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October 6th, 2016, 08:57 AM  #1 
Member Joined: Sep 2013 From: New Delhi Posts: 38 Thanks: 0 Math Focus: Calculus  Notsotough questions for experts, notsosimple for me
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 $\displaystyle a_{n+1}a_n=n(a_n1), n\geq0$ 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 midpoint of the segment PQ. 

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algebra, geometry, number theory 
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