MathDB
Problems
Contests
National and Regional Contests
Moldova Contests
EGMO TST - Moldova
2017 Moldova EGMO TST
2017 Moldova EGMO TST
Part of
EGMO TST - Moldova
Subcontests
(4)
4
1
Hide problems
Moldova egmo 2k17 tst
The points
P
P
P
and
Q
Q
Q
are placed in the interior of the triangle
Δ
A
B
C
\Delta ABC
Δ
A
BC
such that
m
(
∠
P
A
B
)
=
m
(
∠
Q
A
C
)
<
1
2
m
(
∠
B
A
C
)
m(\angle PAB)=m(\angle QAC)<\frac{1}{2}m(\angle BAC)
m
(
∠
P
A
B
)
=
m
(
∠
Q
A
C
)
<
2
1
m
(
∠
B
A
C
)
and similarly for the other
2
2
2
vertices(
P
P
P
and
Q
Q
Q
are isogonal conjugates). Let
P
A
P_{A}
P
A
and
Q
A
Q_{A}
Q
A
be the intersection points of
A
P
AP
A
P
and
A
Q
AQ
A
Q
with the circumcircle of
C
P
B
CPB
CPB
, respectively
C
Q
B
CQB
CQB
. Similarly the pairs of points
(
P
B
,
Q
B
)
(P_{B},Q_{B})
(
P
B
,
Q
B
)
and
(
P
C
,
Q
C
)
(P_{C},Q_{C})
(
P
C
,
Q
C
)
are defined. Let
P
Q
A
∩
Q
P
A
=
{
M
A
}
PQ_{A}\cap QP_{A}=\{M_{A}\}
P
Q
A
∩
Q
P
A
=
{
M
A
}
,
P
Q
B
∩
Q
P
B
=
{
M
B
}
PQ_{B}\cap QP_{B}=\{M_{B}\}
P
Q
B
∩
Q
P
B
=
{
M
B
}
,
P
Q
C
∩
Q
P
C
=
{
M
C
}
PQ_{C}\cap QP_{C}=\{M_{C}\}
P
Q
C
∩
Q
P
C
=
{
M
C
}
. Prove the following statements:
1.
1.
1.
Lines
A
M
A
AM_{A}
A
M
A
,
B
M
B
BM_{B}
B
M
B
,
C
M
C
CM_{C}
C
M
C
concur.
2.
2.
2.
M
A
∈
B
C
M_{A}\in BC
M
A
∈
BC
,
M
B
∈
C
A
M_{B}\in CA
M
B
∈
C
A
,
M
C
∈
A
B
M_{C}\in AB
M
C
∈
A
B
3
1
Hide problems
Egmo moldova tst 2017 easy combinatorics
Let us have
6050
6050
6050
points in the plane, no three collinear. Find the maximum number
k
k
k
of non-overlapping triangles without common vertices in this plane.
2
1
Hide problems
Moldova tst egmo 2017
Let us denote the midpoint of
A
B
AB
A
B
with
O
O
O
. The point
C
C
C
, different from
A
A
A
and
B
B
B
is on the circle
Ω
\Omega
Ω
with center
O
O
O
and radius
O
A
OA
O
A
and the point
D
D
D
is the foot of the perpendicular from
C
C
C
to
A
B
AB
A
B
. The circle with center
C
C
C
and radius
C
D
CD
C
D
and
ω
\omega
ω
intersect at
M
M
M
,
N
N
N
. Prove that
M
N
MN
MN
cuts
C
D
CD
C
D
in two equal segments.
1
1
Hide problems
Cyclic Inequality
Let
a
,
b
,
c
≥
0
a,b,c\geq 0
a
,
b
,
c
≥
0
. Prove:
1
+
a
+
a
2
1
+
b
+
c
2
+
1
+
b
+
b
2
1
+
c
+
a
2
+
1
+
c
+
c
2
1
+
a
+
b
2
≥
3
\frac{1+a+a^{2}}{1+b+c^{2}}+\frac{1+b+b^{2}}{1+c+a^{2}}+\frac{1+c+c^{2}}{1+a+b^{2}}\geq 3
1
+
b
+
c
2
1
+
a
+
a
2
+
1
+
c
+
a
2
1
+
b
+
b
2
+
1
+
a
+
b
2
1
+
c
+
c
2
≥
3