Difference between revisions of "2021 AIME I Problems/Problem 10"
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Revision as of 16:12, 11 March 2021
Problem
Consider the sequence of positive rational numbers defined by
and for
, if
for relatively prime positive integers
and
, then
Determine the sum of all positive integers
such that the rational number
can be written in the form
for some positive integer
.
Solution
We know that when
so
is a possible value of
. Note also that
for
. Then
unless
and
are not relatively prime which happens when
divides
or
divides
, so the least value of
is
and
. We know
. Now
unless
and
are not relatively prime which happens the first time
divides
or
divides
or
, and
. We have
. Now
unless
and
are not relatively prime. This happens the first time
divides
implying
divides
, which is prime so
and
. We have
. We have
, which is always reduced by EA, so the sum of all
is
.
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 |
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