Not-so-tough questions for experts, not-so-simple for me

Sep 2013
New Delhi
Problem 1.
Let p, q, r be integers such that
\(\displaystyle \frac {pq}{r} + \frac {qr}{p} +\frac {rp}{q}\)
is an integer.
Prove that each of the numbers
\(\displaystyle \frac {pq}{r}, \frac {qr}{p}, \frac {rp}{q}\)
is an integer.

Problem 2 .
Two circles \(\displaystyle \Gamma_1\) and \(\displaystyle \Gamma_2\) of radii \(\displaystyle r_1\) and \(\displaystyle r_2\) respectively touch each other externally. The points A, B on \(\displaystyle \Gamma_1\) and C, D on \(\displaystyle \Gamma_2\) are such that AD and BC are common external tangents to the two circles . Prove that a circle can be
inscribed in the quadrilateral ABCD and determine the radius of this circle.

Problem 3.
Let \(\displaystyle a, b, c \geq -1\) be real numbers with \(\displaystyle a^3+b^3+c^3=1\). Prove that \(\displaystyle a+b+c+a^2+b^2+c^2 \leq 4\). When does the equality hold ?

Problem 4.
The sequence \(\displaystyle a_n\) is defined by
\(\displaystyle a_0=3\) and \(\displaystyle a_{n+1}-a_n=n(a_n-1), n\geq0\)
Find all positive integers m for which \(\displaystyle \gcd(m,a_n)=1\) for all \(\displaystyle n\geq0\).

Problem 5 .
Let us consider arrangements of the numbers 1 through 64 on the squares
of an 8 × 8 chessboard, each square containing exactly one number and
each number appearing exactly once.
A number in such an arrangement is called interesting if it is both the
largest number in its row and the smallest number in its column.
Prove or disprove each of the following statements :
(a) Each such arrangement contains at least one interesting number.
(b) Each such arrangement contains at most one interesting number.

Problem 6.
Let ABC be a triangle. Its incircle meets the sides BC, CA and AB in the points D, E and F, respectively. Let P denote the intersection point of ED and the line perpendicular to EF and passing through F, and similarly let Q denote the intersection point of EF and the line perpendicular to ED and passing through D.
Prove that B is the mid-point of the segment PQ.