MathDB
Problems
Contests
National and Regional Contests
Moldova Contests
Moldova Team Selection Test
2012 Moldova Team Selection Test
2012 Moldova Team Selection Test
Part of
Moldova Team Selection Test
Subcontests
(12)
3
1
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the ratio between the areas of triangles $MNP$ and $ABC$ is $\frac{7}{30},$
Let
A
B
C
ABC
A
BC
be an equilateral triangle with
A
B
=
a
AB=a
A
B
=
a
and
M
∈
(
A
B
)
M\in(AB)
M
∈
(
A
B
)
a fixed point. Points
N
∈
(
A
C
)
N\in(AC)
N
∈
(
A
C
)
and
P
∈
(
B
C
)
P\in(BC)
P
∈
(
BC
)
are taken such that the perimeter of
M
N
P
MNP
MNP
is minimal. If the ratio between the areas of triangles
M
N
P
MNP
MNP
and
A
B
C
ABC
A
BC
is
7
30
,
\textstyle\frac{7}{30},
30
7
,
find the perimeter of triangle
M
N
P
.
MNP.
MNP
.
10
1
Hide problems
4a_1-2b_1=7
Let
f
:
R
→
R
,
f
(
x
,
y
)
=
x
2
−
2
y
.
f:\mathbb{R}\rightarrow\mathbb{R}, f(x,y)=x^2-2y.
f
:
R
→
R
,
f
(
x
,
y
)
=
x
2
−
2
y
.
Define the sequences
(
a
n
)
n
≥
1
(a_n)_{n\geq1}
(
a
n
)
n
≥
1
and
(
b
n
)
n
≥
1
(b_n)_{n\geq1}
(
b
n
)
n
≥
1
such that
a
n
+
1
=
f
(
a
n
,
b
n
)
,
b
n
+
1
=
f
(
b
n
,
a
n
)
.
a_{n+1}=f(a_n,b_n), b_{n+1}=f(b_n,a_n).
a
n
+
1
=
f
(
a
n
,
b
n
)
,
b
n
+
1
=
f
(
b
n
,
a
n
)
.
If
4
a
1
−
2
b
1
=
7
:
4a_1-2b_1=7 :
4
a
1
−
2
b
1
=
7
:
a) find the smallest
k
∈
N
k\in\mathbb{N}
k
∈
N
for which the number
p
=
2
k
⋅
(
2
512
a
9
−
b
9
)
p=2^k\cdot(2^{512}a_9-b_9)
p
=
2
k
⋅
(
2
512
a
9
−
b
9
)
is an integer. b) prove that
2
2
10
+
2
2
9
+
1
2^{2^{10}}+2^{2^9}+1
2
2
10
+
2
2
9
+
1
divides
p
.
p.
p
.
11
1
Hide problems
feet of perpendiculars from $ M $ to $AC, AB, BP$ and $CP$ lie on a circle
Let
A
B
C
ABC
A
BC
be an acute triangle,
M
M
M
the foot of the height from
A
A
A
and point
P
∈
(
M
A
)
P\in(MA)
P
∈
(
M
A
)
different from the orthocenter of
A
B
C
.
ABC.
A
BC
.
Prove that the feet of perpendiculars from
M
M
M
to
A
C
,
A
B
,
B
P
AC, AB, BP
A
C
,
A
B
,
BP
and
C
P
CP
CP
lie on a circle.
8
1
Hide problems
S_k=1^k+2^k+...+(p-1)^k
Let
p
≥
5
p\geq5
p
≥
5
be a prime and
S
k
=
1
k
+
2
k
+
.
.
.
+
(
p
−
1
)
k
,
∀
k
∈
N
.
S_k=1^k+2^k+...+(p-1)^k,\forall k\in\mathbb{N}.
S
k
=
1
k
+
2
k
+
...
+
(
p
−
1
)
k
,
∀
k
∈
N
.
Prove that there is an infinity of numbers
n
∈
N
n\in\mathbb{N}
n
∈
N
such that
p
3
p^3
p
3
divides
S
n
S_n
S
n
and
p
p
p
divides
S
n
−
1
S_{n-1}
S
n
−
1
and
S
n
−
2
.
S_{n-2}.
S
n
−
2
.
6
1
Hide problems
\sum_{k\in S}\left[\sqrt{\frac{n}{k}}\right]=n
Let
S
S
S
be the set of positive integers which are not divisible by perfect squares greater than
1.
1.
1.
Prove that for every
n
∈
N
n\in\mathbb{N}
n
∈
N
the following equality is true
∑
k
∈
S
[
n
k
]
=
n
,
\sum_{k\in S}\left[\sqrt{\frac{n}{k}}\right]=n,
k
∈
S
∑
[
k
n
]
=
n
,
where
[
x
]
[x]
[
x
]
is the integer part of
x
∈
R
.
x\in\mathbb{R}.
x
∈
R
.
5
1
Hide problems
\sqrt{m^2-6}<2\sqrt{n}-m<\sqrt{m^2-2}
Find all pairs
(
m
,
n
)
(m, n)
(
m
,
n
)
of integers for which
m
2
−
6
<
2
n
−
m
<
m
2
−
2
.
\sqrt{m^2-6}<2\sqrt{n}-m<\sqrt{m^2-2}.
m
2
−
6
<
2
n
−
m
<
m
2
−
2
.
9
1
Hide problems
\frac{a^4-a^2+1}{b^5}+\frac{b^4-b^2+1}{c^5}+\frac{c^4-c^2+1}{a^5}
Prove that for every numbers
a
,
b
,
c
>
0
a,b,c>0
a
,
b
,
c
>
0
the following inequality is true
a
4
−
a
2
+
1
b
5
+
b
4
−
b
2
+
1
c
5
+
c
4
−
c
2
+
1
a
5
≥
1
a
3
+
1
b
3
+
1
c
3
.
\frac{a^4-a^2+1}{b^5}+\frac{b^4-b^2+1}{c^5}+\frac{c^4-c^2+1}{a^5} \geq \frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}.
b
5
a
4
−
a
2
+
1
+
c
5
b
4
−
b
2
+
1
+
a
5
c
4
−
c
2
+
1
≥
a
3
1
+
b
3
1
+
c
3
1
.
7
1
Hide problems
Externally tangent circles and midpoints
Let
C
(
O
1
)
,
C
(
O
2
)
C(O_1),C(O_2)
C
(
O
1
)
,
C
(
O
2
)
be two externally tangent circles at point
P
P
P
. A line
t
t
t
is tangent to
C
(
O
1
)
C(O_1)
C
(
O
1
)
in point
R
R
R
and intersects
C
(
O
2
)
C(O_2)
C
(
O
2
)
in points
A
,
B
A,B
A
,
B
such that
A
A
A
is closer to
R
R
R
than
B
B
B
is. The line
A
O
1
AO_1
A
O
1
intersects the perpendicular to
t
t
t
in
B
B
B
at point
C
C
C
, the line
P
C
PC
PC
intersects
A
B
AB
A
B
in
Q
Q
Q
. Prove that
Q
O
1
QO_1
Q
O
1
passes through the midpoint of
B
C
BC
BC
.
12
1
Hide problems
Identity
Let
k
∈
N
k \in \mathbb{N}
k
∈
N
. Prove that
(
k
0
)
⋅
(
x
+
k
)
k
−
(
k
1
)
⋅
(
x
+
k
−
1
)
k
+
.
.
.
+
(
−
1
)
k
⋅
(
k
k
)
⋅
x
k
=
k
!
,
∀
k
∈
R
\binom{k}{0} \cdot (x+k)^k - \binom{k}{1} \cdot (x+k-1)^k+...+(-1)^k \cdot \binom{k}{k} \cdot x^k=k! ,\forall k \in \mathbb{R}
(
0
k
)
⋅
(
x
+
k
)
k
−
(
1
k
)
⋅
(
x
+
k
−
1
)
k
+
...
+
(
−
1
)
k
⋅
(
k
k
)
⋅
x
k
=
k
!
,
∀
k
∈
R
4
1
Hide problems
Coins on a circle.
Points
A
1
,
A
2
,
…
,
A
n
A_1, A_2,\ldots, A_n
A
1
,
A
2
,
…
,
A
n
are found on a circle in this order. Each point
A
i
A_i
A
i
has exactly
i
i
i
coins. A move consists in taking two coins from two points (may be the same point) and moving them to adjacent points (one move clockwise and another counter-clockwise). Find all possible values of
n
n
n
for which it is possible after a finite number of moves to obtain a configuration with each point
A
i
A_i
A
i
having
n
+
1
−
i
n+1-i
n
+
1
−
i
coins.
2
1
Hide problems
Minimum value
Positive integers
a
,
b
a,b
a
,
b
are such that
137
137
137
divides
a
+
139
b
a+139b
a
+
139
b
and
139
139
139
divides
a
+
137
b
a+137b
a
+
137
b
. Find the minimal posible value of
a
+
b
a+b
a
+
b
.
1
1
Hide problems
Polynomial factorization
Prove that polynomial
x
8
+
98
x
4
+
1
x^8+98x^4+1
x
8
+
98
x
4
+
1
can be factorized in
Z
[
X
]
Z[X]
Z
[
X
]
.