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International Contests
Tournament Of Towns
2020/2021 Tournament of Towns
2020/2021 Tournament of Towns
Part of
Tournament Of Towns
Subcontests
(7)
P6
1
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Very beautiful NT
Find at least one real number
A
A{}
A
such that for any positive integer
n
n{}
n
the distance between
⌈
A
n
⌉
\lceil A^n\rceil
⌈
A
n
⌉
and the nearest square of an integer is equal to two. Dmitry Krekov
P5
7
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P4
7
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P3
6
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P1
7
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P7
2
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Chessboard Combinatorics
A white bug sits in one corner square of a
1000
1000
1000
×
n
n
n
chessboard, where
n
n
n
is an odd positive integer and
n
>
2020
n > 2020
n
>
2020
. In the two nearest corner squares there are two black chess bishops. On each move, the bug either steps into a square adjacent by side or moves as a chess knight. The bug wishes to reach the opposite corner square by never visiting a square occupied or attacked by a bishop, and visiting every other square exactly once. Show that the number of ways for the bug to attain its goal does not depend on
n
n
n
.
Drawing disjoint arcs on the unit sphere
An integer
n
>
2
n > 2
n
>
2
is given. Peter wants to draw
n
n{}
n
arcs of length
α
\alpha{}
α
of great circles on a unit sphere so that they do not intersect each other. Prove that[*]for all
α
<
π
+
2
π
/
n
\alpha<\pi+2\pi/n
α
<
π
+
2
π
/
n
it is possible; [*]for all
α
>
π
+
2
π
/
n
\alpha>\pi+2\pi/n
α
>
π
+
2
π
/
n
it is impossible;Ilya Bogdanov
P2
6
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