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Problems
Contests
National and Regional Contests
Moldova Contests
Moldova Team Selection Test
2022 Moldova Team Selection Test
2022 Moldova Team Selection Test
Part of
Moldova Team Selection Test
Subcontests
(12)
12
1
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Prove that for every prime number $p$ the number $x_p-1$ is divisible by $3p.$
Let
(
x
n
)
n
≥
1
(x_n)_{n\geq1}
(
x
n
)
n
≥
1
be a sequence that verifies: x_1=1, x_2=7, x_{n+1}=x_n+3x_{n-1}, \forall n \geq 2. Prove that for every prime number
p
p
p
the number
x
p
−
1
x_p-1
x
p
−
1
is divisible by
3
p
.
3p.
3
p
.
11
1
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Prove that $P$, $A$, and $Q$ are collinear.
Let
Ω
\Omega
Ω
be the circumcircle of triangle
A
B
C
ABC
A
BC
such that the tangents to
Ω
\Omega
Ω
in points
B
B
B
and
C
C
C
intersect in
P
P
P
. The squares
A
B
B
1
B
2
ABB_1B_2
A
B
B
1
B
2
and
A
C
C
1
C
2
ACC_1C_2
A
C
C
1
C
2
are constructed on the sides
A
B
AB
A
B
and
A
C
AC
A
C
in the exterior of triangle
A
B
C
ABC
A
BC
, such that the lines
B
1
B
2
B_1B_2
B
1
B
2
and
C
1
C
2
C_1C_2
C
1
C
2
intersect in point
Q
Q
Q
. Prove that
P
P
P
,
A
A
A
, and
Q
Q
Q
are collinear.
10
1
Hide problems
P\left(\frac{x_1}{x_2} \right)^2+P\left(\frac{x_2}{x_3} \right)^2+ ... +P\left(\
Let
P
(
X
)
P(X)
P
(
X
)
be a polynomial with positive coefficients. Show that for every integer
n
≥
2
n \geq 2
n
≥
2
and every
n
n
n
positive numbers
x
1
,
x
2
,
.
.
.
,
x
n
x_1, x_2,..., x_n
x
1
,
x
2
,
...
,
x
n
the following inequality is true:
P
(
x
1
x
2
)
2
+
P
(
x
2
x
3
)
2
+
.
.
.
+
P
(
x
n
x
1
)
2
≥
n
⋅
P
(
1
)
2
.
P\left(\frac{x_1}{x_2} \right)^2+P\left(\frac{x_2}{x_3} \right)^2+ ... +P\left(\frac{x_n}{x_1} \right)^2 \geq n \cdot P(1)^2.
P
(
x
2
x
1
)
2
+
P
(
x
3
x
2
)
2
+
...
+
P
(
x
1
x
n
)
2
≥
n
⋅
P
(
1
)
2
.
When does the equality take place?
9
1
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Find all positive integers $n$, for which the grid always will cointain at least
Let
n
n
n
be a positive integer. A grid of dimensions
n
×
n
n \times n
n
×
n
is divided in
n
2
n^2
n
2
1
×
1
1 \times 1
1
×
1
squares. Every segment of length
1
1
1
(side of a square) from this grid is coloured in blue or red. The number of red segments is not greater than
n
2
n^2
n
2
. Find all positive integers
n
n
n
, for which the grid always will cointain at least one
1
×
1
1 \times 1
1
×
1
square which has at least three blue sides.
8
1
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the union of every $k$ sets contains exactly $k+1$ points
a) Let
n
n
n
(
n
≥
2
)
(n \geq 2)
(
n
≥
2
)
be an integer. On a line there are
n
n
n
distinct (pairwise distinct) sets of points, such that for every integer
k
k
k
(
1
≤
k
≤
n
)
(1 \leq k \leq n)
(
1
≤
k
≤
n
)
the union of every
k
k
k
sets contains exactly
k
+
1
k+1
k
+
1
points. Show that there is always a point that belongs to every set. b) Is the same conclusion true if there is an infinity of distinct sets of points such that for every positive integer
k
k
k
the union of every
k
k
k
sets contains exactly
k
+
1
k+1
k
+
1
points?
7
1
Hide problems
Find the prime number $p$, for which the sum of the digits of the number $f(p^2+
Let
f
:
N
→
N
,
f:\mathbb{N} \rightarrow \mathbb{N},
f
:
N
→
N
,
f
(
n
)
=
n
2
−
69
n
+
2250
f(n)=n^2-69n+2250
f
(
n
)
=
n
2
−
69
n
+
2250
be a function. Find the prime number
p
p
p
, for which the sum of the digits of the number
f
(
p
2
+
32
)
f(p^2+32)
f
(
p
2
+
32
)
is as small as possible.
6
1
Hide problems
Prove that the lines $BP$ and $CQ$ are perpendicular.
Let
A
A
A
be a point outside of the circle
Ω
\Omega
Ω
. Tangents from
A
A
A
touch
Ω
\Omega
Ω
in points
B
B
B
and
C
C
C
. Point
C
C
C
, collinear with
A
A
A
and
P
P
P
, is between
A
A
A
and
P
P
P
, such that the circumcircle of triangle
A
B
P
ABP
A
BP
intersects
Ω
\Omega
Ω
again in point
E
E
E
. Point
Q
Q
Q
is on the segment
B
P
BP
BP
, such that
∠
P
E
Q
=
2
⋅
∠
A
P
B
\angle PEQ=2 \cdot \angle APB
∠
PEQ
=
2
⋅
∠
A
PB
. Prove that the lines
B
P
BP
BP
and
C
Q
CQ
CQ
are perpendicular.
5
1
Hide problems
f(n+2)-2022 \cdot f(n+1)+2021 \cdot f(n)=0
The function
f
:
N
→
N
f:\mathbb{N} \rightarrow \mathbb{N}
f
:
N
→
N
verifies:
1
)
f
(
n
+
2
)
−
2022
⋅
f
(
n
+
1
)
+
2021
⋅
f
(
n
)
=
0
,
∀
n
∈
N
;
1) f(n+2)-2022 \cdot f(n+1)+2021 \cdot f(n)=0, \forall n \in \mathbb{N};
1
)
f
(
n
+
2
)
−
2022
⋅
f
(
n
+
1
)
+
2021
⋅
f
(
n
)
=
0
,
∀
n
∈
N
;
2
)
f
(
2
0
22
)
=
f
(
2
2
20
)
;
2) f(20^{22})=f(22^{20});
2
)
f
(
2
0
22
)
=
f
(
2
2
20
)
;
3
)
f
(
2021
)
=
2022
3) f(2021)=2022
3
)
f
(
2021
)
=
2022
. Find all possible values of
f
(
2022
)
f(2022)
f
(
2022
)
.
4
1
Hide problems
Prove that lines $AH$, $XY$ and $DE$ are concurrent
In the acute triangle
A
B
C
ABC
A
BC
the point
M
M
M
is on the side
B
C
BC
BC
. The inscribed circle of triangle
A
B
M
ABM
A
BM
touches the sides
B
M
BM
BM
,
M
A
MA
M
A
and
A
B
AB
A
B
in points
D
D
D
,
E
E
E
and
F
F
F
, and the inscribed circle of triangle
A
C
M
ACM
A
CM
touches the sides
C
M
CM
CM
,
M
A
MA
M
A
and
A
C
AC
A
C
in points
X
X
X
,
Y
Y
Y
and
Z
Z
Z
. The lines
F
D
FD
F
D
and
Z
X
ZX
ZX
intersect in point
H
H
H
. Prove that lines
A
H
AH
A
H
,
X
Y
XY
X
Y
and
D
E
DE
D
E
are concurrent.
3
1
Hide problems
changing the sign of every number divisible by $m$
Let
n
n
n
be a positive integer. On a board there are written all integers from
1
1
1
to
n
n
n
. Alina does
n
n
n
moves consecutively: for every integer
m
m
m
(
1
≤
m
≤
n
)
(1 \leq m \leq n)
(
1
≤
m
≤
n
)
the move
m
m
m
consists in changing the sign of every number divisible by
m
m
m
. At the end Alina sums the numbers. Find this sum.
2
1
Hide problems
a^4+b^4+c^4+d^4+4(a+b+c+d)^2
Real numbers
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
satisfy
a
2
+
b
2
+
c
2
+
d
2
=
4.
a^2+b^2+c^2+d^2=4.
a
2
+
b
2
+
c
2
+
d
2
=
4.
Find the greatest possible value of
E
(
a
,
b
,
c
,
d
)
=
a
4
+
b
4
+
c
4
+
d
4
+
4
(
a
+
b
+
c
+
d
)
2
.
E(a,b,c,d)=a^4+b^4+c^4+d^4+4(a+b+c+d)^2 .
E
(
a
,
b
,
c
,
d
)
=
a
4
+
b
4
+
c
4
+
d
4
+
4
(
a
+
b
+
c
+
d
)
2
.
1
1
Hide problems
n^{5n-1}+n^{5n-2}+n^{5n-3}+n+1
Show that for every integer
n
≥
2
n \geq 2
n
≥
2
the number
a
=
n
5
n
−
1
+
n
5
n
−
2
+
n
5
n
−
3
+
n
+
1
a=n^{5n-1}+n^{5n-2}+n^{5n-3}+n+1
a
=
n
5
n
−
1
+
n
5
n
−
2
+
n
5
n
−
3
+
n
+
1
is a composite number.