Hanoi Open Mathematics Competition 2007 Senior

What is the last two digits of the number $(11^2+15^2+19^2+\cdots+2007^2)^2$?

Which is largest positive integer $n$ satisfying $n^{2007}\gt (2007)^n$?

Find the number of different positive integer triples $(x,y,z)$ satisfying $x+y-z=1$ and $x^2+y^2-z^2=1$.

List the numbers $\sqrt2$, $\sqrt[3]3$, $\sqrt[4]4$, $\sqrt[5]5$ and $\sqrt[6]6$ in order from greatest to least.

Suppose that $A,B,C,D$ are points on a circle, $AB$ is the diameter, $CD$ is perpendicular to $AB$ and meets $AB$ at $E$, $AB$ and $CD$ are integers and $AE-EB=\sqrt3$. Find $AE$.

Let $P(x)=x^3+ax^2+bx+1$ and $|P(x)|\le1$ for all $x$ such that $|x|\le1$. Prove that $|a|+|b|\le5$.

Find all sequences of integers $x_1,x_2,\ldots,x_n,\ldots$ such that $ij$ divides $x_i+x_j$ for any two distinct positive integers $i$ and $j$.

Let $ABC$ be an equilateral triangle. For a point $M$ inside $\triangle ABC$, let $D,E,F$ be the feet of the perpendiculars from $M$ onto $BC,CA,AB$, respectively. Find the locus of all such points $M$ for which $\angle FDE$ is a right angle.

Let $a_1,a_2,\ldots,a_{2007}$ be real numbers such that $a_1+a_2+\cdots+a_{2007}\ge2007^2$ and $a_1^2+a_2^2+\cdots+a_{2007}^2\le2007^3-1$. Prove that $a_k\in[2006;2008]$ for all $k\in\{1,2,\ldots,2007\}$.

What is the smallest possible value of $x^2+2y^2-x-2y-xy$?

Find all polynomials $P(x)$ satisfying $(2x-1)P(x)=(x-1)P(2x)$ for all $x$.

Calculate the sum $\frac1{2\cdot7\cdot12}+\frac1{7\cdot12\cdot17}+\cdots+\frac1{1997\cdot2002\cdot2007}$.

Let $ABC$ be an acute-angle triangle with $BC\gt CA$. Let $O,H,F$ be the circumcenter, orthocentre and the foot of its altitude $CH$, respectively. Suppose that the perpendicular to $OF$ at $F$ meets the side $CA$ at $P$. Prove $\angle FHP=\angle BAC$.

How many ordered pairs of integers $(x,y)$ satisfy $x^2+y^2+xy=4(x+y)$?

Let $p=\overline{abcd}$ be the 4-digit prime number. Prove that the equation $ax^3+bx^2+cx+d=0$ has no rational roots.