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Problems
Contests
National and Regional Contests
Moldova Contests
Moldova Team Selection Test
1994 Moldova Team Selection Test
1994 Moldova Team Selection Test
Part of
Moldova Team Selection Test
Subcontests
(7)
9
1
Hide problems
$AB+BC=AD+CD$, $BC+CA=BD+AD$, $CA+AB=CD+BD$
Let
O
O{}
O
be the center of the circumscribed sphere of the tetrahedron
A
B
C
D
ABCD
A
BC
D
. Let
L
,
M
,
N
L,M,N
L
,
M
,
N
respectively be the midpoints of the segments
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
. It is known that
A
B
+
B
C
=
A
D
+
C
D
AB+BC=AD+CD
A
B
+
BC
=
A
D
+
C
D
,
B
C
+
C
A
=
B
D
+
A
D
BC+CA=BD+AD
BC
+
C
A
=
B
D
+
A
D
,
C
A
+
A
B
=
C
D
+
B
D
CA+AB=CD+BD
C
A
+
A
B
=
C
D
+
B
D
. Prove that
∠
L
O
M
=
∠
M
O
N
=
∠
N
O
L
\angle LOM=\angle MON=\angle NOL
∠
L
OM
=
∠
MON
=
∠
NO
L
. Find their value.
8
1
Hide problems
f(z)+f(wz+a)=g(z)
Let
g
:
C
→
C
g: \mathbb{C} \rightarrow \mathbb{C}
g
:
C
→
C
be a function,
w
∈
C
w\in\mathbb{C}
w
∈
C
and
w
3
=
1
w^3=1
w
3
=
1
. Show that there exists a function
f
:
C
→
C
f:\mathbb{C} \rightarrow \mathbb{C}
f
:
C
→
C
such that
f
(
z
)
+
f
(
w
z
+
a
)
=
g
(
z
)
,
∀
z
∈
C
f(z)+f(wz+a)=g(z), \forall z\in\mathbb{C}
f
(
z
)
+
f
(
w
z
+
a
)
=
g
(
z
)
,
∀
z
∈
C
. When there is an unique function
f
f
f
with this property? Find it.
6
1
Hide problems
Prove that $\frac{AC}{DD_2}=\frac{AB}{DD_1}+\frac{BC}{DD_3}$.
Inside the triangle
D
D
1
D
3
DD_1D_3
D
D
1
D
3
the cevian
D
D
2
DD_2
D
D
2
is constructed. Perpendiculars from
D
1
,
D
2
D_1, D_2
D
1
,
D
2
and
D
3
D_3
D
3
to lines
D
D
1
,
D
D
2
DD_1, DD_2
D
D
1
,
D
D
2
and
D
D
3
DD_3
D
D
3
, respectively, intersect in points
A
,
B
A,B
A
,
B
and
C
C
C
such that
A
B
⊥
D
D
1
,
A
C
⊥
D
D
2
,
B
C
⊥
D
D
3
AB\perp DD_1, AC\perp DD_2, BC\perp DD_3
A
B
⊥
D
D
1
,
A
C
⊥
D
D
2
,
BC
⊥
D
D
3
. Prove that
A
C
D
D
2
=
A
B
D
D
1
+
B
C
D
D
3
\frac{AC}{DD_2}=\frac{AB}{DD_1}+\frac{BC}{DD_3}
D
D
2
A
C
=
D
D
1
A
B
+
D
D
3
BC
.
5
1
Hide problems
Find the greatest value of $a^{m_1}+a^{m_2}+\ldots+a^{m_p}$
Let
m
m
m
be a positive integer and
a
a
a
a positive real number. Find the greatest value of
a
m
1
+
a
m
2
+
…
+
a
m
p
a^{m_1}+a^{m_2}+\ldots+a^{m_p}
a
m
1
+
a
m
2
+
…
+
a
m
p
where
m
1
+
m
2
+
…
+
m
p
=
m
,
m
i
∈
N
,
i
=
1
,
2
,
…
,
p
;
m_1+m_2+\ldots+m_p=m, m_i\in\mathbb{N},i=1,2,\ldots,p;
m
1
+
m
2
+
…
+
m
p
=
m
,
m
i
∈
N
,
i
=
1
,
2
,
…
,
p
;
1
≤
p
≤
m
,
p
∈
N
1\leq p\leq m, p\in\mathbb{N}
1
≤
p
≤
m
,
p
∈
N
.
4
1
Hide problems
$P(x)$ has integer values for $n+1$ consecutive values of the argument
Let
P
(
x
)
P(x)
P
(
x
)
be a polynomial with at most
n
n{}
n
real coefficeints. Prove that if
P
(
x
)
P(x)
P
(
x
)
has integer values for
n
+
1
n+1
n
+
1
consecutive values of the argument, then
P
(
m
)
∈
Z
,
∀
m
∈
Z
.
P(m)\in\mathbb{Z},\forall m\in\mathbb{Z}.
P
(
m
)
∈
Z
,
∀
m
∈
Z
.
3
1
Hide problems
Triangles $MAB$ and $MA_1B_1$ are similar and have the same orientation
Triangles
M
A
B
MAB{}
M
A
B
and
M
A
1
B
1
MA_1B_1{}
M
A
1
B
1
are similar and have the same orientation. Prove that the circumcircles of these triangles cointain the intersection point of lines
A
A
1
AA_1{}
A
A
1
and
B
B
1
BB_1{}
B
B
1
.
1
1
Hide problems
Find $\max\{P(-2),P(2)\}$
Let
P
(
X
)
=
X
n
+
a
1
X
n
−
1
+
…
+
a
n
P(X)=X^n+a_1X^{n-1}+\ldots+a_n
P
(
X
)
=
X
n
+
a
1
X
n
−
1
+
…
+
a
n
be a plynomial with real roots
x
1
.
x
2
,
…
,
x
n
x_1. x_2,\ldots,x_n
x
1
.
x
2
,
…
,
x
n
. Denote
E
k
=
x
1
k
+
x
2
k
+
…
+
x
n
k
,
∀
k
∈
N
E_k=x_1^k+x_2^k+\ldots+x_n^k, \forall k\in\mathbb{N}
E
k
=
x
1
k
+
x
2
k
+
…
+
x
n
k
,
∀
k
∈
N
. There exists an
m
∈
N
m\in\mathbb{N}
m
∈
N
such that
E
m
=
E
m
+
1
=
E
m
+
2
=
1
E_m=E_{m+1}=E_{m+2}=1
E
m
=
E
m
+
1
=
E
m
+
2
=
1
. Find
max
{
P
(
−
2
)
,
P
(
2
)
}
\max\{P(-2),P(2)\}
max
{
P
(
−
2
)
,
P
(
2
)}
.