Difference between revisions of "2023 AIME II Problems/Problem 13"
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Revision as of 20:11, 16 February 2023
Solution
Denote .
For any
, we have
Next, we compute the first several terms of .
By solving equation , we get
.
Thus,
,
,
,
,
.
In the rest of analysis, we set .
Thus,
Thus, to get an integer, we have
.
In the rest of analysis, we only consider such
. Denote
and
.
Thus,
with initial conditions
,
.
To get the units digit of to be 9, we have
Modulo 2, for , we have
Because , we always have
for all
.
Modulo 5, for , we have
We have ,
,
,
,
,
,
.
Therefore, the congruent values modulo 5 is cyclic with period 3.
To get
, we have
.
From the above analysis with modulus 2 and modulus 5, we require .
For , because
, we only need to count feasible
with
.
The number of feasible
is
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
See also
2023 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.