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Problems
Contests
National and Regional Contests
Moldova Contests
Moldova Team Selection Test
1996 Moldova Team Selection Test
1996 Moldova Team Selection Test
Part of
Moldova Team Selection Test
Subcontests
(11)
11
1
Hide problems
the function $f^{n-2}$ is constant, but $f^{n-3}$ is not
Let
A
A{}
A
be a set with
n
n{}
n
(
n
≥
3
)
(n\geq3)
(
n
≥
3
)
elements. Iterations
f
2
,
f
2
,
…
f^2,f^2,\ldots
f
2
,
f
2
,
…
of the function
f
:
A
→
A
f:A\rightarrow A
f
:
A
→
A
are defined as
f
2
(
x
)
=
f
(
f
(
x
)
)
,
f
i
+
1
=
f
(
f
i
(
x
)
)
,
∀
i
≥
2
f^2(x)=f(f(x)), f^{i+1}=f(f^i(x)), \forall i\geq2
f
2
(
x
)
=
f
(
f
(
x
))
,
f
i
+
1
=
f
(
f
i
(
x
))
,
∀
i
≥
2
. Find the number of functions
f
:
A
→
A
f:A\rightarrow A
f
:
A
→
A
with the property: the function
f
n
−
2
f^{n-2}
f
n
−
2
is constant, but
f
n
−
3
f^{n-3}
f
n
−
3
is not.
8
1
Hide problems
there exist $m$ subsets of $X$ such that each two of them are not disjoint
Let
X
X
X
be set with
n
n{}
n
elements,
n
∈
N
n\in\mathbb{N}
n
∈
N
. Find the greatest integer
m
m
m
(
m
≥
2
)
(m\geq2)
(
m
≥
2
)
for which there exist
m
m
m
subsets of
X
X
X
such that each two of them are not disjoint.
7
1
Hide problems
dihedral angle between the planes $(MA_1C)$ and $(NA_1C)$
Let
A
B
C
D
A
1
B
1
C
1
D
1
ABCDA_1B_1C_1D_1
A
BC
D
A
1
B
1
C
1
D
1
be a cube. On the sides
A
B
AB{}
A
B
and
A
D
AD{}
A
D
there are the points
M
M{}
M
and
N
N{}
N
, respectively, such that
A
M
+
A
N
=
A
B
AM+AN=AB
A
M
+
A
N
=
A
B
. Show that the measure of the dihedral angle between the planes
(
M
A
1
C
)
(MA_1C)
(
M
A
1
C
)
and
(
N
A
1
C
)
(NA_1C)
(
N
A
1
C
)
doe not depend on the positions of
M
M{}
M
and
N
N{}
N
. Find this measure.
6
1
Hide problems
Find the smallest possible perimeter of this triangle.
In triangle
A
B
C
ABC
A
BC
the angle
C
C
C
is obtuse,
m
(
∠
A
)
=
2
m
(
∠
B
)
m(\angle A)=2m(\angle B)
m
(
∠
A
)
=
2
m
(
∠
B
)
and the sidelengths are integers. Find the smallest possible perimeter of this triangle.
5
1
Hide problems
Find all polynomials $P(X)$ of fourth degree with real coefficients that verify
Find all polynomials
P
(
X
)
P(X)
P
(
X
)
of fourth degree with real coefficients that verify the properties: a)
P
(
−
x
)
=
P
(
x
)
,
∀
x
∈
R
;
P(-x)=P(x), \forall x\in\mathbb{R};
P
(
−
x
)
=
P
(
x
)
,
∀
x
∈
R
;
b)
P
(
x
)
≥
0
,
∀
x
∈
R
;
P(x)\geq0, \forall x\in\mathbb{R};
P
(
x
)
≥
0
,
∀
x
∈
R
;
c)
P
(
0
)
=
1
;
P(0)=1;
P
(
0
)
=
1
;
d)
P
(
X
)
P(X)
P
(
X
)
has exactly two local minimums
x
1
x_1
x
1
and
x
2
x_2
x
2
such that
∣
x
1
−
x
2
∣
=
2.
|x_1-x_2|=2.
∣
x
1
−
x
2
∣
=
2.
3
1
Hide problems
Prove that $\frac{\sin(B+C)}{\sin B \cdot \sin C} \geq \frac{2}{3}.$
In triangle
A
B
C
ABC
A
BC
medians from
B
B
B
and
C
C
C
are perpendicular. Prove that
sin
(
B
+
C
)
sin
B
⋅
sin
C
≥
2
3
.
\frac{\sin(B+C)}{\sin B \cdot \sin C} \geq \frac{2}{3}.
s
i
n
B
⋅
s
i
n
C
s
i
n
(
B
+
C
)
≥
3
2
.
2
1
Hide problems
Prove that $\angle AMC=\angle BND$ and $\angle ANC=\angle BMD$
Circles
S
1
S_1{}
S
1
and
S
2
S_2{}
S
2
intersect in
M
M{}
M
and
N
N{}
N
. Line
l
l
l
intersects the circles in points
A
,
B
∈
S
1
A,B\in S_1
A
,
B
∈
S
1
and
C
,
D
∈
S
2
C,D\in S_2
C
,
D
∈
S
2
. Prove that
∠
A
M
C
=
∠
B
N
D
\angle AMC=\angle BND
∠
A
MC
=
∠
BN
D
and
∠
A
N
C
=
∠
B
M
D
\angle ANC=\angle BMD
∠
A
NC
=
∠
BM
D
if the order of points on line
l
l
l
is: a) A,C,B,D; b)
A
,
C
,
D
,
B
.
A,C,D,B.
A
,
C
,
D
,
B
.
1
1
Hide problems
Prove that $2^{n-k}$ divides $n!$
The number
n
n{}
n
cointains
k
k{}
k
units in binary system. Prove that
2
n
−
k
2^{n-k}{}
2
n
−
k
divides
n
!
n!
n
!
.
9
1
Hide problems
nice inequality
Let
x
1
,
x
2
,
.
.
.
,
x
n
∈
[
0
;
1
]
x_1,x_2,...,x_n \in [0;1]
x
1
,
x
2
,
...
,
x
n
∈
[
0
;
1
]
prove that
x
1
(
1
−
x
2
)
+
x
2
(
1
−
x
3
)
+
.
.
.
+
x
n
−
1
(
1
−
x
n
)
+
x
n
(
1
−
x
1
)
≤
[
n
2
]
x_1(1-x_2)+x_2(1-x_3)+...+x_{n-1}(1-x_n)+x_n(1-x_1) \leq [\frac{n}{2}]
x
1
(
1
−
x
2
)
+
x
2
(
1
−
x
3
)
+
...
+
x
n
−
1
(
1
−
x
n
)
+
x
n
(
1
−
x
1
)
≤
[
2
n
]
12
1
Hide problems
Friendly pairs
Suppose that in a certain society, each pair of persons can be classified as either amicable or hostile. We shall say that each member of an amicable pair is a friend of the other, and each member of a hostile pair is a foe of the other. Suppose that the society has
n
\, n \,
n
persons and
q
\, q \,
q
amicable pairs, and that for every set of three persons, at least one pair is hostile. Prove that there is at least one member of the society whose foes include
q
(
1
−
4
q
/
n
2
)
\, q(1 - 4q/n^2) \,
q
(
1
−
4
q
/
n
2
)
or fewer amicable pairs.
10
1
Hide problems
there exists a unique point P equidistant from A and B'
Given an equilateral triangle
A
B
C
ABC
A
BC
and a point
M
M
M
in the plane (
A
B
C
ABC
A
BC
). Let
A
′
,
B
′
,
C
′
A', B', C'
A
′
,
B
′
,
C
′
be respectively the symmetric through
M
M
M
of
A
,
B
,
C
A, B, C
A
,
B
,
C
. I. Prove that there exists a unique point
P
P
P
equidistant from
A
A
A
and
B
′
B'
B
′
, from
B
B
B
and
C
′
C'
C
′
and from
C
C
C
and
A
′
A'
A
′
. II. Let
D
D
D
be the midpoint of the side
A
B
AB
A
B
. When
M
M
M
varies (
M
M
M
does not coincide with
D
D
D
), prove that the circumcircle of triangle
M
N
P
MNP
MNP
(
N
N
N
is the intersection of the line
D
M
DM
D
M
and
A
P
AP
A
P
) pass through a fixed point.