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Integer Approximations

Source: AIME 1985 is here! My favorite problem 8

March 6, 2006
AMCAIME

Problem Statement

The sum of the following seven numbers is exactly 19: a1=2.56,a2=2.61,a3=2.65,a4=2.71,a_1=2.56,\qquad a_2=2.61,\qquad a_3=2.65,\qquad a_4=2.71, a5=2.79,a6=2.82,a7=2.86.a_5=2.79,\qquad a_6=2.82,\qquad a_7=2.86. It is desired to replace each aia_i by an integer approximation AiA_i, 1≤i≤71 \le i \le 7, so that the sum of the AiA_i's is also 19 and so that MM, the maximum of the "errors" ∣Ai−ai∣|A_i - a_i|, is as small as possible. For this minimum MM, what is 100M100M?
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