11

Part of 2022 MMATHS

Problems(2)

2022 MMATHS Team Online p11 - 3 emojis: barbecue, bacon, or burger

Source:

Every time Josh and Ron tap their screens, one of three emojis appears, each with equal probability: barbecue, bacon, or burger. Josh taps his screen until he gets a sequence of barbecue, bacon, and burger consecutively (in that specific order.) Ron taps his screen until he gets a sequence of three bacons in a row. Let

M

N

be the expected number of times Josh and Ron tap their screens, respectively. What is

|M-N|

probabilitycombinatoricsMMATHS

2022 MMATHS Individual p11 - Im(1/w)=Im(k/w^2)=Im(k/w^3) complex

Source:

Re(z)

Im(z)

the real part and imaginary part, respectively, of a complex number

z

z = a + bi

Re(z) = a

Im(z) = b

. Suppose that there exists some real number

k

Im \left( \frac{1}{w} \right) = Im \left( \frac{k}{w^2} \right) = Im \left( \frac{k}{w^3} \right)

for some complex number

w

||w||=\frac{\sqrt3}{2}

Re(w) > 0

Im(w) \ne 0

k

can be expressed as

\frac{\sqrt{a}-b}{c}

a

b

c

a

squarefree, find

a + b + c

MMATHScomplex numbersalgebra