Question bank HOMC 2008

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

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 \(EP+EF+FQ\).