MathDB
Problems
Contests
National and Regional Contests
Moldova Contests
Moldova Team Selection Test
1995 Moldova Team Selection Test
1995 Moldova Team Selection Test
Part of
Moldova Team Selection Test
Subcontests
(8)
9
1
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Find the smallest possible value of $a_{10}.$
For every nonempty set
M
M{}
M
of integers denote
S
(
M
)
S(M)
S
(
M
)
the sum of all its elements. Let
A
=
{
a
1
,
a
2
,
…
,
a
11
}
A=\{a_1,a_2,\ldots,a_{11}\}
A
=
{
a
1
,
a
2
,
…
,
a
11
}
be a set of positive integers with the properties: 1)
a
1
<
a
2
<
…
<
a
11
;
a_1<a_2<\ldots<a_{11};
a
1
<
a
2
<
…
<
a
11
;
2) for every positive integer
n
≤
1500
n\leq 1500
n
≤
1500
there is a subset
M
M{}
M
of
A
A{}
A
for which
S
(
M
)
=
n
.
S(M)=n.
S
(
M
)
=
n
.
Find the smallest possible value of
a
10
.
a_{10}.
a
10
.
8
1
Hide problems
Prove that points $M, N$ and $P$ are collinear.
Each pair of three circles have the common chords
A
A
1
,
B
B
1
AA_1, BB_1
A
A
1
,
B
B
1
and
C
C
1
CC_1{}
C
C
1
such that lines
A
B
AB{}
A
B
and
A
1
B
1
A_1B_1
A
1
B
1
intersect in point
M
M{}
M
,
B
C
BC
BC
and
B
1
C
1
B_1C_1
B
1
C
1
intersect in point
N
N{}
N
,
C
A
CA{}
C
A
and
C
1
A
1
C_1A_1
C
1
A
1
intersect in point
P
P{}
P
. Prove that points
M
,
N
M, N
M
,
N
and
P
P
P
are collinear.
7
1
Hide problems
What are the possible values of $m$?
Let
S
=
{
a
1
,
a
2
,
…
,
a
n
}
S=\{a_1,a_2,\ldots,a_n\}
S
=
{
a
1
,
a
2
,
…
,
a
n
}
of nenul vectors in a plane. Show that
S
S{}
S
can be partitioned in nenul subsets
B
1
,
B
2
,
…
,
B
m
B_1, B_2,\ldots, B_m
B
1
,
B
2
,
…
,
B
m
with the properties: 1) each vector from
S
S{}
S
is part of only on subset; 2) if
a
i
∈
B
j
a_i\in B_j
a
i
∈
B
j
then the angle between vectors
a
i
a_i
a
i
and
c
j
c_j
c
j
, which is the sum of all vectors from
B
j
B_j
B
j
is not greater than
π
2
\frac{\pi}{2}
2
π
; 3) if
i
≠
j
i\neq j
i
=
j
then the angle between vectors
c
i
c_i
c
i
and
c
j
c_j
c
j
, which is the sum of all vectors from
B
i
B_i
B
i
and
B
j
B_j
B
j
, respectively, is greater than
π
2
\frac{\pi}{2}
2
π
. What are the possible values of
m
m
m
?
6
1
Hide problems
Prove that all these perpendiculars are concurrent.
On a spherical surface there is a set
M
M{}
M
with
n
n{}
n
points with the property: for every point
A
A{}
A
from
M
M{}
M
there exist points
B
B
B
and
C
C
C
from
M
M{}
M
such that the triangle
A
B
C
ABC
A
BC
is equilateral. For every equilateral triangle with vertexes in
M
M{}
M
the perpendicular on its plane that goes through the geometric center of the other points from
M
M{}
M
. Prove that all these perpendiculars are concurrent.
3
1
Hide problems
the circumcircles of $BCB_1, CAC_1$ and $ABA_1$ are congruent
Let
A
B
C
ABC
A
BC
be a triangle with the medians
A
A
1
,
B
B
1
AA_1, BB_1
A
A
1
,
B
B
1
and
C
C
1
CC_1{}
C
C
1
. Prove that if the circumcircles of
B
C
B
1
,
C
A
C
1
BCB_1, CAC_1
BC
B
1
,
C
A
C
1
and
A
B
A
1
ABA_1
A
B
A
1
are congruent then
A
B
C
ABC
A
BC
is equilateral.
2
1
Hide problems
Prove that the equation has $x^2-x+3-ps=0$ with $x,s\in\mathbb{Z}$ has solutions
Let
p
p{}
p
be a prime number. Prove that the equation has
x
2
−
x
+
3
−
p
s
=
0
x^2-x+3-ps=0
x
2
−
x
+
3
−
p
s
=
0
with
x
,
s
∈
Z
x,s\in\mathbb{Z}
x
,
s
∈
Z
has solutions if and only if the equation
y
2
−
y
+
25
−
p
t
=
0
y^2-y+25-pt=0
y
2
−
y
+
25
−
pt
=
0
with
y
,
t
∈
Z
y,t\in\mathbb{Z}
y
,
t
∈
Z
has solutions.
1
1
Hide problems
$\sum_{k=1}^{n} cos^{2m} \frac{k\pi}{2n+1}$
Prove that for any positive integers
m
m{}
m
and
n
n{}
n
the number
∑
k
=
1
n
c
o
s
2
m
k
π
2
n
+
1
\sum_{k=1}^{n} cos^{2m} \frac{k\pi}{2n+1}
∑
k
=
1
n
co
s
2
m
2
n
+
1
kπ
is not an integer.
4
1
Hide problems
functional equation on Z
Find all functions
f
:
Z
→
Z
f:\mathbb{Z}\rightarrow \mathbb{Z}
f
:
Z
→
Z
satisfying the following:
i
)
i)
i
)
f
(
1
)
=
1
f(1)=1
f
(
1
)
=
1
;
i
i
)
ii)
ii
)
f
(
m
+
n
)
(
f
(
m
)
−
f
(
n
)
)
=
f
(
m
−
n
)
(
f
(
m
)
+
f
(
n
)
)
f(m+n)(f(m)-f(n))=f(m-n)(f(m)+f(n))
f
(
m
+
n
)
(
f
(
m
)
−
f
(
n
))
=
f
(
m
−
n
)
(
f
(
m
)
+
f
(
n
))
for all
m
,
n
∈
Z
m,n \in \mathbb{Z}
m
,
n
∈
Z
.