MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
2019 Vietnam National Olympiad
2019 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(2)
Day 2
3
Hide problems
Quotient of two polynomials
Consider polynomial
f
(
x
)
=
x
2
−
α
x
+
1
f(x)={{x}^{2}}-\alpha x+1
f
(
x
)
=
x
2
−
αx
+
1
with
α
∈
R
.
\alpha \in \mathbb{R}.
α
∈
R
.
a) For
α
=
15
2
\alpha =\frac{\sqrt{15}}{2}
α
=
2
15
, let write
f
(
x
)
f(x)
f
(
x
)
as the quotient of two polynomials with nonnegative coefficients. b) Find all value of
α
\alpha
α
such that
f
(
x
)
f(x)
f
(
x
)
can be written as the quotient of two polynomials with nonnegative coefficients.
Figure of midpoints and feet of altitude triangle
Let
A
B
C
ABC
A
BC
be an acute, nonisosceles triangle with inscribe in a circle
(
O
)
(O)
(
O
)
and has orthocenter
H
H
H
. Denote
M
,
N
,
P
M,N,P
M
,
N
,
P
as the midpoints of sides
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
and
D
,
E
,
F
D,E,F
D
,
E
,
F
as the feet of the altitudes from vertices
A
,
B
,
C
A,B,C
A
,
B
,
C
of triangle
A
B
C
ABC
A
BC
. Let
K
K
K
as the reflection of
H
H
H
through
B
C
BC
BC
. Two lines
D
E
,
M
P
DE,MP
D
E
,
MP
meet at
X
X
X
; two lines
D
F
,
M
N
DF,MN
D
F
,
MN
meet at
Y
Y
Y
. a) The line
X
Y
XY
X
Y
cut the minor arc
B
C
BC
BC
of
(
O
)
(O)
(
O
)
at
Z
Z
Z
. Prove that
K
,
Z
,
E
,
F
K,Z,E,F
K
,
Z
,
E
,
F
are concyclic. b) Two lines
K
E
,
K
F
KE,KF
K
E
,
K
F
cuts
(
O
)
(O)
(
O
)
second time at
S
,
T
S,T
S
,
T
. Prove that
B
S
,
C
T
,
X
Y
BS,CT,XY
BS
,
CT
,
X
Y
are concurrent.
Painting the paper
There are some papers of the size
5
×
5
5\times 5
5
×
5
with two sides which are divided into unit squares for both sides. One uses
n
n
n
colors to paint each cell on the paper, one cell by one color, such that two cells on the same positions for two sides are painted by the same color. Two painted papers are consider as the same if the color of two corresponding cells are the same. Prove that there are no more than
1
8
(
n
25
+
4
n
15
+
n
13
+
2
n
7
)
\frac{1}{8}\left( {{n}^{25}}+4{{n}^{15}}+{{n}^{13}}+2{{n}^{7}} \right)
8
1
(
n
25
+
4
n
15
+
n
13
+
2
n
7
)
pairwise distinct papers that painted by this way.
Day 1
4
Show problems