MathDB
Problems
Contests
National and Regional Contests
China Contests
China Northern MO
2009 China Northern MO
4
4
Part of
2009 China Northern MO
Problems
(1)
captain and his 3 sailors split 2009 golden coins
Source: China Northern MO 2009 p4 CNMO
12/12/2020
The captain and his three sailors get
2009
2009
2009
golden coins with the same value . The four people decided to divide these coins by the following rules : sailor
1
1
1
,sailor
2
2
2
,sailor
3
3
3
everyone write down an integer
b
1
,
b
2
,
b
3
b_1,b_2,b_3
b
1
,
b
2
,
b
3
, satisfies
b
1
≥
b
2
≥
b
3
b_1\ge b_2\ge b_3
b
1
≥
b
2
≥
b
3
, and
b
1
+
b
2
+
b
3
=
2009
{b_1+b_2+b_3=2009}
b
1
+
b
2
+
b
3
=
2009
; the captain dosen't know what the numbers the sailors have written . He divides
2009
2009
2009
coins into
3
3
3
piles , with number of coins:
a
1
,
a
2
,
a
3
a_1,a_2,a_3
a
1
,
a
2
,
a
3
, and
a
1
≥
a
2
≥
a
3
a_1\ge a_2\ge a_3
a
1
≥
a
2
≥
a
3
. For sailor
k
k
k
(
k
=
1
,
2
,
3
k=1,2,3
k
=
1
,
2
,
3
) , if
b
k
<
a
k
b_k<a_k
b
k
<
a
k
, then he can take
b
k
b_k
b
k
coins from the
k
k
k
th pile ; if
b
k
≥
a
k
b_k\ge a_k
b
k
≥
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
n
n
n
coins . Find the largest possible value of
n
n
n
and prove your conclusion .
combinatorics