MathDB

28

Part of 2015 Online Math Open Problems

Problems(2)

2014-2015 Spring OMO #28

Source:

4/14/2015
Find the number of ordered pairs (P(x),Q(x))(P(x),Q(x)) of polynomials with integer coefficients such that P(x)2+Q(x)2=(x4096−1)2. P(x)^2+Q(x)^2=\left(x^{4096}-1\right)^2.
Proposed by Michael Kural
Online Math Open
2015-2016 Fall OMO #28

Source:

11/18/2015
Let NN be the number of 20152015-tuples of (not necessarily distinct) subsets (S1,S2,…,S2015)(S_1, S_2, \dots, S_{2015}) of {1,2,…,2015}\{1, 2, \dots, 2015 \} such that the number of permutations σ\sigma of {1,2,…,2015}\{1, 2, \dots, 2015 \} satisfying σ(i)∈Si\sigma(i) \in S_i for all 1≤i≤20151 \le i \le 2015 is odd. Let k2,k3k_2, k_3 be the largest integers such that 2k2∣N2^{k_2} | N and 3k3∣N3^{k_3} | N respectively. Find k2+k3.k_2 + k_3.
Proposed by Yang Liu
Online Math Open