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Contests
International Contests
Silk Road
2021 Silk Road
2021 Silk Road
Part of
Silk Road
Subcontests
(3)
3
1
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midpoints and circumcircles
In a triangle
A
B
C
ABC
A
BC
,
M
M
M
is the midpoint of the
A
B
AB
A
B
. A point
B
1
B_1
B
1
is marked on
A
C
AC
A
C
such that
C
B
=
C
B
1
CB=CB_1
CB
=
C
B
1
. Circle
ω
\omega
ω
and
ω
1
\omega_1
ω
1
, the circumcircles of triangles
A
B
C
ABC
A
BC
and
B
M
B
1
BMB_1
BM
B
1
, respectively, intersect again at
K
K
K
. Let
Q
Q
Q
be the midpoint of the arc
A
C
B
ACB
A
CB
on
ω
\omega
ω
. Let
B
1
Q
B_1Q
B
1
Q
and
B
C
BC
BC
intersect at
E
E
E
. Prove that
K
C
KC
K
C
bisects
B
1
E
B_1E
B
1
E
.M. Kungozhin
1
1
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Maximum of a given substring in sequences
Given a sequence
s
s
s
consisting of digits
0
0
0
and
1
1
1
. For any positive integer
k
k
k
, define
v
k
v_k
v
k
the maximum number of ways in any sequence of length
k
k
k
that several consecutive digits can be identified, forming the sequence
s
s
s
. (For example, if
s
=
0110
s=0110
s
=
0110
, then
v
7
=
v
8
=
2
v_7=v_8=2
v
7
=
v
8
=
2
, because in sequences
0110110
0110110
0110110
and
01101100
01101100
01101100
one can find consecutive digits
0110
0110
0110
in two places, and three pairs of
0110
0110
0110
cannot meet in a sequence of length
7
7
7
or
8
8
8
.) It is known that
v
n
<
v
n
+
1
<
v
n
+
2
v_n<v_{n+1}<v_{n+2}
v
n
<
v
n
+
1
<
v
n
+
2
for some positive integer
n
n
n
. Prove that in the sequence
s
s
s
, all the numbers are the same.A. Golovanov
4
1
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Silk Road 2021 P4
Integers
x
,
y
,
z
,
t
x,y,z,t
x
,
y
,
z
,
t
satisfy
x
2
+
y
2
=
z
2
+
t
2
x^2+y^2=z^2+t^2
x
2
+
y
2
=
z
2
+
t
2
and
x
y
=
2
z
t
xy=2zt
x
y
=
2
z
t
prove that
x
y
z
t
=
0
xyzt=0
x
yz
t
=
0
Proposed by
M
.
A
b
d
u
v
a
l
i
e
v
M. Abduvaliev
M
.
A
b
d
uv
a
l
i
e
v