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National and Regional Contests
China Contests
China National Olympiad
2016 China National Olympiad
1
China Mathematical Olympiad 2016 Q1
China Mathematical Olympiad 2016 Q1
Source: China Yingtan ，Dec 16， 2015
December 16, 2015
inequalities
Sequences
Problem Statement
Let
a
1
,
a
2
,
⋯
,
a
31
;
b
1
,
b
2
,
⋯
,
b
31
a_1,a_2,\cdots, a_{31} ;b_1,b_2, \cdots, b_{31}
a
1
,
a
2
,
⋯
,
a
31
;
b
1
,
b
2
,
⋯
,
b
31
be positive integers such that
a
1
<
a
2
<
⋯
<
a
31
≤
2015
a_1< a_2<\cdots< a_{31}\leq2015
a
1
<
a
2
<
⋯
<
a
31
≤
2015
,
b
1
<
b
2
<
⋯
<
b
31
≤
2015
b_1< b_2<\cdots<b_{31}\leq2015
b
1
<
b
2
<
⋯
<
b
31
≤
2015
and
a
1
+
a
2
+
⋯
+
a
31
=
b
1
+
b
2
+
⋯
+
b
31
.
a_1+a_2+\cdots+a_{31}=b_1+b_2+\cdots+b_{31}.
a
1
+
a
2
+
⋯
+
a
31
=
b
1
+
b
2
+
⋯
+
b
31
.
Find the maximum value of
S
=
∣
a
1
−
b
1
∣
+
∣
a
2
−
b
2
∣
+
⋯
+
∣
a
31
−
b
31
∣
.
S=|a_1-b_1|+|a_2-b_2|+\cdots+|a_{31}-b_{31}|.
S
=
∣
a
1
−
b
1
∣
+
∣
a
2
−
b
2
∣
+
⋯
+
∣
a
31
−
b
31
∣.
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