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djfractionman

Source: 2021 AIME I Problem 10

March 11, 2021

AIMEAIME I2021 AIME I

Problem Statement

Consider the sequence

(a_k)_{k\ge 1}

of positive rational numbers defined by

a_1 = \frac{2020}{2021}

and for

a_{k+1} = \frac{m + 18}{n+19}.

, if

a_k = \frac{m}{n}

for relatively prime positive integers

m

and

n

, then

a_{k+1} = \frac{m + 18}{n+19}.

Determine the sum of all positive integers

j

such that the rational number

a_j

can be written in the form

\frac{t}{t+1}

for some positive integer

t

.

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