Hanoi Open Mathematics Competition 2011 Senior

An integer is called “octal” if it is divisible by 8 or if at least one of its digits is 8. How many integers between 1 and 100 are octal?

What is the smallest number?

What is the largest integer less than \(\sqrt[3]{2011^3+3\cdot2011^2+4\cdot2011+5}\)?

Prove that \(1+x+x^2+x^3+\cdots+x^{2011}\ge0\) for every \(x\ge-1\).

Let \(a,b,c\) be positive integers such that \(a+2b+3c=100\). Find the greatest value of \(M=abc\).

Find all pairs \((x,y)\) of real numbers satisfying \(x+y=2\) and \(x^4-y^4=5x-3y\).

How many positive integers \(a\lt 100\) satisfy that \(4a^2+3a+5\) is divisible by 6?

Find the minimum value of \(S=|x+1|+|x+5|+|x+14|+|x+97|+|x+1920|\).

For every pair of positive integers \((x,y)\), define \(f(x,y)\) by
\[f(x,1)=x,\qquad f(x,y)=0\quad\text{if }y\gt x,\]
\[f(x+1,y)=y[f(x,y)+f(x,y-1)].\]
Evaluate \(f(5,5)\).

Editorial boundary convention: use the recurrence for \(y\ge2\); the first formula gives the \(y=1\) boundary values. The source does not state that recurrence restriction explicitly.

Two angle bisectors \(BD\) and \(CE\) of triangle \(ABC\) intersect at \(O\). Suppose \(BD\cdot CE=2BO\cdot OC\). Let \(H\) be the point on \(BC\) such that \(OH\perp BC\). Prove \(AB\cdot AC=2HB\cdot HC\).

Consider a right-angled triangle \(ABC\) with \(\angle A=90^\circ\), \(AB=c\) and \(AC=b\). Let \(P\in AC\) and \(Q\in AB\) satisfy \(\angle APQ=\angle ABC\) and \(\angle AQP=\angle ACB\). Calculate \(PQ+PE+QF\), where \(E,F\) are the perpendicular projections of \(P,Q\) onto \(BC\), respectively.

Suppose \(|ax^2+bx+c|\ge|x^2-1|\) for all real numbers \(x\). Prove that \(|b^2-4ac|\ge4\).