INMO 2004

Consider a convex quadrilateral $A B C D$, in which $K, L, M, N$ are the midpoints of the sides $A B$, $B C, C D, D A$ respectively. Suppose

(a) $B D$ bisects $K M$ at $Q$;

(b) $Q A=Q B=Q C=Q D$; and

(c) $L K / L M=C D / C B$.

Prove that $A B C D$ is a square.

Suppose $p$ is a prime greater than 3 . Find all pairs of integers $(a, b)$ satisfying the equation

$$a^{2}+3 a b+2 p(a+b)+p^{2}=0$$

If $\alpha$ is a real root of the equation $x^{5}-x^{3}+x-2=0$, prove that $\left[\alpha^{6}\right]=3$. (For any real number $a$, we denote by $[a]$ the greatest integer not exceeding $a$.)

Let $R$ denote the circumradius of a triangle $A B C ; a, b, c$ its sides $B C, C A, A B$; and $r_{a}, r_{b}, r_{c}$ its exradii opposite $A, B, C$. If $2 R \leq r_{a}$, prove that

(i) $a>b$ and $a>c$;

(ii) $2 R>r_{b}$ and $2 R>r_{c}$.

Let $S$ denote the set of all 6-tuples $(a, b, c, d, e, f)$ of positive integers such that $a^{2}+b^{2}+c^{2}+d^{2}+e^{2}=$ $f^{2}$. Consider the set

$$T=\{a b c d e f:(a, b, c, d, e, f) \in S\}$$

Find the greatest common divisor of all the members of $T$.

Prove that the number of 5 -tuples of positive integers $(a, b, c, d, e)$ satisfying the equation

$$a b c d e=5(b c d e+a c d e+a b d e+a b c e+a b c d)$$

is an odd integer.