MathDB
Problems
Contests
National and Regional Contests
Moldova Contests
Moldova Team Selection Test
2020 Moldova Team Selection Test
2020 Moldova Team Selection Test
Part of
Moldova Team Selection Test
Subcontests
(12)
12
1
Hide problems
Tournament combi
In a chess tournament each player played one match with every other player. It is known that all players have different scores. The player who is on the last place got
k
k
k
points. What is the smallest number of wins that the first placed player got? (For the win
1
1
1
point is given, for loss
0
0
0
and for a draw both players get
0
,
5
0,5
0
,
5
points.)
10
1
Hide problems
Find the greatest value
Let
n
n
n
be a positive integer. Positive numbers
a
a
a
,
b
b
b
,
c
c
c
satisfy
1
a
+
1
b
+
1
c
=
1
\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1
a
1
+
b
1
+
c
1
=
1
. Find the greatest possible value of
E
(
a
,
b
,
c
)
=
a
n
a
2
n
+
1
+
b
2
n
⋅
c
+
b
⋅
c
2
n
+
b
n
b
2
n
+
1
+
c
2
n
⋅
a
+
c
⋅
a
2
n
+
c
n
c
2
n
+
1
+
a
2
n
⋅
b
+
a
⋅
b
2
n
E(a,b,c)=\frac{a^{n}}{a^{2n+1}+b^{2n} \cdot c + b \cdot c^{2n}}+\frac{b^{n}}{b^{2n+1}+c^{2n} \cdot a + c \cdot a^{2n}}+\frac{c^{n}}{c^{2n+1}+a^{2n} \cdot b + a \cdot b^{2n}}
E
(
a
,
b
,
c
)
=
a
2
n
+
1
+
b
2
n
⋅
c
+
b
⋅
c
2
n
a
n
+
b
2
n
+
1
+
c
2
n
⋅
a
+
c
⋅
a
2
n
b
n
+
c
2
n
+
1
+
a
2
n
⋅
b
+
a
⋅
b
2
n
c
n
8
1
Hide problems
Show that DE and AC are perpendicular
In
Δ
A
B
C
\Delta ABC
Δ
A
BC
the angles
A
B
C
ABC
A
BC
and
A
C
B
ACB
A
CB
are acute. Let
M
M
M
be the midpoint of
A
B
AB
A
B
. Point
D
D
D
is on the half-line
(
C
B
(CB
(
CB
such that
B
∈
(
C
D
)
B \in (CD)
B
∈
(
C
D
)
and
∠
D
A
B
=
∠
B
C
M
\angle DAB= \angle BCM
∠
D
A
B
=
∠
BCM
. Perpendicular from
B
B
B
to line
C
D
CD
C
D
intersects the line bisector of
A
B
AB
A
B
in
E
E
E
. Prove that
D
E
DE
D
E
and
A
C
AC
A
C
are perpendicular.
6
1
Hide problems
Induction with divisibility
Let
n
n
n
,
(
n
≥
3
)
(n \geq3)
(
n
≥
3
)
be a positive integer and the polynomial
f
(
x
)
=
(
1
+
x
)
⋅
(
1
+
2
x
)
⋅
(
1
+
3
x
)
⋅
.
.
.
⋅
(
1
+
n
x
)
f(x)=(1+x) \cdot (1+2x) \cdot (1+3x) \cdot ... \cdot (1+nx)
f
(
x
)
=
(
1
+
x
)
⋅
(
1
+
2
x
)
⋅
(
1
+
3
x
)
⋅
...
⋅
(
1
+
n
x
)
=
a
0
+
a
1
⋅
x
+
a
2
⋅
x
2
+
a
3
⋅
x
3
+
.
.
.
+
a
n
⋅
x
n
= a_0+a_1 \cdot x+a_2 \cdot x^2+a_3 \cdot x^3+...+a_n \cdot x^n
=
a
0
+
a
1
⋅
x
+
a
2
⋅
x
2
+
a
3
⋅
x
3
+
...
+
a
n
⋅
x
n
. Show that the number
a
3
a_3
a
3
divides the number
k
=
C
n
+
1
2
⋅
(
2
⋅
C
n
2
⋅
C
n
+
1
2
−
3
⋅
a
2
)
.
k=C^2_{n+1} \cdot (2 \cdot C^2_n \cdot C^2_{n+1}-3 \cdot a_2).
k
=
C
n
+
1
2
⋅
(
2
⋅
C
n
2
⋅
C
n
+
1
2
−
3
⋅
a
2
)
.
5
1
Hide problems
Trigonometry with natural parameter
Let
n
n
n
be a natural number. Find all solutions
x
x
x
of the system of equations
{
s
i
n
x
+
c
o
s
x
=
n
2
t
g
x
2
=
n
−
2
3
\left\{\begin{matrix} sinx+cosx=\frac{\sqrt{n}}{2}\\tg\frac{x}{2}=\frac{\sqrt{n}-2}{3}\end{matrix}\right.
{
s
in
x
+
cos
x
=
2
n
t
g
2
x
=
3
n
−
2
On interval
[
0
,
π
4
)
.
\left[0,\frac{\pi}{4}\right).
[
0
,
4
π
)
.
7
1
Hide problems
Inequality
Show that for any positive real numbers
a
a
a
,
b
b
b
,
c
c
c
the following inequality takes place
a
7
a
2
+
b
2
+
c
2
+
b
a
2
+
7
b
2
+
c
2
+
c
a
2
+
b
2
+
7
c
2
≤
1.
\frac{a}{\sqrt{7a^2+b^2+c^2}}+\frac{b}{\sqrt{a^2+7b^2+c^2}}+\frac{c}{\sqrt{a^2+b^2+7c^2}} \leq 1.
7
a
2
+
b
2
+
c
2
a
+
a
2
+
7
b
2
+
c
2
b
+
a
2
+
b
2
+
7
c
2
c
≤
1.
4
1
Hide problems
Geometry with orthocenters
Let
Δ
A
B
C
\Delta ABC
Δ
A
BC
be an acute triangle and
H
H
H
its orthocenter.
B
1
B_1
B
1
and
C
1
C_1
C
1
are the feet of heights from
B
B
B
and
C
C
C
,
M
M
M
is the midpoint of
A
H
AH
A
H
. Point
K
K
K
is on the segment
B
1
C
1
B_1C_1
B
1
C
1
, but isn't on line
A
H
AH
A
H
. Line
A
K
AK
A
K
intersects the lines
M
B
1
MB_1
M
B
1
and
M
C
1
MC_1
M
C
1
in
E
E
E
and
F
F
F
, the lines
B
E
BE
BE
and
C
F
CF
CF
intersect at
N
N
N
. Prove that
K
K
K
is the orthocenter of
Δ
N
B
C
\Delta NBC
Δ
NBC
.
3
1
Hide problems
Combi with arrangements
Let
n
n
n
,
(
n
≥
3
)
(n \geq 3)
(
n
≥
3
)
be a positive integer and the set
A
A
A
={
1
,
2
,
.
.
.
,
n
1,2,...,n
1
,
2
,
...
,
n
}. All the elements of
A
A
A
are randomly arranged in a sequence
(
a
1
,
a
2
,
.
.
.
,
a
n
)
(a_1,a_2,...,a_n)
(
a
1
,
a
2
,
...
,
a
n
)
. The pair
(
a
i
,
a
j
)
(a_i,a_j)
(
a
i
,
a
j
)
forms an
i
n
v
e
r
s
i
o
n
inversion
in
v
ers
i
o
n
if
1
≤
i
≤
j
≤
n
1 \leq i \leq j \leq n
1
≤
i
≤
j
≤
n
and
a
i
>
a
j
a_i > a_j
a
i
>
a
j
. In how many different ways all the elements of the set
A
A
A
can be arranged in a sequence that contains exactly
3
3
3
inversions?
9
1
Hide problems
Show that angle BMN=90
Let
Δ
A
B
C
\Delta ABC
Δ
A
BC
be an acute triangle and
Ω
\Omega
Ω
its circumscribed circle, with diameter
A
P
AP
A
P
. Points
E
E
E
and
F
F
F
are the orthogonal projections from
B
B
B
on
A
C
AC
A
C
and
A
P
AP
A
P
, points
M
M
M
and
N
N
N
are the midpoints of segments
E
F
EF
EF
and
C
P
CP
CP
. Prove that
∠
B
M
N
=
90
\angle BMN=90
∠
BMN
=
90
.
1
1
Hide problems
Number theory with series
All members of geometrical progression
(
b
n
)
n
≥
1
(b_n)_{n\geq1}
(
b
n
)
n
≥
1
are members of some arithmetical progression. It is known that
b
1
b_1
b
1
is an integer. Prove that all members of this geometrical progression are integers. (progression is infinite)
2
1
Hide problems
Interesting inequality
Show that for any positive real numbers
a
a
a
,
b
b
b
,
c
c
c
the following inequality takes place
a
b
+
b
c
+
c
a
+
a
+
b
+
c
a
2
+
b
2
+
c
2
≥
3
+
3
\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{a+b+c}{\sqrt{a^2+b^2+c^2}} \geq 3+\sqrt{3}
b
a
+
c
b
+
a
c
+
a
2
+
b
2
+
c
2
a
+
b
+
c
≥
3
+
3
11
1
Hide problems
Very beautiful functional equation involving trigonometry
Find all functions
f
:
[
−
1
,
1
]
→
R
,
f:[-1,1] \rightarrow \mathbb{R},
f
:
[
−
1
,
1
]
→
R
,
which satisfy
f
(
sin
x
)
+
f
(
cos
x
)
=
2020
f(\sin{x})+f(\cos{x})=2020
f
(
sin
x
)
+
f
(
cos
x
)
=
2020
for any real number
x
.
x.
x
.