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Problems
Contests
National and Regional Contests
Russia Contests
Russian Team Selection Tests
Russian TST 2019
Russian TST 2019
Part of
Russian Team Selection Tests
Subcontests
(3)
P3
1
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Tangent circumcircles
Let
H
H{}
H
be the orthocenter of the acute-angled triangle
A
B
C
ABC
A
BC
. In the triangle
B
H
C
BHC
B
H
C
, the median
H
M
HM
H
M
and the symedian
H
L
HL
H
L
are drawn. The point
K
K{}
K
is marked on the line
L
H
LH
L
H
so that
∠
A
K
L
=
9
0
∘
\angle AKL=90^\circ
∠
A
K
L
=
9
0
∘
. Prove that the circumcircles of the triangles
A
B
C
ABC
A
BC
and
K
L
M
KLM
K
L
M
are tangent.
P1
6
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P2
2
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Congruence modulo p^2
Prove that for every odd prime number
p
p{}
p
, the following congruence holds
∑
n
=
1
p
−
1
n
p
−
1
≡
(
p
−
1
)
!
+
p
(
m
o
d
p
2
)
.
\sum_{n=1}^{p-1}n^{p-1}\equiv (p-1)!+p\pmod{p^2}.
n
=
1
∑
p
−
1
n
p
−
1
≡
(
p
−
1
)!
+
p
(
mod
p
2
)
.
Permutations problem
For each permutation
σ
\sigma
σ
of the set
{
1
,
2
,
…
,
N
}
\{1, 2, \ldots , N\}
{
1
,
2
,
…
,
N
}
we define its correctness as the number of triples
1
⩽
i
<
j
<
k
⩽
N
1 \leqslant i < j < k \leqslant N
1
⩽
i
<
j
<
k
⩽
N
such that the number
σ
(
j
)
\sigma(j)
σ
(
j
)
lies between the numbers
σ
(
i
)
\sigma(i)
σ
(
i
)
and
σ
(
k
)
\sigma(k)
σ
(
k
)
. Find the difference between the number of permutations with even correctness and the number of permutations with odd correctness if a)
N
=
2018
N = 2018
N
=
2018
and b)
N
=
2019
N = 2019
N
=
2019
.