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Contests
National and Regional Contests
Tajikistan Contests
Tajikistan Team Selection Test
2014 Tajikistan Team Selection Test
2014 Tajikistan Team Selection Test
Part of
Tajikistan Team Selection Test
Subcontests
(5)
1
1
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Polynomial with integer coefficients
Given the polynomial
p
(
x
)
=
x
2
+
x
−
70
p(x) = x^2 + x - 70
p
(
x
)
=
x
2
+
x
−
70
, do there exist integers
0
<
m
<
n
0<m<n
0
<
m
<
n
, so that
p
(
m
)
p(m)
p
(
m
)
is divisible by
n
n
n
and
p
(
m
+
1
)
p(m+1)
p
(
m
+
1
)
is divisible by
n
+
1
n+1
n
+
1
?Proposed by Nairy Sedrakyan
5
1
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Graphs in combinatorics
There are
12
12
12
delegates in a mathematical conference. It is known that every two delegates share a common friend. Prove that there is a delegate who has at least five friends in that conference.Proposed by Nairy Sedrakyan
2
1
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Inequality in Geometry 2
Let
M
M
M
be an interior point of triangle
A
B
C
ABC
A
BC
. Let the line
A
M
AM
A
M
intersect the circumcircle of the triangle
M
B
C
MBC
MBC
for the second time at point
D
D
D
, the line
B
M
BM
BM
intersect the circumcircle of the triangle
M
C
A
MCA
MC
A
for the second time at point
E
E
E
, and the line
C
M
CM
CM
intersect the circumcircle of the triangle
M
A
B
MAB
M
A
B
for the second time at point
F
F
F
. Prove that
A
D
M
D
+
B
E
M
E
+
C
F
M
F
≥
9
2
\frac{AD}{MD} + \frac{BE}{ME} + \frac{CF}{MF} \geq \frac{9}{2}
M
D
A
D
+
ME
BE
+
MF
CF
≥
2
9
.Proposed by Nairy Sedrakyan
4
1
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Convex hexagon
In a convex hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
the diagonals
A
D
,
B
E
,
C
F
AD,BE,CF
A
D
,
BE
,
CF
intersect at a point
M
M
M
. It is known that the triangles
A
B
M
,
B
C
M
,
C
D
M
,
D
E
M
,
E
F
M
,
F
A
M
ABM,BCM,CDM,DEM,EFM,FAM
A
BM
,
BCM
,
C
D
M
,
D
EM
,
EFM
,
F
A
M
are acute. It is also known that the quadrilaterals
A
B
D
E
,
B
C
E
F
,
C
D
F
A
ABDE,BCEF,CDFA
A
B
D
E
,
BCEF
,
C
D
F
A
have the same area. Prove that the circumcenters of triangles
A
B
M
,
B
C
M
,
C
D
M
,
D
E
M
,
E
F
M
,
F
A
M
ABM,BCM,CDM,DEM,EFM,FAM
A
BM
,
BCM
,
C
D
M
,
D
EM
,
EFM
,
F
A
M
are concyclic.Proposed by Nairy Sedrakyan
3
1
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Inequality in Geometry
Let
a
a
a
,
b
b
b
,
c
c
c
be side length of a triangle. Prove the inequality \begin{align*} \sqrt{a^2 + ab + b^2} + \sqrt{b^2 + bc + c^2} + \sqrt{c^2 + ca + a^2} \leq \sqrt{5a^2 + 5b^2 + 5c^2 + 4ab + 4 bc + 4ca}.\end{align*}