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captain and his 3 sailors split 2009 golden coins

Source: China Northern MO 2009 p4 CNMO

December 12, 2020
combinatorics

Problem Statement

The captain and his three sailors get 20092009 golden coins with the same value . The four people decided to divide these coins by the following rules : sailor 11,sailor 22,sailor 33 everyone write down an integer b1,b2,b3b_1,b_2,b_3 , satisfies b1≥b2≥b3b_1\ge b_2\ge b_3 , and b1+b2+b3=2009{b_1+b_2+b_3=2009}; the captain dosen't know what the numbers the sailors have written . He divides 20092009 coins into 33 piles , with number of coins: a1,a2,a3a_1,a_2,a_3 , and a1≥a2≥a3a_1\ge a_2\ge a_3 . For sailor kk (k=1,2,3k=1,2,3) , if bk<akb_k<a_k , then he can take bkb_k coins from the kkth pile ; if bk≥akb_k\ge a_k , then he can't take any coins away . At last , the captain own the rest of the coins .If no matter what the numbers the sailors write , the captain can make sure that he always gets nn coins . Find the largest possible value of nn and prove your conclusion .
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