15

Part of 2023 AIME

Problems(2)

Prime Numbers, Triangle Inequality, Complex… Oh My!

Source: 2023 AIME I #15

Find the largest prime number

p<1000

for which there exists a complex number

z

satisfying [*] the real and imaginary part of

z

are both integers; [*]

|z|=\sqrt{p}

, and [*] there exists a triangle whose three side lengths are

p

, the real part of

z^{3}

, and the imaginary part of

z^{3}

.

multiples of 23 mod 2^n

Source: 2023 AIME II/15

For each positive integer

n

a_n

be the least positive integer multiple of

23

a_n\equiv1\pmod{2^n}

. Find the number of positive integers

n

less than or equal to

1000

a_n=a_{n+1}

AMCAIMEmodular arithmetic