MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam Team Selection Test
2021 Vietnam TST
2021 Vietnam TST
Part of
Vietnam Team Selection Test
Subcontests
(6)
6
1
Hide problems
Residues over a "quite" large prime.
Let
n
≥
3
n \geq 3
n
≥
3
be a positive integers and
p
p
p
be a prime number such that
p
>
6
n
−
1
−
2
n
+
1
p > 6^{n-1} - 2^n + 1
p
>
6
n
−
1
−
2
n
+
1
. Let
S
S
S
be the set of
n
n
n
positive integers with different residues modulo
p
p
p
. Show that there exists a positive integer
c
c
c
such that there are exactly two ordered triples
(
x
,
y
,
z
)
∈
S
3
(x,y,z) \in S^3
(
x
,
y
,
z
)
∈
S
3
with distinct elements, such that
x
−
y
+
z
−
c
x-y+z-c
x
−
y
+
z
−
c
is divisible by
p
p
p
.
5
1
Hide problems
Goes through fixed points
Given a fixed circle
(
O
)
(O)
(
O
)
and two fixed points
B
,
C
B, C
B
,
C
on that circle, let
A
A
A
be a moving point on
(
O
)
(O)
(
O
)
such that
△
A
B
C
\triangle ABC
△
A
BC
is acute and scalene. Let
I
I
I
be the midpoint of
B
C
BC
BC
and let
A
D
,
B
E
,
C
F
AD, BE, CF
A
D
,
BE
,
CF
be the three heights of
△
A
B
C
\triangle ABC
△
A
BC
. In two rays
F
A
→
,
E
A
→
\overrightarrow{FA}, \overrightarrow{EA}
F
A
,
E
A
, we pick respectively
M
,
N
M,N
M
,
N
such that
F
M
=
C
E
,
E
N
=
B
F
FM = CE, EN = BF
FM
=
CE
,
EN
=
BF
. Let
L
L
L
be the intersection of
M
N
MN
MN
and
E
F
EF
EF
, and let
G
≠
L
G \neq L
G
=
L
be the second intersection of
(
L
E
N
)
(LEN)
(
L
EN
)
and
(
L
F
M
)
(LFM)
(
L
FM
)
.a) Show that the circle
(
M
N
G
)
(MNG)
(
MNG
)
always goes through a fixed point. b) Let
A
D
AD
A
D
intersects
(
O
)
(O)
(
O
)
at
K
≠
A
K \neq A
K
=
A
. In the tangent line through
D
D
D
of
(
D
K
I
)
(DKI)
(
DK
I
)
, we pick
P
,
Q
P,Q
P
,
Q
such that
G
P
∥
A
B
,
G
Q
∥
A
C
GP \parallel AB, GQ \parallel AC
GP
∥
A
B
,
GQ
∥
A
C
. Let
T
T
T
be the center of
(
G
P
Q
)
(GPQ)
(
GPQ
)
. Show that
G
T
GT
GT
always goes through a fixed point.
4
1
Hide problems
Inequality
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
are non-negative numbers such that
2
(
a
2
+
b
2
+
c
2
)
+
3
(
a
b
+
b
c
+
c
a
)
=
5
(
a
+
b
+
c
)
2(a^2+b^2+c^2)+3(ab+bc+ca)=5(a+b+c)
2
(
a
2
+
b
2
+
c
2
)
+
3
(
ab
+
b
c
+
c
a
)
=
5
(
a
+
b
+
c
)
then prove that
4
(
a
2
+
b
2
+
c
2
)
+
2
(
a
b
+
b
c
+
c
a
)
+
7
a
b
c
≤
25
4(a^2+b^2+c^2)+2(ab+bc+ca)+7abc\le 25
4
(
a
2
+
b
2
+
c
2
)
+
2
(
ab
+
b
c
+
c
a
)
+
7
ab
c
≤
25
3
1
Hide problems
Reflection and concurrence
Let
A
B
C
ABC
A
BC
be a triangle and
N
N
N
be a point that differs from
A
,
B
,
C
A,B,C
A
,
B
,
C
. Let
A
b
A_b
A
b
be the reflection of
A
A
A
through
N
B
NB
NB
, and
B
a
B_a
B
a
be the reflection of
B
B
B
through
N
A
NA
N
A
. Similarly, we define
B
c
,
C
b
,
A
c
,
C
a
B_c, C_b, A_c, C_a
B
c
,
C
b
,
A
c
,
C
a
. Let
m
a
m_a
m
a
be the line through
N
N
N
and perpendicular to
B
c
C
b
B_cC_b
B
c
C
b
. Define similarly
m
b
,
m
c
m_b, m_c
m
b
,
m
c
. a) Assume that
N
N
N
is the orthocenter of
△
A
B
C
\triangle ABC
△
A
BC
, show that the respective reflection of
m
a
,
m
b
,
m
c
m_a, m_b, m_c
m
a
,
m
b
,
m
c
through the bisector of angles
∠
B
N
C
,
∠
C
N
A
,
∠
A
N
B
\angle BNC, \angle CNA, \angle ANB
∠
BNC
,
∠
CN
A
,
∠
A
NB
are the same line. b) Assume that
N
N
N
is the nine-point center of
△
A
B
C
\triangle ABC
△
A
BC
, show that the respective reflection of
m
a
,
m
b
,
m
c
m_a, m_b, m_c
m
a
,
m
b
,
m
c
through
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
concur.
2
1
Hide problems
Choosing maximal subset of squares.
In a board of
2021
×
2021
2021 \times 2021
2021
×
2021
grids, we pick
k
k
k
unit squares such that every picked square shares vertice(s) with at most
1
1
1
other picked square. Determine the maximum of
k
k
k
.
1
1
Hide problems
Inequality on sequence
Define the sequence
(
a
n
)
(a_n)
(
a
n
)
as
a
1
=
1
a_1 = 1
a
1
=
1
,
a
2
n
=
a
n
a_{2n} = a_n
a
2
n
=
a
n
and
a
2
n
+
1
=
a
n
+
1
a_{2n+1} = a_n + 1
a
2
n
+
1
=
a
n
+
1
for all
n
≥
1
n\geq 1
n
≥
1
.a) Find all positive integers
n
n
n
such that
a
k
n
=
a
n
a_{kn} = a_n
a
kn
=
a
n
for all integers
1
≤
k
≤
n
1 \leq k \leq n
1
≤
k
≤
n
. b) Prove that there exist infinitely many positive integers
m
m
m
such that
a
k
m
≥
a
m
a_{km} \geq a_m
a
km
≥
a
m
for all positive integers
k
k
k
.