Difference between revisions of "2018 IMO Problems/Problem 5"
(Created page with "Let <math>a_1, a_2, \dots</math> be an infinite sequence of positive integers. Suppose that there is an integer<math> N > 1</math> such that, for each <math>n \geq N</math>, t...") |
|||
| Line 1: | Line 1: | ||
| + | ==Problem== | ||
| + | |||
Let <math>a_1, a_2, \dots</math> be an infinite sequence of positive integers. Suppose that there is an integer<math> N > 1</math> such that, for each <math>n \geq N</math>, the number <math>\frac{a_1}{a_2}+\frac{a_2}{a_3}+\dots +\frac{a_{n-1}}{a_n}+\frac{a_n}{a_1}</math> is an integer. Prove that there is a positive integer <math>M</math> such that <math>a_m = a_{m+1}</math> for all <math>m \geq M.</math> | Let <math>a_1, a_2, \dots</math> be an infinite sequence of positive integers. Suppose that there is an integer<math> N > 1</math> such that, for each <math>n \geq N</math>, the number <math>\frac{a_1}{a_2}+\frac{a_2}{a_3}+\dots +\frac{a_{n-1}}{a_n}+\frac{a_n}{a_1}</math> is an integer. Prove that there is a positive integer <math>M</math> such that <math>a_m = a_{m+1}</math> for all <math>m \geq M.</math> | ||
| + | |||
| + | ==Solution== | ||
| + | {{solution}} | ||
| + | |||
| + | ==See Also== | ||
| + | |||
| + | {{IMO box|year=2018|num-b=4|num-a=6}} | ||
Latest revision as of 00:46, 19 November 2023
Problem
Let 
be an infinite sequence of positive integers. Suppose that there is an integer
such that, for each 
, the number 
is an integer. Prove that there is a positive integer 
such that 
for all 
Solution
This problem needs a solution. If you have a solution for it, please help us out by adding it.
See Also
| 2018 IMO (Problems) • Resources | ||
| Preceded by Problem 4 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 6 |
| All IMO Problems and Solutions | ||
