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Problems
Contests
National and Regional Contests
Moldova Contests
Moldova Team Selection Test
2024 Moldova Team Selection Test
2024 Moldova Team Selection Test
Part of
Moldova Team Selection Test
Subcontests
(5)
7
1
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Tricky Shapiro-like inequality
Prove that
a
=
2
a=2
a
=
2
is the greatest real number for which the inequality:
x
1
x
n
+
x
2
+
x
2
x
1
+
x
3
+
⋯
+
x
n
x
n
−
1
+
x
1
≥
a
\frac{x_1}{x_n+x_2}+\frac{x_2}{x_1+x_3}+\dots+\frac{x_n}{x_{n-1}+x_1} \ge a
x
n
+
x
2
x
1
+
x
1
+
x
3
x
2
+
⋯
+
x
n
−
1
+
x
1
x
n
≥
a
holds true for any
n
≥
4
n \ge 4
n
≥
4
and any positive real numbers
x
1
,
x
2
,
…
,
x
n
x_1, x_2,\dots,x_n
x
1
,
x
2
,
…
,
x
n
.
6
1
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triple radius of incircle
Prove that in any triangle the length of the shortest bisector does not exceed three times the radius of the incircle.
5
1
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CG with convex polygon inside a square and weak bound
Consider a natural number
n
≥
3
n \ge 3
n
≥
3
. A convex polygon with
n
n
n
sides is entirely placed inside a square with side length 1. Prove that we can always find three vertices of this polygon, the triangle formed by which has area not greater than
8
n
2
\frac{8}{n^2}
n
2
8
.
2
1
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Perpendiculars to AD and tangent circles
In the acute-angled triangle
A
B
C
ABC
A
BC
, let
A
D
AD
A
D
,
D
∈
B
C
D \in BC
D
∈
BC
be the
A
A
A
-angle bisector. The perpenducular to
B
C
BC
BC
through
D
D
D
and the perpendicular to
A
D
AD
A
D
through
A
A
A
meet at
I
I
I
. The circle with center
I
I
I
and radius
I
D
ID
I
D
, intersects
A
B
AB
A
B
and
A
C
AC
A
C
at
F
F
F
and
E
E
E
respectively. On the arc
F
E
FE
FE
, which does not contain
A
A
A
, of the circumcircle of
A
F
E
AFE
A
FE
, consider a point
X
X
X
, such that
X
F
X
E
=
A
F
A
E
\frac{XF}{XE}=\frac{AF}{AE}
XE
XF
=
A
E
A
F
. Prove that the circumcircles of triangles
A
F
E
AFE
A
FE
and
B
X
C
BXC
BXC
are tangent.
1
1
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abc perfect cube
If
a
b
+
b
c
+
c
a
\frac{a }{b}+ \frac{b}{c}+ \frac{c}{a}
b
a
+
c
b
+
a
c
is integer. show that
a
b
c
abc
ab
c
is perfect cube.