2009 IMO Problems/Problem 3

Problem

Suppose that $s_1,s_2,s_3,\ldots$ is a strictly increasing sequence of positive integers such that the subsequences

$s_{s_1},s_{s_2},s_{s_3},\ldots$ and $s_{s_1+1},s_{s_2+1},s_{s_3+1},\ldots$

are both arithmetic progressions. Prove that the sequence $s_1,s_2,s_3,\ldots$ is itself an arithmetic progression.

Author: Gabriel Carroll, USA

then

S(s2) - S(s1) = S(s3) - S(s2)
2S(s2) = S(s1) + S(s3)

i.s.w.

2S(s2+1) = S(s1+1) + S(s3+1)
2S(s2) = S(s1) + S(s3)

put S(3) = b, S(2) = a, S(1) = k

--> 2S(k+1) = S(a+1) + S(b+1)
    2S(k) = S(a) + S(b)

- if there is S(x) = a,b,k --> we have 2S(k) = S(a) + S(b) ... (1) , 2S(k+1) = S(a+1) + S(b+1) ...(2)

  but also, by (2), we have 2S(k+2) = S(a+2) + S(b+2) 
                                     ....
                     we get 2S(b+a) = S(z) + S(y)

therefore,

every S(n) is an arithmetic sequence.
Q.E.D.