Hanoi Open Mathematics Competition 2009 Senior

Let $a,b,c$ be three distinct numbers from $\{1,2,3,4,5,6\}$. Show that $7$ divides $abc+(7-a)(7-b)(7-c)$.

Show that there is a natural number $n$ such that the number $a=n!$ ends exactly in $2009$ zeros.

Let $a,b,c$ be positive integers with no common factor and satisfying $\frac1a+\frac1b=\frac1c$. Prove that $a+b$ is a square.

Suppose that $a=2^b$, where $b=2^{10n+1}$. Prove that $a$ is divisible by $23$ for any positive integer $n$.

Prove that $m^7-m$ is divisible by $42$ for any positive integer $m$.

Suppose that real numbers $a,b,c,d$ satisfy $a^2+b^2=4$, $c^2+d^2=4$, and $ac+bd=2$. Find the set of all possible values of $M=ab+cd$.

Let $a,b,c,d$ be positive integers such that $a+b+c+d=99$. Find the smallest and greatest values of $P=abcd$.

Find all pairs of positive integers such that the product of the numbers of any pair plus half of one of the numbers plus one third of the other number is three times less than $1004$.

Given an acute-angled triangle $ABC$ with area $S$, define $A^{\prime}$ as the intersection of the altitude line from $A$ to $BC$ with the outward-facing semicircle on diameter $BC$. Define $B^{\prime},C^{\prime}$ cyclically. Evaluate $T=[BCA^{\prime}]^2+[CAB^{\prime}]^2+[ABC^{\prime}]^2$, where brackets denote triangle area.

Prove that $d^2+(a-b)^2\lt c^2$, where $d$ is the diameter of the inscribed circle of triangle $ABC$.

Let $A=\{1,2,\ldots,100\}$ and let $B$ be a subset of $A$ with $48$ elements. Show that $B$ has two distinct elements $x,y$ whose sum is divisible by $11$.