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2010 Algebra #2: Number Ranks

Source:

July 15, 2012

Problem Statement

The rank of a rational number

q

is the unique

k

for which

q=\frac{1}{a_1}+\cdots+\frac{1}{a_k}

, where each

a_i

is the smallest positive integer

q

such that

q\geq \frac{1}{a_1}+\cdots+\frac{1}{a_i}

. Let

q

be the largest rational number less than

\frac{1}{4}

with rank

3

, and suppose the expression for

q

is

\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}

. Find the ordered triple

(a_1,a_2,a_3)

.

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