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Problems
Contests
National and Regional Contests
Russia Contests
Russian Team Selection Tests
Russian TST 2017
Russian TST 2017
Part of
Russian Team Selection Tests
Subcontests
(3)
P3
2
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From a Graph to the Plane
Let
K
=
(
V
,
E
)
K=(V, E)
K
=
(
V
,
E
)
be a finite, simple, complete graph. Let
ϕ
:
E
→
R
2
\phi: E \to \mathbb{R}^2
ϕ
:
E
→
R
2
be a map from the edge set to the plane, such that the preimage of any point in the range defines a connected graph on the entire vertex set
V
V
V
, and the points assigned to the edges of any triangle are collinear. Show that the range of
ϕ
\phi
ϕ
is contained in a line.
Similar to Bulgaria 2000
Let
a
1
,
…
,
a
p
−
2
a_1,\ldots , a_{p-2}{}
a
1
,
…
,
a
p
−
2
be nonzero residues modulo an odd prime
p
p{}
p
. For every
d
∣
p
−
1
d\mid p - 1
d
∣
p
−
1
there are at least
⌊
(
p
−
2
)
/
d
⌋
\lfloor(p - 2)/d\rfloor
⌊(
p
−
2
)
/
d
⌋
indices
i
i{}
i
for which
p
p{}
p
does not divide
a
i
d
−
1
a_i^d-1
a
i
d
−
1
. Prove that the product of some of
a
1
,
…
,
a
p
−
2
a_1,\ldots , a_{p-2}
a
1
,
…
,
a
p
−
2
gives the remainder two modulo
p
p{}
p
.
P2
4
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P1
5
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