MathDB

4

Part of 2001 Mexico National Olympiad

Problems(1)

f(n,m)=a_1-a_2+a_3 -a_4+ ...+a _{2001} & a_{k+1} is residue of a_k^2 mod n

Source: Mexican Mathematical Olympiad 2001 OMM P4

7/30/2018
For positive integers n,mn, m define f(n,m)f(n,m) as follows. Write a list of 2001 2001 numbers aia_i, where a1=ma_1 = m, and ak+1a_{k+1} is the residue of ak2a_k^2 mod nmod \, n (for k=1,2,...,2000k = 1, 2,..., 2000). Then put f(n,m)=a1−a2+a3−a4+a5−...+a2001f(n,m) = a_1-a_2 + a_3 -a_4 + a_5- ... + a_{2001}. For which n≥5n \ge 5 can we find m such that 2≤m≤n/22 \le m \le n/2 and f(m,n)>0f(m,n) > 0?
number theoryNumber theoretic functionsresidue