MathDB
Problems
Contests
National and Regional Contests
Moldova Contests
Moldova Team Selection Test
2014 Moldova Team Selection Test
2014 Moldova Team Selection Test
Part of
Moldova Team Selection Test
Subcontests
(4)
4
3
Hide problems
Moldova TST, greatest prime divisor of the sum
Define
p
(
n
)
p(n)
p
(
n
)
to be th product of all non-zero digits of
n
n
n
. For instance
p
(
5
)
=
5
p(5)=5
p
(
5
)
=
5
,
p
(
27
)
=
14
p(27)=14
p
(
27
)
=
14
,
p
(
101
)
=
1
p(101)=1
p
(
101
)
=
1
and so on. Find the greatest prime divisor of the following expression:
p
(
1
)
+
p
(
2
)
+
p
(
3
)
+
.
.
.
+
p
(
999
)
.
p(1)+p(2)+p(3)+...+p(999).
p
(
1
)
+
p
(
2
)
+
p
(
3
)
+
...
+
p
(
999
)
.
How many distinct midpoints
Consider
n
≥
2
n \geq 2
n
≥
2
distinct points in the plane
A
1
,
A
2
,
.
.
.
,
A
n
A_1,A_2,...,A_n
A
1
,
A
2
,
...
,
A
n
. Color the midpoints of the segments determined by each pair of points in red. What is the minimum number of distinct red points?
Cyclic sums
On a circle
n
≥
1
n \geq 1
n
≥
1
real numbers are written, their sum is
n
−
1
n-1
n
−
1
. Prove that one can denote these numbers as
x
1
,
x
2
,
.
.
.
,
x
n
x_1, x_2, ..., x_n
x
1
,
x
2
,
...
,
x
n
consecutively, starting from a number and moving clockwise, such that for any
k
k
k
(
1
≤
k
≤
n
1\leq k \leq n
1
≤
k
≤
n
)
x
1
+
x
2
+
.
.
.
+
x
k
≥
k
−
1
x_1 + x_2+...+x_k \geq k-1
x
1
+
x
2
+
...
+
x
k
≥
k
−
1
.
1
3
Hide problems
Moldova TST, equation in positive integers
Find all pairs of non-negative integers
(
x
,
y
)
(x,y)
(
x
,
y
)
such that
x
+
y
−
x
−
y
+
2
=
0.
\sqrt{x+y}-\sqrt{x}-\sqrt{y}+2=0.
x
+
y
−
x
−
y
+
2
=
0.
Sequence inequality
Consider
n
≥
2
n \geq 2
n
≥
2
positive numbers
0
<
x
1
≤
x
2
≤
.
.
.
≤
x
n
0<x_1 \leq x_2 \leq ... \leq x_n
0
<
x
1
≤
x
2
≤
...
≤
x
n
, such that
x
1
+
x
2
+
.
.
.
+
x
n
=
1
x_1 + x_2 + ... + x_n = 1
x
1
+
x
2
+
...
+
x
n
=
1
. Prove that if
x
n
≤
2
3
x_n \leq \dfrac{2}{3}
x
n
≤
3
2
, then there exists a positive integer
1
≤
k
≤
n
1 \leq k \leq n
1
≤
k
≤
n
such that
1
3
≤
x
1
+
x
2
+
.
.
.
+
x
k
<
2
3
\dfrac{1}{3} \leq x_1+x_2+...+x_k < \dfrac{2}{3}
3
1
≤
x
1
+
x
2
+
...
+
x
k
<
3
2
.
Distance of 4 points
Prove that there do not exist
4
4
4
points in the plane such that the distances between any pair of them is an odd integer.
3
3
Hide problems
Moldova TST triangle geometry
Let
△
A
B
C
\triangle ABC
△
A
BC
be an acute triangle and
A
D
AD
A
D
the bisector of the angle
∠
B
A
C
\angle BAC
∠
B
A
C
with
D
∈
(
B
C
)
D\in(BC)
D
∈
(
BC
)
. Let
E
E
E
and
F
F
F
denote feet of perpendiculars from
D
D
D
to
A
B
AB
A
B
and
A
C
AC
A
C
respectively. If
B
F
∩
C
E
=
K
BF\cap CE=K
BF
∩
CE
=
K
and
⊙
A
K
E
∩
B
F
=
L
\odot AKE\cap BF=L
⊙
A
K
E
∩
BF
=
L
prove that
D
L
⊥
B
F
DL\perp BF
D
L
⊥
BF
.
Cyclic quadrilateral bisectors
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral. The bisectors of angles
B
A
D
BAD
B
A
D
and
B
C
D
BCD
BC
D
intersect in point
K
K
K
such that
K
∈
B
D
K \in BD
K
∈
B
D
. Let
M
M
M
be the midpoint of
B
D
BD
B
D
. A line passing through point
C
C
C
and parallel to
A
D
AD
A
D
intersects
A
M
AM
A
M
in point
P
P
P
. Prove that triangle
△
D
P
C
\triangle DPC
△
D
PC
is isosceles.
Geometry
Let
△
A
B
C
\triangle ABC
△
A
BC
be a triangle with
∠
A
\angle A
∠
A
-acute. Let
P
P
P
be a point inside
△
A
B
C
\triangle ABC
△
A
BC
such that
∠
B
A
P
=
∠
A
C
P
\angle BAP = \angle ACP
∠
B
A
P
=
∠
A
CP
and
∠
C
A
P
=
∠
A
B
P
\angle CAP =\angle ABP
∠
C
A
P
=
∠
A
BP
. Let
M
,
N
M, N
M
,
N
be the centers of the incircle of
△
A
B
P
\triangle ABP
△
A
BP
and
△
A
C
P
\triangle ACP
△
A
CP
, and
R
R
R
the radius of the circumscribed circle of
△
A
M
N
\triangle AMN
△
A
MN
. Prove that
1
R
=
1
A
B
+
1
A
C
+
1
A
P
.
\displaystyle \frac{1}{R}=\frac{1}{AB}+\frac{1}{AC}+\frac{1}{AP}.
R
1
=
A
B
1
+
A
C
1
+
A
P
1
.
2
3
Hide problems
Moldova TST, easy inequality
Let
a
,
b
∈
R
+
a,b\in\mathbb{R}_+
a
,
b
∈
R
+
such that
a
+
b
=
1
a+b=1
a
+
b
=
1
. Find the minimum value of the following expression:
E
(
a
,
b
)
=
3
1
+
2
a
2
+
2
40
+
9
b
2
.
E(a,b)=3\sqrt{1+2a^2}+2\sqrt{40+9b^2}.
E
(
a
,
b
)
=
3
1
+
2
a
2
+
2
40
+
9
b
2
.
Interesting Inequality
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
. Determine the minimum value of
E
(
a
,
b
,
c
)
=
∑
a
3
+
5
a
3
(
b
+
c
)
E(a,b,c) = \sum \dfrac{a^3+5}{a^3(b+c)}
E
(
a
,
b
,
c
)
=
∑
a
3
(
b
+
c
)
a
3
+
5
.
Function on real
Find all functions
f
:
R
→
R
f:R \rightarrow R
f
:
R
→
R
, which satisfy the equality for any
x
,
y
∈
R
x,y \in R
x
,
y
∈
R
:
f
(
x
f
(
y
)
+
y
)
+
f
(
x
y
+
x
)
=
f
(
x
+
y
)
+
2
x
y
f(xf(y)+y)+f(xy+x)=f(x+y)+2xy
f
(
x
f
(
y
)
+
y
)
+
f
(
x
y
+
x
)
=
f
(
x
+
y
)
+
2
x
y
,