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Purple Comet 2017 MS Problem 17

Source:

April 14, 2018

Purple Comet

Problem Statement

Let

a_0

,

a_1

, ...,

a_6

be real numbers such that

a_0 + a_1 + ... + a_6 = 1

and

a_0 + a_2 + a_3 + a_4 + a_5 + a_6 =\frac{1}{2}

a_0 + a_1 + a_3 + a_4 + a_5 + a_6 = \frac{2}{3}

a_0 + a_1 + a_2 + a_4 + a_5 + a_6 =\frac{7}{8}

a_0 + a_1 + a_2 + a_3 + a_5 + a_6 =\frac{29}{30}

a_0 + a_1 + a_2 + a_3 + a_4 + a_6 =\frac{143}{144}

a_0 + a_1 + a_2 + a_3 + a_4 + a_5 =\frac{839}{840}

The value of

a_0

is

\frac{m}{n}

, where

m

and

n

are relatively prime positive integers. Find

m + n

.

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