2023 AIME II Problems/Problem 13
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)