12

Part of 2019 AIME Problems

Problems(2)

geometry in complex numbers

Source: 2019 AIME I #12

f(z) = z^2-19z

, there are complex numbers

z

with the property that

z

f(z)

f(f(z))

are the vertices of a right triangle in the complex plane with a right angle at

f(z)

. There are positive integers

m

n

such that one such value of

z

m+\sqrt{n}+11i

m+n

geometryAMCAIMEAIME Icomplex numbers2019 AIME I2019 AMC

Progressive Sequences

Source: 2019 AIME II #12

n \ge 1

call a finite sequence

(a_1, a_2 \ldots a_n)

of positive integers progressive if

a_i < a_{i+1}

a_i

a_{i+1}

1 \le i \le n-1

. Find the number of progressive sequences such that the sum of the terms in the sequence is equal to

360

AMCAIMEAIME II2019 AIME II2019 AMCSequences