EGMO 2015, Problem 4. Determine whether there exists an infinite sequence \(a_1, a_2, a_3, \dots\) of positive integers which satisfies the equality \[a_{n+2}=a_{n+1}+\sqrt{a_{n+1}+a_{n}} \] for every positive integer \(n\).
EGMO 2015, Problem 4. Determine whether there exists an infinite sequence \(a_1, a_2, a_3, \dots\) of positive integers which satisfies the equality \[a_{n+2}=a_{n+1}+\sqrt{a_{n+1}+a_{n}} \] for every positive integer \(n\).
EGMO 2014, Problem 3. We denote the number of positive divisors of a positive integer \(m\) by \(d(m)\) and the number of distinct prime divisors of \(m\) by \(\omega(m)\). Let \(k\) be a positive integer. Prove that there exist infinitely many positive integers \(n\) such that \(\omega(n) = k\) and \(d(n)\) does not divide \(d(a^2+b^2)\) for any positive integers \(a, b\) satisfying \(a + b = n\).
EGMO 2013, problem 4. Find all positive integers \(a\) and \(b\) for which there are three consecutive integers at which the polynomial \[P(n)=\frac{n^5+a}{b}\] takes integer values.
EGMO 2015, Problem 5. Let \(m, n\) be positive integers with \(m > 1\). Anastasia partitions the integers \(1, 2, \dots , 2m\) into \(m\) pairs. Boris then chooses one integer from each pair and finds the sum of these chosen integers. Prove that Anastasia can select the pairs so that Boris cannot make his sum equal to \(n.\)