2023 AIME II Problems/Problem 13
Problem
Let be an acute angle such that Find the number of positive integers less than or equal to such that is a positive integer whose units digit is
Solution 1
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)
Solution 2 (Simple)
It is clear, that is not integer if
Denote
The condition is satisfied iff or
If then the number of possible n is
For we get
vladimir.shelomovskii@gmail.com, vvsss
Video Solution
~MathProblemSolvingSkills.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.