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December 14, 2016

Regional Math Olympiad (India) Number Theory Problems

Here is the post for the Regional Mathematics Olympiad (India) RMO Number Theory Problems. These are problems from previous year papers.

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

RMO Number Theory Problems:

  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.  
  3. 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)
  4.  
  5. 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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