Difference between revisions of "2018 IMO Problems/Problem 2"
(Created page with "Find all numbers <math>n \ge 3</math> for which there exists real numbers <math>a_1, a_2, ..., a_{n+2}</math> satisfying <math>a_{n+1} = a_1, a_{n+2} = a_2</math> and <cmath>...") |
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<cmath>a_{i}a_{i+1} + 1 = a_{i+2}</cmath> | <cmath>a_{i}a_{i+1} + 1 = a_{i+2}</cmath> | ||
for <math>i = 1, 2, ..., n.</math> | for <math>i = 1, 2, ..., n.</math> | ||
| + | |||
| + | ==Solution== | ||
| + | We find at least one series of real numbers for <math>n = 3,</math> for each <math>n = 3k</math> and we prove that if <math>n = 3k ± 1,</math> then the series does not exist. | ||
Revision as of 08:23, 15 August 2022
Find all numbers 
for which there exists real numbers 
satisfying 
and
![\[a_{i}a_{i+1} + 1 = a_{i+2}\]](http://latex.artofproblemsolving.com/d/b/3/db3ac32b874d4123e08da8e59006ab7b4d64e2b7.png)
for 
Solution
We find at least one series of real numbers for 
for each 
and we prove that if 
then the series does not exist.
