SMO 2008 Senior Round 2

Let $ABCD$ be a trapezium with $AD\parallel BC$. Suppose $K$ and $L$ are, respectively, points on the sides $AB$ and $CD$ such that $\angle BAL=\angle CDK$. Prove that $\angle BLA=\angle CKD$.

Determine all primes $p$ such that $5^p+4p^4$ is a perfect square, that is, the square of an integer.

Find all functions $f:\mathbb R\to\mathbb R$ such that

(i) $f(2u)=f(u+v)f(v-u)+f(u-v)f(-u-v)$ for all $u,v\in\mathbb R$, and

(ii) $f(u)\ge0$ for all $u\in\mathbb R$.

There are $11$ committees in a club. Each committee has $5$ members and every two committees have a member in common. Show that there is a member who belongs to $4$ committees.

Let $a,b,c\ge0$. Prove that $$\frac{(1+a^2)(1+b^2)(1+c^2)}{(1+a)(1+b)(1+c)}\ge\frac12(1+abc).$$