MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam Team Selection Test
2018 Vietnam Team Selection Test
2018 Vietnam Team Selection Test
Part of
Vietnam Team Selection Test
Subcontests
(6)
6
1
Hide problems
2018 VNTST Problem 6
Triangle
A
B
C
ABC
A
BC
circumscribed
(
O
)
(O)
(
O
)
has
A
A
A
-excircle
(
J
)
(J)
(
J
)
that touches
A
B
,
B
C
,
A
C
AB,\ BC,\ AC
A
B
,
BC
,
A
C
at
F
,
D
,
E
F,\ D,\ E
F
,
D
,
E
, resp.a.
L
L
L
is the midpoint of
B
C
BC
BC
. Circle with diameter
L
J
LJ
L
J
cuts
D
E
,
D
F
DE,\ DF
D
E
,
D
F
at
K
,
H
K,\ H
K
,
H
. Prove that
(
B
D
K
)
,
(
C
D
H
)
(BDK),\ (CDH)
(
B
DK
)
,
(
C
DH
)
has an intersecting point on
(
J
)
(J)
(
J
)
. b. Let
E
F
∩
B
C
=
{
G
}
EF\cap BC =\{G\}
EF
∩
BC
=
{
G
}
and
G
J
GJ
G
J
cuts
A
B
,
A
C
AB,\ AC
A
B
,
A
C
at
M
,
N
M,\ N
M
,
N
, resp.
P
∈
J
B
P\in JB
P
∈
J
B
and
Q
∈
J
C
Q\in JC
Q
∈
J
C
such that
∠
P
A
B
=
∠
Q
A
C
=
90
∘
.
\angle PAB=\angle QAC=90{}^\circ .
∠
P
A
B
=
∠
Q
A
C
=
90
∘
.
P
M
∩
Q
N
=
{
T
}
PM\cap QN=\{T\}
PM
∩
QN
=
{
T
}
and
S
S
S
is the midpoint of the larger
B
C
BC
BC
-arc of
(
O
)
(O)
(
O
)
.
(
I
)
(I)
(
I
)
is the incircle of
A
B
C
ABC
A
BC
. Prove that
S
I
∩
A
T
∈
(
O
)
SI\cap AT\in (O)
S
I
∩
A
T
∈
(
O
)
.
5
1
Hide problems
2018 VNTST Problem 5
In a
m
×
n
m\times n
m
×
n
square grid, with top-left corner is
A
A
A
, there is route along the edges of the grid starting from
A
A
A
and visits all lattice points (called "nodes") exactly once and ending also at
A
A
A
.a. Prove that this route exists if and only if at least one of
m
,
n
m,\ n
m
,
n
is odd. b. If such a route exists, then what is the least possible of turning points?*A turning point is a node that is different from
A
A
A
and if two edges on the route intersect at the node are perpendicular.
4
1
Hide problems
2018 VNTST Problem 4
Let
a
∈
[
1
2
,
3
2
]
a\in\left[ \tfrac{1}{2},\ \tfrac{3}{2}\right]
a
∈
[
2
1
,
2
3
]
be a real number. Sequences
(
u
n
)
,
(
v
n
)
(u_n),\ (v_n)
(
u
n
)
,
(
v
n
)
are defined as follows:
u
n
=
3
2
n
+
1
⋅
(
−
1
)
⌊
2
n
+
1
a
⌋
,
v
n
=
3
2
n
+
1
⋅
(
−
1
)
n
+
⌊
2
n
+
1
a
⌋
.
u_n=\frac{3}{2^{n+1}}\cdot (-1)^{\lfloor2^{n+1}a\rfloor},\ v_n=\frac{3}{2^{n+1}}\cdot (-1)^{n+\lfloor 2^{n+1}a\rfloor}.
u
n
=
2
n
+
1
3
⋅
(
−
1
)
⌊
2
n
+
1
a
⌋
,
v
n
=
2
n
+
1
3
⋅
(
−
1
)
n
+
⌊
2
n
+
1
a
⌋
.
a. Prove that
(
u
0
+
u
1
+
⋯
+
u
2018
)
2
+
(
v
0
+
v
1
+
⋯
+
v
2018
)
2
≤
72
a
2
−
48
a
+
10
+
2
4
2019
.
{{({{u}_{0}}+{{u}_{1}}+\cdots +{{u}_{2018}})}^{2}}+{{({{v}_{0}}+{{v}_{1}}+\cdots +{{v}_{2018}})}^{2}}\le 72{{a}^{2}}-48a+10+\frac{2}{{{4}^{2019}}}.
(
u
0
+
u
1
+
⋯
+
u
2018
)
2
+
(
v
0
+
v
1
+
⋯
+
v
2018
)
2
≤
72
a
2
−
48
a
+
10
+
4
2019
2
.
b. Find all values of
a
a
a
in the equality case.
3
1
Hide problems
2018 VNTST Problem 3
For every positive integer
n
≥
3
n\ge 3
n
≥
3
, let
ϕ
n
\phi_n
ϕ
n
be the set of all positive integers less than and coprime to
n
n
n
. Consider the polynomial:
P
n
(
x
)
=
∑
k
∈
ϕ
n
x
k
−
1
.
P_n(x)=\sum_{k\in\phi_n} {x^{k-1}}.
P
n
(
x
)
=
k
∈
ϕ
n
∑
x
k
−
1
.
a. Prove that
P
n
(
x
)
=
(
x
r
n
+
1
)
Q
n
(
x
)
P_n(x)=(x^{r_n}+1)Q_n(x)
P
n
(
x
)
=
(
x
r
n
+
1
)
Q
n
(
x
)
for some positive integer
r
n
r_n
r
n
and polynomial
Q
n
(
x
)
∈
Z
[
x
]
Q_n(x)\in\mathbb{Z}[x]
Q
n
(
x
)
∈
Z
[
x
]
(not necessary non-constant polynomial). b. Find all
n
n
n
such that
P
n
(
x
)
P_n(x)
P
n
(
x
)
is irreducible over
Z
[
x
]
\mathbb{Z}[x]
Z
[
x
]
.
2
1
Hide problems
2018 VNTST Problem 2
For every positive integer
m
m
m
, a
m
×
2018
m\times 2018
m
×
2018
rectangle consists of unit squares (called "cell") is called complete if the following conditions are met: i. In each cell is written either a "
0
0
0
", a "
1
1
1
" or nothing; ii. For any binary string
S
S
S
with length
2018
2018
2018
, one may choose a row and complete the empty cells so that the numbers in that row, if read from left to right, produce
S
S
S
(In particular, if a row is already full and it produces
S
S
S
in the same manner then this condition ii. is satisfied). A complete rectangle is called minimal, if we remove any of its rows and then making it no longer complete.a. Prove that for any positive integer
k
≤
2018
k\le 2018
k
≤
2018
there exists a minimal
2
k
×
2018
2^k\times 2018
2
k
×
2018
rectangle with exactly
k
k
k
columns containing both
0
0
0
and
1
1
1
. b. A minimal
m
×
2018
m\times 2018
m
×
2018
rectangle has exactly
k
k
k
columns containing at least some
0
0
0
or
1
1
1
and the rest of columns are empty. Prove that
m
≤
2
k
m\le 2^k
m
≤
2
k
.
1
1
Hide problems
2018 VNTST Problem 1
Let
A
B
C
ABC
A
BC
be a acute, non-isosceles triangle.
D
,
E
,
F
D,\ E,\ F
D
,
E
,
F
are the midpoints of sides
A
B
,
B
C
,
A
C
AB,\ BC,\ AC
A
B
,
BC
,
A
C
, resp. Denote by
(
O
)
,
(
O
′
)
(O),\ (O')
(
O
)
,
(
O
′
)
the circumcircle and Euler circle of
A
B
C
ABC
A
BC
. An arbitrary point
P
P
P
lies inside triangle
D
E
F
DEF
D
EF
and
D
P
,
E
P
,
F
P
DP,\ EP,\ FP
D
P
,
EP
,
FP
intersect
(
O
′
)
(O')
(
O
′
)
at
D
′
,
E
′
,
F
′
D',\ E',\ F'
D
′
,
E
′
,
F
′
, resp. Point
A
′
A'
A
′
is the point such that
D
′
D'
D
′
is the midpoint of
A
A
′
AA'
A
A
′
. Points
B
′
,
C
′
B',\ C'
B
′
,
C
′
are defined similarly. a. Prove that if
P
O
=
P
O
′
PO=PO'
PO
=
P
O
′
then
O
∈
(
A
′
B
′
C
′
)
O\in(A'B'C')
O
∈
(
A
′
B
′
C
′
)
; b. Point
A
′
A'
A
′
is mirrored by
O
D
OD
O
D
, its image is
X
X
X
.
Y
,
Z
Y,\ Z
Y
,
Z
are created in the same manner.
H
H
H
is the orthocenter of
A
B
C
ABC
A
BC
and
X
H
,
Y
H
,
Z
H
XH,\ YH,\ ZH
X
H
,
Y
H
,
Z
H
intersect
B
C
,
A
C
,
A
B
BC, AC, AB
BC
,
A
C
,
A
B
at
M
,
N
,
L
M,\ N,\ L
M
,
N
,
L
resp. Prove that
M
,
N
,
L
M,\ N,\ L
M
,
N
,
L
are collinear.