MathDB
Problems
Contests
National and Regional Contests
Moldova Contests
Moldova Team Selection Test
2019 Moldova Team Selection Test
2019 Moldova Team Selection Test
Part of
Moldova Team Selection Test
Subcontests
(12)
10
1
Hide problems
Prove that the incenters of triangles $PEF, PCA$, and $PBA$ are collinear.
The circle
Ω
\Omega
Ω
with center
O
O
O
is circumscribed to acute triangle
A
B
C
ABC
A
BC
. Let
P
P
P
be a point on the circumscribed circle of
O
B
C
OBC
OBC
, such that
P
P
P
is inside
A
B
C
ABC
A
BC
and is different from
B
B
B
and
C
C
C
. Bisectors of angles
B
P
A
BPA
BP
A
and
C
P
A
CPA
CP
A
intersect the sides
A
B
AB
A
B
and
A
C
AC
A
C
in points
E
E
E
and
F
.
F.
F
.
Prove that the incenters of triangles
P
E
F
,
P
C
A
PEF, PCA
PEF
,
PC
A
and
P
B
A
PBA
PB
A
are collinear.
5
1
Hide problems
This is not ISL 2005 G5
Point
H
H
H
is the orthocenter of the scalene triangle
A
B
C
.
ABC.
A
BC
.
A line, which passes through point
H
H
H
, intersect the sides
A
B
AB
A
B
and
A
C
AC
A
C
at points
D
D
D
and
E
E
E
, respectively, such that
A
D
=
A
E
.
AD=AE.
A
D
=
A
E
.
Let
M
M
M
be the midpoint of side
B
C
.
BC.
BC
.
Line
M
H
MH
M
H
intersects the circumscribed circle of triangle
A
B
C
ABC
A
BC
at point
K
K
K
, which is on the smaller arc
A
B
AB
A
B
. Prove that Nibab can draw a circle through
A
,
D
,
E
A, D, E
A
,
D
,
E
and
K
.
K.
K
.
7
1
Hide problems
another incredible polynomial
Let
P
(
X
)
=
a
2
n
+
1
X
2
n
+
1
+
a
2
n
X
2
n
+
.
.
.
+
a
1
X
+
a
0
P(X)=a_{2n+1}X^{2n+1}+a_{2n}X^{2n}+...+a_1X+a_0
P
(
X
)
=
a
2
n
+
1
X
2
n
+
1
+
a
2
n
X
2
n
+
...
+
a
1
X
+
a
0
be a polynomial with all positive coefficients. Prove that there exists a permutation
(
b
2
n
+
1
,
b
2
n
,
.
.
.
,
b
1
,
b
0
)
(b_{2n+1},b_{2n},...,b_1,b_0)
(
b
2
n
+
1
,
b
2
n
,
...
,
b
1
,
b
0
)
of numbers
(
a
2
n
+
1
,
a
2
n
,
.
.
.
,
a
1
,
a
0
)
(a_{2n+1},a_{2n},...,a_1,a_0)
(
a
2
n
+
1
,
a
2
n
,
...
,
a
1
,
a
0
)
such that the polynomial
Q
(
X
)
=
b
2
n
+
1
X
2
n
+
1
+
b
2
n
X
2
n
+
.
.
.
+
b
1
X
+
b
0
Q(X)=b_{2n+1}X^{2n+1}+b_{2n}X^{2n}+...+b_1X+b_0
Q
(
X
)
=
b
2
n
+
1
X
2
n
+
1
+
b
2
n
X
2
n
+
...
+
b
1
X
+
b
0
has exactly one real root.
8
1
Hide problems
very beautiful number theory
For any positive integer
k
k
k
denote by
S
(
k
)
S(k)
S
(
k
)
the number of solutions
(
x
,
y
)
∈
Z
+
×
Z
+
(x,y)\in \mathbb{Z}_+ \times \mathbb{Z}_+
(
x
,
y
)
∈
Z
+
×
Z
+
of the system
{
⌈
x
⋅
d
y
⌉
⋅
x
d
=
⌈
(
y
+
1
)
2
⌉
∣
x
−
y
∣
=
k
,
\begin{cases} \left\lceil\frac{x\cdot d}{y}\right\rceil\cdot \frac{x}{d}=\left\lceil\left(\sqrt{y}+1\right)^2\right\rceil \\ \mid x-y\mid =k , \end{cases}
{
⌈
y
x
⋅
d
⌉
⋅
d
x
=
⌈
(
y
+
1
)
2
⌉
∣
x
−
y
∣=
k
,
where
d
d
d
is the greatest common divisor of positive integers
x
x
x
and
y
.
y.
y
.
Determine
S
(
k
)
S(k)
S
(
k
)
as a function of
k
k
k
. (Here
⌈
z
⌉
\lceil z\rceil
⌈
z
⌉
denotes the smalles integer number which is bigger or equal than
z
.
z.
z
.
)
11
1
Hide problems
okay this combinatorics is epic
Let
n
≥
2
,
n\ge 2,
n
≥
2
,
be a positive integer. Numbers
{
1
,
2
,
3
,
.
.
.
,
n
}
\{1,2,3, ...,n\}
{
1
,
2
,
3
,
...
,
n
}
are written in a row in an arbitrary order. Determine the smalles positive integer
k
k
k
with the property: everytime it is possible to delete
k
k
k
numbers from those written on the table, such that the remained numbers are either in an increasing or decreasing order.
9
1
Hide problems
incredible polynomial, 10/10
Find all polynomials
P
(
X
)
P(X)
P
(
X
)
with real coefficients such that if real numbers
x
,
y
x,y
x
,
y
and
z
z
z
satisfy
x
+
y
+
z
=
0
,
x+y+z=0,
x
+
y
+
z
=
0
,
then the points
(
x
,
P
(
x
)
)
,
(
y
,
P
(
y
)
)
,
(
z
,
P
(
z
)
)
\left(x,P(x)\right), \left(y,P(y)\right), \left(z,P(z)\right)
(
x
,
P
(
x
)
)
,
(
y
,
P
(
y
)
)
,
(
z
,
P
(
z
)
)
are all colinear.
12
1
Hide problems
number theory problem
Let
p
≥
5
p\ge 5
p
≥
5
be a prime number. Prove that there exist positive integers
m
m
m
and
n
n
n
with
m
+
n
≤
p
+
1
2
m+n\le \frac{p+1}{2}
m
+
n
≤
2
p
+
1
for which
p
p
p
divides
2
n
⋅
3
m
−
1.
2^n\cdot 3^m-1.
2
n
⋅
3
m
−
1.
6
1
Hide problems
Schur inequality
Let
a
,
b
,
c
≥
0
a,b,c \ge 0
a
,
b
,
c
≥
0
such that
a
+
b
+
c
=
1
a+b+c=1
a
+
b
+
c
=
1
and
s
≥
5
s \ge 5
s
≥
5
. Prove that
s
(
a
2
+
b
2
+
c
2
)
≤
3
(
s
−
3
)
(
a
3
+
b
3
+
c
3
)
+
1
s(a^2+b^2+c^2) \le 3(s-3)(a^3+b^3+c^3)+1
s
(
a
2
+
b
2
+
c
2
)
≤
3
(
s
−
3
)
(
a
3
+
b
3
+
c
3
)
+
1
3
1
Hide problems
Ez pz combinatorics
On the table there are written numbers
673
,
674
,
⋯
,
2018
,
2019.
673, 674, \cdots, 2018, 2019.
673
,
674
,
⋯
,
2018
,
2019.
Nibab chooses arbitrarily three numbers
a
,
b
a,b
a
,
b
and
c
c
c
, erases them and writes the number
min
(
a
,
b
,
c
)
3
\frac{\min(a,b,c)}{3}
3
m
i
n
(
a
,
b
,
c
)
, then he continues in an analogous way. After Nibab performed this operation
673
673
673
times, on the table remained a single number
k
k
k
. Prove that
k
∈
(
0
,
1
)
.
k\in(0,1).
k
∈
(
0
,
1
)
.
1
1
Hide problems
Ez pz tn
Let
S
S
S
be the set of all natural numbers with the property: the sum of the biggest three divisors of number
n
n
n
, different from
n
n
n
, is bigger than
n
n
n
. Determine the largest natural number
k
k
k
, which divides any number from
S
S
S
. (A natural number is a positive integer)
2
1
Hide problems
Ez pz trig
Prove that
E
n
=
arccos
n
−
1
n
arccot
2
n
−
1
E_n=\frac{\arccos {\frac{n-1}{n}} } {\text{arccot} {\sqrt{2n-1} }}
E
n
=
arccot
2
n
−
1
a
r
c
c
o
s
n
n
−
1
is a natural number for any natural number
n
n
n
. (A natural number is a positive integer)
4
1
Hide problems
Hardcore geo
Quadrilateral
A
B
C
D
ABCD
A
BC
D
is inscribed in circle
Γ
\Gamma
Γ
with center
O
O
O
. Point
I
I
I
is the incenter of triangle
A
B
C
ABC
A
BC
, and point
J
J
J
is the incenter of the triangle
A
B
D
ABD
A
B
D
. Line
I
J
IJ
I
J
intersects segments
A
D
,
A
C
,
B
D
,
B
C
AD, AC, BD, BC
A
D
,
A
C
,
B
D
,
BC
at points
P
,
M
,
N
P, M, N
P
,
M
,
N
and, respectively
Q
Q
Q
. The perpendicular from
M
M
M
to line
A
C
AC
A
C
intersects the perpendicular from
N
N
N
to line
B
D
BD
B
D
at point
X
X
X
. The perpendicular from
P
P
P
to line
A
D
AD
A
D
intersects the perpendicular from
Q
Q
Q
to line
B
C
BC
BC
at point
Y
Y
Y
. Prove that
X
,
O
,
Y
X, O, Y
X
,
O
,
Y
are colinear.