7

Part of 2024 AIME

Problems(2)

Complex Expression

Source: 2024 AIME I #7

Find the largest possible real part of

(75+117i)z+\frac{96+144i}{z}

z

is a complex number with

|z|=4

Digit Replacements

Source: 2024 AIME II #7

N

be the greatest four-digit integer with the property that whenever one of its digits is changed to

1

, the resulting number is divisible by

7

Q

R

be the quotient and remainder, respectively, when

N

1000

Q+R