Hanoi Open Mathematics Competition 2008 Senior

How many integers are there in \((b,2008b]\), where \(b>0\) is given?

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

Show that \(x^2+8z=3+2y^2\) has no solutions in positive integers \(x,y,z\).

Prove that there are infinitely many relatively prime pairs \((m,n)\) of positive integers such that \(x^3-nx+mn=0\) has three distinct integer roots.

Find all polynomials \(P(x)\) of degree 1 such that \(\max_{a\le x\le b}P(x)-\min_{a\le x\le b}P(x)=b-a\) for every real \(a<b\).

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 \(a^2+b^2+c^2\)?

Find all triples \((a,b,c)\) of consecutive odd positive integers with \(a<b<c\) such that \(a^2+b^2+c^2\) is a four-digit number whose digits are all equal.

Consider a convex quadrilateral \(ABCD\). Let \(O=AC\cap BD\). Let \(M,N\) be the centroids of \(\triangle AOB,\triangle COD\), and \(P,Q\) the orthocenters of \(\triangle BOC,\triangle DOA\), respectively. Prove that \(MN\perp PQ\).

Consider a triangle \(ABC\). For every \(M\in BC\), define \(N\in CA\), \(P\in AB\) such that \(APMN\) is a parallelogram. Let \(O=BN\cap CP\). Find \(M\in BC\) such that \(\angle PMO=\angle OMN\).

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 \(EP+EF+FQ\).