There are 2011 positive numbers with both their sum and the sum of their reciprocals equal to 2012. Let be one of these numbers, if the maximum value of can be expressed as for coprime positive integers, submit your answer as .
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L e t p 1 , p 2 . . . . p 2 0 1 0 b e t h e 2 0 1 0 n u m b e r s d i s t i n c t f r o m x . T h e n , p 1 + p 2 + . . . . . . + p 2 0 1 0 = 2 0 1 2 − x a n d p 1 1 + p 2 1 + . . . . + p 2 0 1 0 1 = 2 0 1 2 − x 1 A p p l y i n g C a u c h y − S c h w a r z / A M − G M I n e q u a l i t y , ( i = 1 ∑ 2 0 1 0 p i ) ( i = 1 ∑ 2 0 1 0 p i 1 ) = ( 2 0 1 2 − x ) ( 2 0 1 2 − x 1 ) ≥ 2 0 1 0 2 ⟹ 2 0 1 2 2 − 2 0 1 2 ( x + x 1 ) + 1 − 2 0 1 0 2 ≥ 0 ⟹ ( x + x 1 ) ≤ 2 0 1 2 8 0 4 5 T h u s , 8 0 4 5 − 2 0 1 2 = 6 0 3 3