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Contests
National and Regional Contests
Moldova Contests
Moldova Team Selection Test
2013 Moldova Team Selection Test
2013 Moldova Team Selection Test
Part of
Moldova Team Selection Test
Subcontests
(4)
3
3
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Geometric inequality
Consider the obtuse-angled triangle
△
A
B
C
\triangle ABC
△
A
BC
and its side lengths
a
,
b
,
c
a,b,c
a
,
b
,
c
. Prove that
a
3
cos
∠
A
+
b
3
cos
∠
B
+
c
3
cos
∠
C
<
a
b
c
a^3\cos\angle A +b^3\cos\angle B + c^3\cos\angle C < abc
a
3
cos
∠
A
+
b
3
cos
∠
B
+
c
3
cos
∠
C
<
ab
c
.
Known geometry problem
The diagonals of a trapezoid
A
B
C
D
ABCD
A
BC
D
with
A
D
∥
B
C
AD \parallel BC
A
D
∥
BC
intersect at point
P
P
P
. Point
Q
Q
Q
lies between the parallel lines
A
D
AD
A
D
and
B
C
BC
BC
such that the line
C
D
CD
C
D
separates points
P
P
P
and
Q
Q
Q
, and
∠
A
Q
D
=
∠
C
Q
B
\angle AQD=\angle CQB
∠
A
Q
D
=
∠
CQB
. Prove that
∠
B
Q
P
=
∠
D
A
Q
\angle BQP = \angle DAQ
∠
BQP
=
∠
D
A
Q
.
Reflection across midpoint
Consider the triangle
△
A
B
C
\triangle ABC
△
A
BC
with
A
B
≠
A
C
AB \not = AC
A
B
=
A
C
. Let point
O
O
O
be the circumcenter of
△
A
B
C
\triangle ABC
△
A
BC
. Let the angle bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
intersect
B
C
BC
BC
at point
D
D
D
. Let
E
E
E
be the reflection of point
D
D
D
across the midpoint of the segment
B
C
BC
BC
. The lines perpendicular to
B
C
BC
BC
in points
D
,
E
D,E
D
,
E
intersect the lines
A
O
,
A
D
AO,AD
A
O
,
A
D
at the points
X
,
Y
X,Y
X
,
Y
respectively. Prove that the quadrilateral
B
,
X
,
C
,
Y
B,X,C,Y
B
,
X
,
C
,
Y
is cyclic.
4
3
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Limit of a sequence
Consider a positive real number
a
a
a
and a positive integer
m
m
m
. The sequence
(
x
k
)
k
∈
Z
+
(x_k)_{k\in \mathbb{Z}^{+}}
(
x
k
)
k
∈
Z
+
is defined as:
x
1
=
1
x_1=1
x
1
=
1
,
x
2
=
a
x_2=a
x
2
=
a
,
x
n
+
2
=
x
n
+
1
m
x
n
m
+
1
x_{n+2}=\sqrt[m+1]{x_{n+1}^mx_n}
x
n
+
2
=
m
+
1
x
n
+
1
m
x
n
.
a
)
a)
a
)
Prove that the sequence is converging.
b
)
b)
b
)
Find
lim
n
→
∞
x
n
\lim_{n\rightarrow \infty}{x_n}
lim
n
→
∞
x
n
.
Nine number inequality
Prove that for any positive real numbers
a
i
,
b
i
,
c
i
a_i,b_i,c_i
a
i
,
b
i
,
c
i
with
i
=
1
,
2
,
3
i=1,2,3
i
=
1
,
2
,
3
,
(
a
1
3
+
b
1
3
+
c
1
3
+
1
)
(
a
2
3
+
b
2
3
+
c
2
3
+
1
)
(
a
3
3
+
b
3
3
+
c
3
3
+
1
)
≥
3
4
(
a
1
+
b
1
+
c
1
)
(
a
2
+
b
2
+
c
2
)
(
a
3
+
b
3
+
c
3
)
(a_1^3+b_1^3+c_1^3+1)(a_2^3+b_2^3+c_2^3+1)(a_3^3+b_3^3+c_3^3+1)\geq \frac{3}{4} (a_1+b_1+c_1)(a_2+b_2+c_2)(a_3+b_3+c_3)
(
a
1
3
+
b
1
3
+
c
1
3
+
1
)
(
a
2
3
+
b
2
3
+
c
2
3
+
1
)
(
a
3
3
+
b
3
3
+
c
3
3
+
1
)
≥
4
3
(
a
1
+
b
1
+
c
1
)
(
a
2
+
b
2
+
c
2
)
(
a
3
+
b
3
+
c
3
)
Number Theory
p
p
p
is a 4k+3 prime. Prove that there are infinite
p
p
p
which satisfies
p
∣
2
n
y
+
1
p|2^ny+1
p
∣
2
n
y
+
1
.
y
y
y
is an random integer.
2
4
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1
4
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