Regional Math Olympiad (India) Number Theory Problems

(This is a work in progress. More problems will be added soon).

  1. Find all triples (p, q, r) of primes such that pq = r + 1 and 2(p 2 + q 2 ) = r 2 + 1 (RMO 2013, Mumbai Region)
  2. Let a1, b1, c1 be natural numbers. We define a2 = gcd(b1, c1), b2 = gcd(c1, a1), c2 = gcd(a1, b1), and a3 = lcm(b2, c2), b3 = lcm(c2, a2), c3 = lcm(a2, b2). Show that gcd(b3, c3) = a2. (RMO 2013, Mumbai Region)
  3. A natural number n is chosen strictly between two consecutive perfect squares. The smaller of these two squares is obtained by subtracting k from n and the larger one is obtained by adding l to n. Prove that n − kl is a perfect square. (RMO 2011)

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