Math Gold Medalist

Lor

2023 AIME II

Problem 13

Let $A$ be an acute angle such that $\tan A = 2 \cos A.$ Find the number of positive integers $n$ less than or equal to $1000$ such that $\sec^n A + \tan^n A$ is a positive integer whose units digit is $9.$

Trigonometry Formula

Recursive Sequence

Small Example

an=(secA)^n + (tanA)^n

Find recursive sequence of an

Calculate x and y

Caculate:

a0

a1

a2

a3

a4

a5

a6

a7

a8

a4q

a0

a4

a8

a12

a16

a20

a24

…

a1000

.

   Solution