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Problems
Contests
National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
2012 Korea Junior Math Olympiad
4
4
Part of
2012 Korea Junior Math Olympiad
Problems
(1)
n students shaking hands
Source: KJMO 2012 p4
5/4/2019
There are
n
n
n
students
A
1
,
A
2
,
.
.
.
,
A
n
A_1,A_2,...,A_n
A
1
,
A
2
,
...
,
A
n
and some of them shaked hands with each other. (
A
i
A_i
A
i
and
A
−
j
A-j
A
−
j
can shake hands more than one time.) Let the student
A
i
A_i
A
i
shaked hands
d
i
d_i
d
i
times. Suppose
d
1
+
d
2
+
.
.
.
+
d
n
>
0
d_1 + d_2 +... + d_n > 0
d
1
+
d
2
+
...
+
d
n
>
0
. Prove that there exist
1
≤
i
<
j
≤
n
1 \le i < j \le n
1
≤
i
<
j
≤
n
satisfying the following conditions: (a) Two students
A
i
A_i
A
i
and
A
j
A_j
A
j
shaked hands each other. (b)
(
d
1
+
d
2
+
.
.
.
+
d
n
)
2
n
2
≤
d
i
d
j
\frac{(d_1 + d_2 +... + d_n)^2}{n^2}\le d_id_j
n
2
(
d
1
+
d
2
+
...
+
d
n
)
2
≤
d
i
d
j
combinatorics