Hanoi Open Mathematics Competition 2011 Junior

Three lines are drawn in a plane. Which of the following could NOT be the total number of points of intersections?

The last digit of the number \(A=7^{2011}\) is

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

Among the four statements on real numbers below, how many of them are correct?

1. If \(a\lt b\lt 0\), then \(a\lt b^2\).
2. If \(0\lt a\lt b\), then \(a\lt b^2\).
3. If \(a^3\lt b^3\), then \(a\lt b\).
4. If \(a^2\lt b^2\), then \(a\lt b\).
5. If \(|a|\lt |b|\), then \(a\lt b\).

Editorial reading note: the source says “four statements” but actually displays the five statements above. Count all five displayed statements.

Let \(M=7!\cdot8!\cdot9!\cdot10!\cdot11!\cdot12!\). How many factors of \(M\) are perfect squares?

Find all positive integers \((m,n)\) such that \(m^2+n^2+3=4(m+n)\).

Find all pairs \((x,y)\) of real numbers satisfying \(x+y=3\) and \(x^4-y^4=8x-y\).

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

Solve the equation \(1+x+x^2+x^3+\cdots+x^{2011}=0\).

Editorial domain convention: solve over the real numbers; the source does not explicitly specify a domain.

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.

Given a quadrilateral \(ABCD\) with \(AB=BC=3\text{ cm}\), \(CD=4\text{ cm}\), \(DA=8\text{ cm}\), and \(\angle DAB+\angle ABC=180^\circ\), calculate its area.

Editorial convention: \(ABCD\) is the ordinary simple convex quadrilateral with vertices in that order.

Suppose \(a\gt 0,b\gt 0,a+b\le1\). Determine the minimum of \(M=\frac1{ab}+\frac1{a^2+ab}+\frac1{ab+b^2}+\frac1{a^2+b^2}\).