Difference between revisions of "2021 AIME I Problems/Problem 10"
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==Problem== | ==Problem== | ||
| − | + | Consider the sequence <math>(a_k)_{k\ge 1}</math> of positive rational numbers defined by <math>a_1 = \frac{2020}{2021}</math> and for <math>k\ge 1</math>, if <math>a_k = \frac{m}{n}</math> for relatively prime positive integers <math>m</math> and <math>n</math>, then | |
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| + | <cmath>a_{k+1} = \frac{m + 18}{n+19}.</cmath>Determine the sum of all positive integers <math>j</math> such that the rational number <math>a_j</math> can be written in the form <math>\frac{t}{t+1}</math> for some positive integer <math>t</math>. | ||
==Solution== | ==Solution== | ||
Revision as of 15:49, 11 March 2021
Problem
Consider the sequence 
of positive rational numbers defined by 
and for 
, if 
for relatively prime positive integers 
and 
, then
![\[a_{k+1} = \frac{m + 18}{n+19}.\]](http://latex.artofproblemsolving.com/f/8/3/f8375cf3493310402a251be2df8e930251bd5a5d.png)
Determine the sum of all positive integers 
such that the rational number 
can be written in the form 
for some positive integer 
.
Solution
See also
| 2021 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 9 |
Followed by Problem 11 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
