Difference between revisions of "2014 IMO Problems/Problem 1"
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− | It is | + | It is more convenient to work with differences <math>d_i=a_i-a_{i-1}</math>, <math>i\ge 1</math>. <math>d_i\ge 1</math>. Instead of using <math>a_i=a_0+d_1+\ldots+d_i</math> the inequalities can be rewritten in terms of <math>d_i</math> as |
<cmath> 0 < V_n \le nd_{n+1}</cmath> | <cmath> 0 < V_n \le nd_{n+1}</cmath> | ||
where <math>V_n=a_0-d_2-2d_3-3d_4-\ldots - (n-1)d_n</math>. <math>V_n</math> is strictly monotonically decreasing. <math>V_1=a_0 >0</math>. That is the left inequality is satisfied for <math>n=1</math>. Lets take a look at the time step <math>(n+1)</math> which is right after <math>n</math>: <math>n \rightarrow n+1</math> | where <math>V_n=a_0-d_2-2d_3-3d_4-\ldots - (n-1)d_n</math>. <math>V_n</math> is strictly monotonically decreasing. <math>V_1=a_0 >0</math>. That is the left inequality is satisfied for <math>n=1</math>. Lets take a look at the time step <math>(n+1)</math> which is right after <math>n</math>: <math>n \rightarrow n+1</math> |
Revision as of 10:59, 8 July 2023
Problem
Let be an infinite sequence of positive integers, Prove that there exists a unique integer
such that
Solution
Define . (In particular,
) Notice that because
, we have
Thus,
; i.e.,
is monotonic decreasing. Therefore, because
, there exists a unique
such that
. In other words,
This rearranges to give
Define
. Then because
, we have
Therefore,
is also monotonic decreasing. Note that
from our inequality, and so
for all
. Thus, the given inequality, which requires that
, cannot be satisfied for
, and so
is the unique solution to this inequality.
--Suli 22:38, 7 February 2015 (EST)
Alternative Solution
It is more convenient to work with differences ,
.
. Instead of using
the inequalities can be rewritten in terms of
as
where
.
is strictly monotonically decreasing.
. That is the left inequality is satisfied for
. Lets take a look at the time step
which is right after
:
The condition
for breaking the left inequality at some step
is exactly the condition for satisfying right inequality at step
. Once left inequality is broken at step
it will remain broken for future steps as
is strictly decreasing. The right inequality will be satisfied for some
as
is strictly decreasing integer sequence and the right hand side
of the right inequality is bounded by
from below. That is the left inequality is satisfied initially and as soon as the right inequality is satisfied, which will happen at some point, the left inequality will break at the next step and will remain broken for all future steps. That is
when both inequalities are satisfied exists and unique.
--alexander_skabelin 9:24, 7 July 2023 (EST)
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
See Also
2014 IMO (Problems) • Resources | ||
Preceded by First Problem |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 2 |
All IMO Problems and Solutions |