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Arithmetic progression [Iran TST 2010]

Source:

June 9, 2010
functionlogarithmsnumber theoryprime numbersarithmetic sequencenumber theory proposed

Problem Statement

Let f:N→Nf:\mathbb N\rightarrow\mathbb N be a non-decreasing function and let nn be an arbitrary natural number. Suppose that there are prime numbers p1,p2,…,pnp_1,p_2,\dots,p_n and natural numbers s1,s2,…,sns_1,s_2,\dots,s_n such that for each 1≤i≤n1\leq i\leq n the set {f(pir+si)∣r=1,2,… }\{f(p_ir+s_i)|r=1,2,\dots\} is an infinite arithmetic progression. Prove that there is a natural number aa such that f(a+1),f(a+2),…,f(a+n)f(a+1), f(a+2), \dots, f(a+n) form an arithmetic progression.
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