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2015-2016 Spring OMO #26

Source:

March 29, 2016
Online Math Open

Problem Statement

Let SS be the set of all pairs (a,b)(a, b) of integers satisfying 0≤a,b≤2014.0 \le a, b \le 2014. For any pairs s1=(a1,b1),s2=(a2,b2)∈Ss_1 = (a_1, b_1), s_2 = (a_2, b_2) \in S, define s1+s2=((a1+a2)2015,(b1+b2)2015) and s1×s2=((a1a2+2b1b2)2015,(a1b2+a2b1)2015),s_1 + s_2 = ((a_1 + a_2)_{2015}, (b_1 + b_2)_{2015}) \\ \text { and } \\ s_1 \times s_2 = ((a_1a_2 + 2b_1b_2)_{2015}, (a_1b_2 + a_2b_1)_{2015}), where n2015n_{2015} denotes the remainder when an integer nn is divided by 2015.2015.
Compute the number of functions f:S→Sf : S \rightarrow S satisfying f(s1+s2)=f(s1)+f(s2) and f(s1×s2)=f(s1)×f(s2) f(s_1 + s_2) = f(s_1) + f(s_2) \text{ and } f(s_1 \times s_2) = f(s_1) \times f(s_2) for all s1,s2∈S.s_1, s_2 \in S.
Proposed by Yang Liu
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