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Problems
Contests
National and Regional Contests
Moldova Contests
Moldova Team Selection Test
2023 Moldova Team Selection Test
2023 Moldova Team Selection Test
Part of
Moldova Team Selection Test
Subcontests
(12)
11
1
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if $m+n\in A$ then $m\cdot n\in A.$
Find all sets
A
A
A
of nonnegative integers with the property: if for the nonnegative intergers
m
m
m
and
n
n
n
we have
m
+
n
∈
A
m+n\in A
m
+
n
∈
A
then
m
⋅
n
∈
A
.
m\cdot n\in A.
m
⋅
n
∈
A
.
10
1
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Prove that line $QM$ passes through the midpoint of $AC.$
Let
A
B
C
ABC
A
BC
be a triangle with
∠
A
C
B
=
90
\angle ACB=90
∠
A
CB
=
90
and
A
C
>
B
C
.
AC>BC.
A
C
>
BC
.
Let
Ω
\Omega
Ω
be the circumcircle of
A
B
C
.
ABC.
A
BC
.
Point
D
D
D
is the midpoint of small arc
A
C
AC
A
C
of
Ω
.
\Omega.
Ω.
Point
M
M
M
is symmetric with
A
A
A
with respect to
D
.
D.
D
.
Point
N
N
N
is the midpoint of
M
C
.
MC.
MC
.
Line
A
N
AN
A
N
intersects
Ω
\Omega
Ω
in point
P
P
P
and line
B
P
BP
BP
intersects line
D
N
DN
D
N
in point
Q
.
Q.
Q
.
Prove that line
Q
M
QM
QM
passes through the midpoint of
A
C
.
AC.
A
C
.
9
1
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E=\frac{a_1}{1+a_1^2}+\frac{a_2}{1+a_2^2}+\ldots+\frac{a_n}{1+a_n^2}
Let
n
n
n
(
n
≥
2
)
(n\geq2)
(
n
≥
2
)
be an integer. Find the greatest possible value of the expression
E
=
a
1
1
+
a
1
2
+
a
2
1
+
a
2
2
+
…
+
a
n
1
+
a
n
2
E=\frac{a_1}{1+a_1^2}+\frac{a_2}{1+a_2^2}+\ldots+\frac{a_n}{1+a_n^2}
E
=
1
+
a
1
2
a
1
+
1
+
a
2
2
a
2
+
…
+
1
+
a
n
2
a
n
if the positive real numbers
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\ldots,a_n
a
1
,
a
2
,
…
,
a
n
satisfy
a
1
+
a
2
+
…
+
a
n
=
n
2
.
a_1+a_2+\ldots+a_n=\frac{n}{2}.
a
1
+
a
2
+
…
+
a
n
=
2
n
.
What are the values of
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\ldots,a_n
a
1
,
a
2
,
…
,
a
n
when the greatest value is achieved?
8
1
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Prove that the lines $EG, DF$ and $BC$ are concurrent
Let
A
B
C
ABC
A
BC
be an acute triangle with orthocenter
H
H
H
and
A
B
<
A
C
.
AB<AC.
A
B
<
A
C
.
Let
Ω
1
\Omega_1
Ω
1
be a circle with diameter
A
C
AC
A
C
and
Ω
2
\Omega_2
Ω
2
a circle with diameter
A
B
.
AB.
A
B
.
Line
B
H
BH
B
H
intersects
Ω
1
\Omega_1
Ω
1
in points
D
D
D
and
E
E
E
such that
E
E
E
is not on segment
B
H
.
BH.
B
H
.
Line
C
H
CH
C
H
intersects
Ω
2
\Omega_2
Ω
2
in points
F
F
F
and
G
G
G
such that
G
G
G
is not on segment
C
H
.
CH.
C
H
.
Prove that the lines
E
G
,
D
F
EG, DF
EG
,
D
F
and
B
C
BC
BC
are concurrent.
7
1
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one of the disks contains the center of another disk
Find all integers
n
n
n
(
n
≥
2
)
(n\geq2)
(
n
≥
2
)
with the property: for every
n
n
n
distinct disks in a plane with at least a common point one of the disks contains the center of another disk.
6
1
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x_1^2+3x_2^2+5x_3^2+\cdots+(2\cdot2023-1)\cdot x^2_{2023}
Show that if
2023
2023
2023
real numbers
x
1
,
x
2
,
…
,
x
2023
x_1,x_2,\dots,x_{2023}
x
1
,
x
2
,
…
,
x
2023
satisfy
x
1
≥
x
2
≥
⋯
≥
x
2023
≥
0
,
x_1\geq x_2\geq\dots\geq x_{2023}\geq0,
x
1
≥
x
2
≥
⋯
≥
x
2023
≥
0
,
then
x
1
2
+
3
x
2
2
+
5
x
3
2
+
⋯
+
(
2
⋅
2023
−
1
)
⋅
x
2023
2
≤
(
x
1
+
x
2
+
⋯
+
x
2023
)
2
.
x_1^2+3x_2^2+5x_3^2+\cdots+(2\cdot2023-1)\cdot x^2_{2023}\leq(x_1+x_2+\cdots+x_{2023})^2.
x
1
2
+
3
x
2
2
+
5
x
3
2
+
⋯
+
(
2
⋅
2023
−
1
)
⋅
x
2023
2
≤
(
x
1
+
x
2
+
⋯
+
x
2023
)
2
.
When does the equality take place?
5
1
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$m=1^{2k+1}+2^{2k+1}+\cdots+n^{2k+1}$
Find all pairs of positive integers
(
n
,
k
)
(n,k)
(
n
,
k
)
for which the number
m
=
1
2
k
+
1
+
2
2
k
+
1
+
⋯
+
n
2
k
+
1
m=1^{2k+1}+2^{2k+1}+\cdots+n^{2k+1}
m
=
1
2
k
+
1
+
2
2
k
+
1
+
⋯
+
n
2
k
+
1
is divisible by
n
+
2.
n+2.
n
+
2.
3
1
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A sequence $(a_1,a_2,\ldots,a_n)$ of length is called $balanced$
Let
n
n
n
be a positive integer. A sequence
(
a
1
,
a
2
,
…
,
a
n
)
(a_1,a_2,\ldots,a_n)
(
a
1
,
a
2
,
…
,
a
n
)
of length is called
b
a
l
a
n
c
e
d
balanced
ba
l
an
ce
d
if for every
k
k
k
(
1
≤
k
≤
n
)
(1\leq k\leq n)
(
1
≤
k
≤
n
)
the term
a
k
a_k
a
k
is equal with the number of distinct numbers from the subsequence
(
a
1
,
a
2
,
…
,
a
k
)
.
(a_1,a_2,\ldots,a_k).
(
a
1
,
a
2
,
…
,
a
k
)
.
a) How many balanced sequences
(
a
1
,
a
2
,
…
,
a
n
)
(a_1,a_2,\ldots,a_n)
(
a
1
,
a
2
,
…
,
a
n
)
of length
n
n
n
do exist? b) For every positive integer
m
m
m
find how many balanced sequences
(
a
1
,
a
2
,
…
,
a
n
)
(a_1,a_2,\ldots,a_n)
(
a
1
,
a
2
,
…
,
a
n
)
of length
n
n
n
exist such that
a
n
=
m
.
a_n=m.
a
n
=
m
.
4
1
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P_n(X)=P_{n-1}(X)+3P_{n-1}(X)\cdot P_{n-2}(X)+P_{n-2}(X)
Polynomials
(
P
n
(
X
)
)
n
∈
N
(P_n(X))_{n\in\mathbb{N}}
(
P
n
(
X
)
)
n
∈
N
are defined as: P_0(X)=0, P_1(X)=X+2, P_n(X)=P_{n-1}(X)+3P_{n-1}(X)\cdot P_{n-2}(X)+P_{n-2}(X), (\forall) n\geq2. Show that if
k
k
k
divides
m
m
m
then
P
k
(
X
)
P_k(X)
P
k
(
X
)
divides
P
m
(
X
)
.
P_m(X).
P
m
(
X
)
.
12
1
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Arithmetic sequence
The sequence
(
a
n
)
\left(a_n \right)
(
a
n
)
is defined by
a
1
=
1
,
a
2
=
2
a_1=1, \ a_2=2
a
1
=
1
,
a
2
=
2
and
a
n
+
2
=
2
a
n
+
1
−
p
a
n
,
∀
n
≥
1
,
a_{n+2} = 2a_{n+1}-pa_n, \ \forall n \ge 1,
a
n
+
2
=
2
a
n
+
1
−
p
a
n
,
∀
n
≥
1
,
for some prime
p
.
p.
p
.
Find all
p
p
p
for which there exists
m
m
m
such that
a
m
=
−
3.
a_m=-3.
a
m
=
−
3.
1
1
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All-Russian Olympiad Day 1 Problem 10.2.
Let
△
A
B
C
\triangle ABC
△
A
BC
be an acute-angled triangle with
A
B
<
A
C
AB<AC
A
B
<
A
C
. Let
M
M
M
and
N
N
N
be the midpoints of
A
B
AB
A
B
and
A
C
AC
A
C
, respectively; let
A
D
AD
A
D
be an altitude in this triangle. A point
K
K
K
is chosen on the segment
M
N
MN
MN
so that
B
K
=
C
K
BK=CK
B
K
=
C
K
. The ray
K
D
KD
KD
meets the circumcircle
Ω
\Omega
Ω
of
A
B
C
ABC
A
BC
at
Q
Q
Q
. Prove that
C
,
N
,
K
,
Q
C, N, K, Q
C
,
N
,
K
,
Q
are concyclic.
2
1
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NUMBER THORY
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be distinct positive integers and let
r
,
s
,
t
r,s,t
r
,
s
,
t
be positive integers such that:
a
b
+
1
=
r
2
,
a
c
+
1
=
s
2
,
b
c
+
1
=
t
2
ab+1=r^2,ac+1=s^2,bc+1=t^2
ab
+
1
=
r
2
,
a
c
+
1
=
s
2
,
b
c
+
1
=
t
2
Prove that it is not possible that all three fractions
r
t
s
,
r
s
t
,
s
t
r
\frac{rt}{s}, \frac{rs}{t}, \frac{st}{r}
s
r
t
,
t
rs
,
r
s
t
are integers.