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Problems
Contests
National and Regional Contests
China Contests
XES Mathematics Olympiad
the 6th XMO
the 6th XMO
Part of
XES Mathematics Olympiad
Subcontests
(2)
5
1
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(ASG) tangent to (J)
As shown in the figure,
⊙
O
\odot O
⊙
O
is the circumcircle of
△
A
B
C
\vartriangle ABC
△
A
BC
,
⊙
J
\odot J
⊙
J
is inscribed in
⊙
O
\odot O
⊙
O
and is tangent to
A
B
AB
A
B
,
A
C
AC
A
C
at points
D
D
D
and E respectively, line segment
F
G
FG
FG
and
⊙
O
\odot O
⊙
O
are tangent to point
A
A
A
, and
A
F
=
A
G
=
A
D
AF =AG=AD
A
F
=
A
G
=
A
D
, the circumscribed circle of
△
A
F
B
\vartriangle AFB
△
A
FB
intersects
⊙
J
\odot J
⊙
J
at point
S
S
S
. Prove that the circumscribed circle of
△
A
S
G
\vartriangle ASG
△
A
SG
is tangent to
⊙
J
\odot J
⊙
J
. https://cdn.artofproblemsolving.com/attachments/a/a/62d44e071ea9903ebdd68b43943ba1d93b4138.png
2
1
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max of complex numbers' sum |z_i-z_j|^2 if |z_i-z_j| <=1
Assume that complex numbers
z
1
,
z
2
,
.
.
.
,
z
n
z_1,z_2,...,z_n
z
1
,
z
2
,
...
,
z
n
satisfy
∣
z
i
−
z
j
∣
≤
1
|z_i-z_j| \le 1
∣
z
i
−
z
j
∣
≤
1
for any
1
≤
i
<
j
≤
n
1 \le i <j \le n
1
≤
i
<
j
≤
n
. Let
S
=
∑
1
≤
i
<
j
≤
n
∣
z
i
−
z
j
∣
2
.
S= \sum_{1 \le i <j \le n} |z_i-z_j|^2.
S
=
1
≤
i
<
j
≤
n
∑
∣
z
i
−
z
j
∣
2
.
(1) If
n
=
6063
n = 6063
n
=
6063
, find the maximum value of
S
S
S
. (2) If
n
=
2021
n= 2021
n
=
2021
, find the maximum value of
S
S
S
.