MathDB

4

Part of 2022 AIME Problems

Problems(2)

Complex Number Combo

Source: 2022 AIME I #4

2/9/2022
Let w=3+i2w = \frac{\sqrt{3}+i}{2} and z=−1+i32z=\frac{-1+i\sqrt{3}}{2}, where i=−1i=\sqrt{-1}. Find the number of ordered pairs (r,s)(r, s) of positive integers not exceeding 100100 that satisfy the equation i⋅wr=zsi\cdot w^r=z^s.
AMCAIMEAIME Icomplex numbers
L O G A R I T H M S

Source:

2/17/2022
There is a positive real number xx not equal to either 120\tfrac{1}{20} or 12\tfrac{1}{2} such that log⁡20x(22x)=log⁡2x(202x).\log_{20x} (22x)=\log_{2x} (202x). The value log⁡20x(22x)\log_{20x} (22x) can be written as log⁡10(mn)\log_{10} (\tfrac{m}{n}), where mm and nn are relatively prime positive integers. Find m+nm+n.
AIME II