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International Contests
Silk Road
2022 Silk Road
2022 Silk Road
Part of
Silk Road
Subcontests
(4)
4
1
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Combinatorics Problem
In a language
,
,
,
an alphabet with
25
25
25
letters is used
;
;
;
words are exactly all sequences of
(
(
(
not necessarily different
)
)
)
letters of length
17.
17.
17.
Two ends of a paper strip are glued so that the strip forms a ring
;
;
;
the strip bears a sequence of
5
18
5^{18}
5
18
letters
.
.
.
Say that a word is singular if one can cut a piece bearing exactly that word from the strip
,
,
,
but one cannot cut out two such non-overlapping pieces
.
.
.
It is known that one can cut out
5
16
5^{16}
5
16
non-overlapping pieces each containing the same word
.
.
.
Determine the largest possible number of singular words
.
.
.
(Bogdanov I.)
3
1
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Algebra Problem
In an infinite sequence
{
α
}
,
{
α
2
}
,
{
α
3
}
,
⋯
\{\alpha\}, \{\alpha^2\}, \{\alpha^3\}, \cdots
{
α
}
,
{
α
2
}
,
{
α
3
}
,
⋯
there are finitely many distinct values
.
.
.
Show that
α
\alpha
α
is an integer
.
(
{
x
}
. (\{x\}
.
({
x
}
denotes the fractional part of
x
.
)
x.)
x
.
)
(Golovanov A.S.)
2
1
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Number Theory Problem
Distinct positive integers
A
A
A
and
B
B
B
are given
.
.
.
Prove that there exist infinitely many positive integers that can be represented both as
x
1
2
+
A
y
1
2
x_{1}^2+Ay_{1}^2
x
1
2
+
A
y
1
2
for some positive coprime integers
x
1
x_{1}
x
1
and
y
1
,
y_{1},
y
1
,
and as
x
2
2
+
B
y
2
2
x_{2}^2+By_{2}^2
x
2
2
+
B
y
2
2
for some positive coprime integers
x
2
x_{2}
x
2
and
y
2
.
y_{2}.
y
2
.
(Golovanov A.S.)
1
1
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Geometry Problem
Convex quadrilateral
A
B
C
D
ABCD
A
BC
D
is inscribed in circle
w
.
w.
w
.
Rays
A
B
AB
A
B
and
D
C
DC
D
C
intersect at
K
.
L
K.\ L
K
.
L
is chosen on the diagonal
B
D
BD
B
D
so that
∠
B
A
C
=
∠
D
A
L
.
M
\angle BAC= \angle DAL.\ M
∠
B
A
C
=
∠
D
A
L
.
M
is chosen on the segment
K
L
KL
K
L
so that
C
M
∣
∣
B
D
.
CM \mid\mid BD.
CM
∣∣
B
D
.
Prove that line
B
M
BM
BM
touches
w
.
w.
w
.
(Kungozhin M.)