Difference between revisions of "2009 IMO Problems/Problem 3"

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     2S(k) = S(a) + S(b)
 
     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)
+
**if there is S(x) = a,b,k (which is an arithmetic sequence) --> 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)  
 
   but also, by (2), we have 2S(k+2) = S(a+2) + S(b+2)  
 
                                       ....
 
                                       ....
 
                       we get 2S(b+a) = S(z) + S(y)
 
                       we get 2S(b+a) = S(z) + S(y)
 
therefore,  
 
therefore,  
every S(n) is an arithmetic sequence.
+
every S(n) is an arithmetic sequence.
Q.E.D.
+
Q.E.D.
 +
 
 +
==See Also==
 +
 
 +
{{IMO box|year=2009|num-b=2|num-a=4}}

Latest revision as of 00:16, 19 November 2023

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

Solution

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 (which is an arithmetic sequence) --> 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.

See Also

2009 IMO (Problems) • Resources
Preceded by
Problem 2
1 2 3 4 5 6 Followed by
Problem 4
All IMO Problems and Solutions