Hanoi Open Mathematics Competition 2008 Junior

How many integers from 1 to 2008 have the sum of their digits divisible by 5?

How many integers belong to \((a,2008a)\), where \(a>0\) is given?

Find the coefficient of \(x\) in \((1+x)(1-2x)(1+3x)(1-4x)\cdots(1-2008x)\).

Find all pairs \((m,n)\) of positive integers such that \(m^2+n^2=3(m+n)\).

Suppose \(x,y,z,t\) are real numbers such that
\[|x+y+z-t|\le1,\quad |y+z+t-x|\le1,\quad |z+t+x-y|\le1,\quad |t+x+y-z|\le1.\]
Prove that \(x^2+y^2+z^2+t^2\le1\).

Let \(P(x)\) be a polynomial such that \(P(x^2-1)=x^4-3x^2+3\). Find \(P(x^2+1)\).

The figure \(ABCDE\) is a convex pentagon. Find \(\angle DAC+\angle EBD+\angle ACE+\angle BDA+\angle CEB\).

The sides of a rhombus have length \(a\) and the area is \(S\). What is the length of the shorter diagonal?

Let \(ABC\) be a right-angled triangle with \(\angle A=90^\circ\), \(AB=c\), \(AC=b\). Let \(E\in AC\), \(F\in AB\) satisfy \(\angle AEF=\angle ABC\), \(\angle AFE=\angle ACB\). Let \(P,Q\in BC\) satisfy \(EP\perp BC\), \(FQ\perp BC\). Determine \(EP+EF+PQ\).

Let \(a,b,c\in[1,3]\) satisfy \(\max\{a,b,c\}\ge2\) and \(a+b+c=5\). What is the smallest possible value of \(a^2+b^2+c^2\)?