SMO 2008 Junior Round 2

In $\triangle ABC$, $\angle ACB=90^\circ$. Let $D$ be the foot of the altitude from $C$ to $AB$, and let $E$ be a point on the side $BC$ such that $CE=BD/2$. Prove that $AD+CE=AE$.

Let $a,b,c,d$ be positive real numbers such that $cd=1$. Prove that there is an integer $n$ such that $ab\le n^2\le(a+c)(b+d)$.

In the quadrilateral $PQRS$, $A$, $B$, $C$ and $D$ are the midpoints of the sides $PQ$, $QR$, $RS$ and $SP$ respectively, and $M$ is the midpoint of $CD$. Suppose $H$ is the point on the line $AM$ such that $HC=BC$. Prove that $\angle BHM=90^\circ$.

Six distinct positive integers $a,b,c,d,e,f$ are given. Jack and Jill calculated the sums of each pair of these numbers. Jack claims that he has $10$ prime numbers while Jill claims that she has $9$ prime numbers among the sums. Who has the correct claim?

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