment $AP_{n-2}$ and the triangle $P_n P_{n-1} P_{n-2}$ is isosceles with $P_n P_{n-1}=P_n P_{n-2}$ for all $n \ge 2$ . $(a)$ Prove that for each value of $\alpha$ there is a unique point $P_1$ for which the sequence $P_1,P_2,...,P_n,...$ does not terminate. $(b)$ Suppose that the sequence $P_1,P_2,...$ does not terminate and that the length of the polygonal line $P_0 P_1 P_2 ... P_k$ tends to $5$ when $k \rightarrow \infty$ . Compute the length of $P_0 P_1$ .

Attachments:

A real function $f$ satisfies $4f(f(x))-2f(x)-3x=0$ for all real numbers $x$ . Prove that $f(0)=0$ .

Three real numbers $a,b,c$ satisfy the equations $a+b+c=3, a^2+b^2+c^2=9, a^3+b^3+c^3=24.$ Find $a^4+b^4+c^4$ .

Let $H$ be the orthocenter, $O$ the circumcenter, and $R$ the circumradius of an acute-angled triangle $ABC$ . Consider the circles $k_a,k_b,k_c,k_h,k$ , all with radius $R$ , centered at $A,B,C,H,M,$ respectively. Circles $k_a$ and $k_b$ meet at $M$ and $F$ ; $k_a$ and $k_c$ meet at $M$ and $E$ ; and $k_b$ and $k_c$ meet at $M$ and $D$ . $(a)$ Prove that the points $D,E,F$ lie on the circle $k_h$ . $(b)$ Prove that the set of the points inside $k_h$ that are inside exactly one of the circles $k_a,k_b,k_c$ has the area twice the area of $\triangle ABC$ .