MathDB
Problems
Contests
National and Regional Contests
Moldova Contests
Moldova Team Selection Test
2021 Moldova Team Selection Test
2021 Moldova Team Selection Test
Part of
Moldova Team Selection Test
Subcontests
(10)
10
1
Hide problems
On a board there are written the integers from 1 to 119
On a board there are written the integers from
1
1
1
to
119
119
119
. Two players,
A
A
A
and
B
B
B
, make a move by turn. A
m
o
v
e
move
m
o
v
e
consists in erasing
9
9
9
numbers from the board. The player after whose move two numbers remain on the board wins and his score is equal with the positive difference of the two remaining numbers. The player
A
A
A
makes the first move. Find the highest integer
k
k
k
, such that the player
A
A
A
can be sure that his score is not smaller than
k
k
k
.
12
1
Hide problems
$n!\cdot(n+1)!\cdot(n+2)!$ divides $(3n)!$
Prove that
n
!
⋅
(
n
+
1
)
!
⋅
(
n
+
2
)
!
n!\cdot(n+1)!\cdot(n+2)!
n
!
⋅
(
n
+
1
)!
⋅
(
n
+
2
)!
divides
(
3
n
)
!
(3n)!
(
3
n
)!
for every integer
n
≥
3
n \geq 3
n
≥
3
.
11
1
Hide problems
Prove that the lines KH and CD are perpendicular
In a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
the angles
B
A
D
BAD
B
A
D
and
B
C
D
BCD
BC
D
are equal. Points
M
M
M
and
N
N
N
lie on the sides
(
A
B
)
(AB)
(
A
B
)
and
(
B
C
)
(BC)
(
BC
)
such that the lines
M
N
MN
MN
and
A
D
AD
A
D
are parallel and
M
N
=
2
A
D
MN=2AD
MN
=
2
A
D
. The point
H
H
H
is the orthocenter of the triangle
A
B
C
ABC
A
BC
and the point
K
K
K
is the midpoint of
M
N
MN
MN
. Prove that the lines
K
H
KH
KH
and
C
D
CD
C
D
are perpendicular.
9
1
Hide problems
\frac{a^3}{1-a^2}+\frac{b^3}{1-b^2}+\frac{c^3}{1-c^2}
Positive real numbers
a
a
a
,
b
b
b
,
c
c
c
satisfy
a
+
b
+
c
=
1
a+b+c=1
a
+
b
+
c
=
1
. Find the smallest possible value of
E
(
a
,
b
,
c
)
=
a
3
1
−
a
2
+
b
3
1
−
b
2
+
c
3
1
−
c
2
.
E(a,b,c)=\frac{a^3}{1-a^2}+\frac{b^3}{1-b^2}+\frac{c^3}{1-c^2}.
E
(
a
,
b
,
c
)
=
1
−
a
2
a
3
+
1
−
b
2
b
3
+
1
−
c
2
c
3
.
6
1
Hide problems
What is the highest possible number of draws in the tournament
There are
14
14
14
players participating at a chess tournament, each playing one game with every other player. After the end of the tournament, the players were ranked in descending order based on their points. The sum of the points of the first three players is equal with the sum of the points of the last nine players. What is the highest possible number of draws in the tournament.(For a victory the player gets
1
1
1
point, for a loss
0
0
0
points, in a draw both players get
0
,
5
0,5
0
,
5
points.)
7
1
Hide problems
\frac{a+1}{\sqrt{a+bc}}+\frac{b+1}{\sqrt{b+ca}}+\frac{c+1}{\sqrt{c+ab}}
Positive real numbers
a
a
a
,
b
b
b
,
c
c
c
satisfy
a
+
b
+
c
=
1
a+b+c=1
a
+
b
+
c
=
1
. Show that
a
+
1
a
+
b
c
+
b
+
1
b
+
c
a
+
c
+
1
c
+
a
b
≥
2
a
2
+
b
2
+
c
2
.
\frac{a+1}{\sqrt{a+bc}}+\frac{b+1}{\sqrt{b+ca}}+\frac{c+1}{\sqrt{c+ab}} \geq \frac{2}{a^2+b^2+c^2}.
a
+
b
c
a
+
1
+
b
+
c
a
b
+
1
+
c
+
ab
c
+
1
≥
a
2
+
b
2
+
c
2
2
.
When does the equality take place?
5
1
Hide problems
f(M)=MA^n+MB^n+MC^n
Let
A
B
C
ABC
A
BC
be an equilateral triangle. Find all positive integers
n
n
n
, for which the function
f
f
f
, defined on all points
M
M
M
from the circle
S
S
S
circumscribed to triangle
A
B
C
ABC
A
BC
, defined by the formula
f
:
S
→
R
,
f
(
M
)
=
M
A
n
+
M
B
n
+
M
C
n
f:S \rightarrow R, f(M)=MA^n+MB^n+MC^n
f
:
S
→
R
,
f
(
M
)
=
M
A
n
+
M
B
n
+
M
C
n
, is a constant function.
3
1
Hide problems
Show that the circumcenter of triangle MDN lies on the line BC
Acute triangle
A
B
C
ABC
A
BC
with
A
B
>
B
C
AB>BC
A
B
>
BC
is inscribed in circle
Ω
\Omega
Ω
. Points
D
D
D
and
E
E
E
, that lie on
(
B
C
)
(BC)
(
BC
)
and
(
A
B
)
(AB)
(
A
B
)
are the feet of altitudes from
A
A
A
and
C
C
C
in triangle
A
B
C
ABC
A
BC
, and
M
M
M
is the midpoint of the segment
D
E
DE
D
E
. Half-line
(
A
M
(AM
(
A
M
intersects the circle
Ω
\Omega
Ω
for the second time in
N
N
N
. Show that the circumcenter of triangle
M
D
N
MDN
M
D
N
lies on the line
B
C
BC
BC
.
4
1
Hide problems
Let $n$ be a positive integer. A panel of dimenisions $2n\times2n$ is divided in
Let
n
n
n
be a positive integer. A panel of dimenisions
2
n
×
2
n
2n\times2n
2
n
×
2
n
is divided in
4
n
2
4n^2
4
n
2
squares with dimensions
1
×
1
1\times1
1
×
1
. What is the highest possible number of diagonals that can be drawn in
1
×
1
1\times1
1
×
1
squares, such that each two diagonals have no common points.
2
1
Hide problems
p+p^2+p^3+...+p^q=q+q^2+q^3+...q^p , p=q
Prove that if
p
p
p
and
q
q
q
are two prime numbers, such that
p
+
p
2
+
p
3
+
.
.
.
+
p
q
=
q
+
q
2
+
q
3
+
.
.
.
+
q
p
,
p+p^2+p^3+...+p^q=q+q^2+q^3+...+q^p,
p
+
p
2
+
p
3
+
...
+
p
q
=
q
+
q
2
+
q
3
+
...
+
q
p
,
then
p
=
q
p=q
p
=
q
.