MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam Team Selection Test
2019 Vietnam TST
2019 Vietnam TST
Part of
Vietnam Team Selection Test
Subcontests
(6)
P4
1
Hide problems
Diophantine Equation
Find all triplets of positive integers
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
such that
2
x
+
1
=
7
y
+
2
z
2^x+1=7^y+2^z
2
x
+
1
=
7
y
+
2
z
.
P2
1
Hide problems
$P(x)$ has real roots
For each positive integer
n
n
n
, show that the polynomial:
P
n
(
x
)
=
∑
k
=
0
n
2
k
(
2
n
2
k
)
x
k
(
x
−
1
)
n
−
k
P_n(x)=\sum _{k=0}^n2^k\binom{2n}{2k}x^k(x-1)^{n-k}
P
n
(
x
)
=
k
=
0
∑
n
2
k
(
2
k
2
n
)
x
k
(
x
−
1
)
n
−
k
has
n
n
n
real roots.
P6
1
Hide problems
Bug jumps in the coordinate axis
In the real axis, there is bug standing at coordinate
x
=
1
x=1
x
=
1
. Each step, from the position
x
=
a
x=a
x
=
a
, the bug can jump to either
x
=
a
+
2
x=a+2
x
=
a
+
2
or
x
=
a
2
x=\frac{a}{2}
x
=
2
a
. Show that there are precisely
F
n
+
4
−
(
n
+
4
)
F_{n+4}-(n+4)
F
n
+
4
−
(
n
+
4
)
positions (including the initial position) that the bug can jump to by at most
n
n
n
steps.Recall that
F
n
F_n
F
n
is the
n
t
h
n^{th}
n
t
h
element of the Fibonacci sequence, defined by
F
0
=
F
1
=
1
F_0=F_1=1
F
0
=
F
1
=
1
,
F
n
+
1
=
F
n
+
F
n
−
1
F_{n+1}=F_n+F_{n-1}
F
n
+
1
=
F
n
+
F
n
−
1
for all
n
≥
1
n\geq 1
n
≥
1
.
P1
1
Hide problems
Licensing airways for airlines
In a country there are
n
≥
2
n\geq 2
n
≥
2
cities. Any two cities has exactly one two-way airway. The government wants to license several airlines to take charge of these airways with such following conditions:i) Every airway can be licensed to exactly one airline. ii) By choosing one arbitrary airline, we can move from a city to any other cities, using only flights from this airline.What is the maximum number of airlines that the government can license to satisfy all of these conditions?
P3
1
Hide problems
Point with the same power to four circles.
Given an acute scalene triangle
A
B
C
ABC
A
BC
inscribed in circle
(
O
)
(O)
(
O
)
. Let
H
H
H
be its orthocenter and
M
M
M
be the midpoint of
B
C
BC
BC
. Let
D
D
D
lie on the opposite rays of
H
A
HA
H
A
so that
B
C
=
2
D
M
BC=2DM
BC
=
2
D
M
. Let
D
′
D'
D
′
be the reflection of
D
D
D
through line
B
C
BC
BC
and
X
X
X
be the intersection of
A
O
AO
A
O
and
M
D
MD
M
D
.a) Show that
A
M
AM
A
M
bisects
D
′
X
D'X
D
′
X
.b) Similarly, we define the points
E
,
F
E,F
E
,
F
like
D
D
D
and
Y
,
Z
Y,Z
Y
,
Z
like
X
X
X
. Let
S
S
S
be the intersection of tangent lines from
B
,
C
B,C
B
,
C
with respect to
(
O
)
(O)
(
O
)
. Let
G
G
G
be the projection of the midpoint of
A
S
AS
A
S
to the line
A
O
AO
A
O
. Show that there exists a point with the same power to all the circles
(
B
E
Y
)
,
(
C
F
Z
)
,
(
S
G
O
)
(BEY),(CFZ),(SGO)
(
BE
Y
)
,
(
CFZ
)
,
(
SGO
)
and
(
O
)
(O)
(
O
)
.
P5
1
Hide problems
Median perpendicular to line.
Given a scalene triangle
A
B
C
ABC
A
BC
inscribed in the circle
(
O
)
(O)
(
O
)
. Let
(
I
)
(I)
(
I
)
be its incircle and
B
I
,
C
I
BI,CI
B
I
,
C
I
cut
A
C
,
A
B
AC,AB
A
C
,
A
B
at
E
,
F
E,F
E
,
F
respectively. A circle passes through
E
E
E
and touches
O
B
OB
OB
at
B
B
B
cuts
(
O
)
(O)
(
O
)
again at
M
M
M
. Similarly, a circle passes through
F
F
F
and touches
O
C
OC
OC
at
C
C
C
cuts
(
O
)
(O)
(
O
)
again at
N
N
N
.
M
E
,
N
F
ME,NF
ME
,
NF
cut
(
O
)
(O)
(
O
)
again at
P
,
Q
P,Q
P
,
Q
. Let
K
K
K
be the intersection of
E
F
EF
EF
and
B
C
BC
BC
and let
P
Q
PQ
PQ
cuts
B
C
BC
BC
and
E
F
EF
EF
at
G
,
H
G,H
G
,
H
, respectively. Show that the median correspond to
G
G
G
of the triangle
G
H
K
GHK
G
HK
is perpendicular to
I
O
IO
I
O
.