MathDB
Problems
Contests
National and Regional Contests
Moldova Contests
Moldova Team Selection Test
2015 Moldova Team Selection Test
2015 Moldova Team Selection Test
Part of
Moldova Team Selection Test
Subcontests
(4)
4
3
Hide problems
Coloring the vertices of a n-gon
In how many ways can we color exactly
k
k
k
vertices of an
n
n
n
-gon in red such that any
2
2
2
consecutive vertices are not both red. (Vertices are considered to be labeled)
Counting sets that contain their cardinality
Consider a positive integer
n
n
n
and
A
=
{
1
,
2
,
.
.
.
,
n
}
A = \{ 1,2,...,n \}
A
=
{
1
,
2
,
...
,
n
}
. Call a subset
X
⊆
A
X \subseteq A
X
⊆
A
perfect if
∣
X
∣
∈
X
|X| \in X
∣
X
∣
∈
X
. Call a perfect subset
X
X
X
minimal if it doesn't contain another perfect subset. Find the number of minimal subsets of
A
A
A
.
find number of functions...
Let
n
n
n
and
k
k
k
be positive integers, and let be the sets
X
=
{
1
,
2
,
3
,
.
.
.
,
n
}
X=\{1,2,3,...,n\}
X
=
{
1
,
2
,
3
,
...
,
n
}
and
Y
=
{
1
,
2
,
3
,
.
.
.
,
k
}
Y=\{1,2,3,...,k\}
Y
=
{
1
,
2
,
3
,
...
,
k
}
. Let
P
P
P
be the set of all the subsets of the set
X
X
X
. Find the number of functions
f
:
P
→
Y
f: P \to Y
f
:
P
→
Y
that satisfy
f
(
A
∩
B
)
=
min
(
f
(
A
)
,
f
(
B
)
)
f(A \cap B)=\min(f(A),f(B))
f
(
A
∩
B
)
=
min
(
f
(
A
)
,
f
(
B
))
for all
A
,
B
∈
P
A,B \in P
A
,
B
∈
P
.
3
3
Hide problems
Minimum value of expression with prime
Let
p
p
p
be a fixed odd prime. Find the minimum positive value of
E
p
(
x
,
y
)
=
2
p
−
x
−
y
E_{p}(x,y) = \sqrt{2p}-\sqrt{x}-\sqrt{y}
E
p
(
x
,
y
)
=
2
p
−
x
−
y
where
x
,
y
∈
Z
+
x,y \in \mathbb{Z}_{+}
x
,
y
∈
Z
+
.
Concurrency of lines involving altitudes
Consider an acute triangle
A
B
C
ABC
A
BC
, points
E
,
F
E,F
E
,
F
are the feet of the perpendiculars from
B
B
B
and
C
C
C
in
△
A
B
C
\triangle ABC
△
A
BC
. Points
I
I
I
and
J
J
J
are the projections of points
F
,
E
F,E
F
,
E
on the line
B
C
BC
BC
, points
K
,
L
K,L
K
,
L
are on sides
A
B
,
A
C
AB,AC
A
B
,
A
C
respectively such that
I
K
∥
A
C
IK \parallel AC
I
K
∥
A
C
and
J
L
∥
A
B
JL \parallel AB
J
L
∥
A
B
. Prove that the lines
I
E
IE
I
E
,
J
F
JF
J
F
,
K
L
KL
K
L
are concurrent.
a cute geometric inequality
The tangents to the inscribed circle of
△
A
B
C
\triangle ABC
△
A
BC
, which are parallel to the sides of the triangle and do not coincide with them, intersect the sides of the triangle in the points
M
,
N
,
P
,
Q
,
R
,
S
M,N,P,Q,R,S
M
,
N
,
P
,
Q
,
R
,
S
such that
M
,
S
∈
(
A
B
)
M,S\in (AB)
M
,
S
∈
(
A
B
)
,
N
,
P
∈
(
A
C
)
N,P\in (AC)
N
,
P
∈
(
A
C
)
,
Q
,
R
∈
(
B
C
)
Q,R\in (BC)
Q
,
R
∈
(
BC
)
. The interior angle bisectors of
△
A
M
N
\triangle AMN
△
A
MN
,
△
B
S
R
\triangle BSR
△
BSR
and
△
C
P
Q
\triangle CPQ
△
CPQ
, from points
A
,
B
A,B
A
,
B
and respectively
C
C
C
have lengths
l
1
l_{1}
l
1
,
l
2
l_{2}
l
2
and
l
3
l_{3}
l
3
.\\ Prove the inequality:
1
l
1
2
+
1
l
2
2
+
1
l
3
2
≥
81
p
2
\frac {1}{l^{2}_{1}}+\frac {1}{l^{2}_{2}}+\frac {1}{l^{2}_{3}} \ge \frac{81}{p^{2}}
l
1
2
1
+
l
2
2
1
+
l
3
2
1
≥
p
2
81
where
p
p
p
is the semiperimeter of
△
A
B
C
\triangle ABC
△
A
BC
.
2
3
Hide problems
Inequality with product equal to 1
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers such that
a
b
c
=
1
abc=1
ab
c
=
1
. Prove the following inequality: \\
a
3
+
b
3
+
c
3
+
a
b
a
2
+
b
2
+
b
c
b
2
+
c
2
+
c
a
c
2
+
a
2
≥
9
2
a^3+b^3+c^3+\frac{ab}{a^2+b^2}+\frac{bc}{b^2+c^2}+\frac{ca}{c^2+a^2} \geq \frac{9}{2}
a
3
+
b
3
+
c
3
+
a
2
+
b
2
ab
+
b
2
+
c
2
b
c
+
c
2
+
a
2
c
a
≥
2
9
.
Collinearity of points on angle bisectors
Consider a triangle
△
A
B
C
\triangle ABC
△
A
BC
, let the incircle centered at
I
I
I
touch the sides
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
at points
D
,
E
,
F
D,E,F
D
,
E
,
F
respectively. Let the angle bisector of
∠
B
I
C
\angle BIC
∠
B
I
C
meet
B
C
BC
BC
at
M
M
M
, and the angle bisector of
∠
E
D
F
\angle EDF
∠
E
D
F
meet
E
F
EF
EF
at
N
N
N
. Prove that
A
,
M
,
N
A,M,N
A
,
M
,
N
are collinear.
Nice trigonometry problem
Prove the equality:\\
tan
(
3
π
7
)
−
4
sin
(
π
7
)
=
7
\tan (\frac{3\pi}{7})-4\sin (\frac{\pi}{7})= \sqrt{7}
tan
(
7
3
π
)
−
4
sin
(
7
π
)
=
7
.
1
3
Hide problems
Function on positive integers
Find all functions
f
:
Z
+
→
Z
+
f : \mathbb{Z}_{+} \rightarrow \mathbb{Z}_{+}
f
:
Z
+
→
Z
+
that satisfy
f
(
m
f
(
n
)
)
=
n
+
f
(
2015
m
)
f(mf(n)) = n+f(2015m)
f
(
m
f
(
n
))
=
n
+
f
(
2015
m
)
for all
m
,
n
∈
Z
+
m,n \in \mathbb{Z}_{+}
m
,
n
∈
Z
+
.
Simple inequality involving sine function
Let
c
∈
(
0
,
π
2
)
,
a
=
(
1
s
i
n
(
c
)
)
1
c
o
s
2
(
c
)
,
b
=
(
1
c
o
s
(
c
)
)
1
s
i
n
2
(
c
)
c\in \Big(0,\dfrac{\pi}{2}\Big) , a = \Big(\dfrac{1}{sin(c)}\Big)^{\dfrac{1}{cos^2 (c)}}, b = \Big(\dfrac{1}{cos(c)}\Big)^{\dfrac{1}{sin^2 (c)}}
c
∈
(
0
,
2
π
)
,
a
=
(
s
in
(
c
)
1
)
co
s
2
(
c
)
1
,
b
=
(
cos
(
c
)
1
)
s
i
n
2
(
c
)
1
. \\Prove that at least one of
a
,
b
a,b
a
,
b
is bigger than
2015
11
\sqrt[11]{2015}
11
2015
.
Find all polynomials...
Find all polynomials
P
(
x
)
P(x)
P
(
x
)
with real coefficients which satisfies \\
P
(
2015
)
=
2025
P(2015)=2025
P
(
2015
)
=
2025
and
P
(
x
)
−
10
=
P
(
x
2
+
3
)
−
13
P(x)-10=\sqrt{P(x^{2}+3)-13}
P
(
x
)
−
10
=
P
(
x
2
+
3
)
−
13
for every
x
≥
0
x\ge 0
x
≥
0
.