Difference between revisions of "2017 IMO Problems/Problem 1"
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==Solution== | ==Solution== | ||
| + | First we notice the following: | ||
| + | |||
| + | When we start with <math>a_0=3</math>, we get <math>a_1=6</math>, <math>a_2=9</math>, <math>a_3=3</math> and the pattern repeats. | ||
| + | |||
| + | When we start with <math>a_0=6</math>, we get <math>a_1=9</math>, <math>a_2=3</math>, <math>a_3=6</math> and the pattern repeats. | ||
| + | |||
| + | When we start with <math>a_0=9</math>, we get <math>a_1=3</math>, <math>a_2=6</math>, <math>a_3=9</math> and the pattern repeats. | ||
| + | |||
| + | So, | ||
| + | |||
| + | |||
| + | |||
| + | ~Tomas Diaz. orders@tomasdiaz.com | ||
{{alternate solutions}} | {{alternate solutions}} | ||
Revision as of 02:23, 19 November 2023
Problem
For each integer 
, define the sequence 
for 
as
![\[a_{n+1} = \begin{cases} \sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer,} \\ a_n + 3 & \text{otherwise.} \end{cases}\]](http://latex.artofproblemsolving.com/7/e/5/7e5005fa83196dcdb7e5270b0dc20e1f13e22bd4.png)
Determine all values of 
such that there exists a number 
such that 
for infinitely many values of 
.
Solution
First we notice the following:
When we start with 
, we get 
, 
, 
and the pattern repeats.
When we start with 
, we get 
, 
, 
and the pattern repeats.
When we start with 
, we get 
, 
, 
and the pattern repeats.
So,
~Tomas Diaz. orders@tomasdiaz.com
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
See Also
| 2017 IMO (Problems) • Resources | ||
| Preceded by First Problem |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 2 |
| All IMO Problems and Solutions | ||
