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Source: IrMO 1988 paper 2 Q2

November 14, 2015
algebra

Problem Statement

2. Let x1,...,xnx_1, . . . , x_n be nn integers, and let pp be a positive integer, with p<np < n. Put S1=x1+x2+...+xpS_1 = x_1 + x_2 + . . . + x_p T1=xp+1+xp+2+...+xnT_1 = x_{p+1} + x_{p+2} + . . . + x_n S2=x2+x3+...+xp+1S_2 = x_2 + x_3 + . . . + x_{p+1} T2=xp+2+xp+3+...+xn+x1T_2 = x_{p+2} + x_{p+3} + . . . + x_n + x_1 ...... Sn=xn+x1+...+xp−1S_n=x_n+x_1+...+x_{p-1} Tn=xp+xp+1+...+xn−1T_n=x_p+x_{p+1}+...+x_{n-1} For a=0,1,2,3a = 0, 1, 2, 3, and b=0,1,2,3b = 0, 1, 2, 3, let m(a,b)m(a, b) be the number of numbers ii, 1≤i≤n1 \leq i \leq n, such that SiS_i leaves remainder aa on division by 44 and TiT_i leaves remainder bb on division by 44. Show that m(1,3)m(1, 3) and m(3,1)m(3, 1) leave the same remainder when divided by 44 if, and only if, m(2,2)m(2, 2) is even.
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