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Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam Team Selection Test
2017 Vietnam Team Selection Test
2017 Vietnam Team Selection Test
Part of
Vietnam Team Selection Test
Subcontests
(3)
3
2
Hide problems
Vietnam TST 2017 problem 3
Triangle
A
B
C
ABC
A
BC
with incircle
(
I
)
(I)
(
I
)
touches the sides
A
B
,
B
C
,
A
C
AB, BC, AC
A
B
,
BC
,
A
C
at
F
,
D
,
E
F, D, E
F
,
D
,
E
, res.
I
b
,
I
c
I_b, I_c
I
b
,
I
c
are
B
B
B
- and
C
−
C-
C
−
excenters of
A
B
C
ABC
A
BC
.
P
,
Q
P, Q
P
,
Q
are midpoints of
I
b
E
,
I
c
F
I_bE, I_cF
I
b
E
,
I
c
F
.
(
P
A
C
)
∩
A
B
=
{
A
,
R
}
(PAC)\cap AB=\{ A, R\}
(
P
A
C
)
∩
A
B
=
{
A
,
R
}
,
(
Q
A
B
)
∩
A
C
=
{
A
,
S
}
(QAB)\cap AC=\{ A,S\}
(
Q
A
B
)
∩
A
C
=
{
A
,
S
}
. a. Prove that
P
R
,
Q
S
,
A
I
PR, QS, AI
PR
,
QS
,
A
I
are concurrent. b.
D
E
,
D
F
DE, DF
D
E
,
D
F
cut
I
b
I
c
I_bI_c
I
b
I
c
at
K
,
J
K, J
K
,
J
, res.
E
J
∩
F
K
=
{
M
}
EJ\cap FK=\{ M\}
E
J
∩
F
K
=
{
M
}
.
P
E
,
Q
F
PE, QF
PE
,
QF
cut
(
P
A
C
)
,
(
Q
A
B
)
(PAC), (QAB)
(
P
A
C
)
,
(
Q
A
B
)
at
X
,
Y
X, Y
X
,
Y
res. Prove that
B
Y
,
C
X
,
A
M
BY, CX, AM
B
Y
,
CX
,
A
M
are concurrent.
Vietnam TST 2017 problem 6
For each integer
n
>
0
n>0
n
>
0
, a permutation
a
1
,
a
2
,
…
,
a
2
n
a_1,a_2,\dots ,a_{2n}
a
1
,
a
2
,
…
,
a
2
n
of
1
,
2
,
…
2
n
1,2,\dots 2n
1
,
2
,
…
2
n
is called beautiful if for every
1
≤
i
<
j
≤
2
n
1\leq i<j \leq 2n
1
≤
i
<
j
≤
2
n
,
a
i
+
a
n
+
i
=
2
n
+
1
a_i+a_{n+i}=2n+1
a
i
+
a
n
+
i
=
2
n
+
1
and
a
i
−
a
i
+
1
≢
a
j
−
a
j
+
1
a_i-a_{i+1}\not \equiv a_j-a_{j+1}
a
i
−
a
i
+
1
≡
a
j
−
a
j
+
1
(mod
2
n
+
1
2n+1
2
n
+
1
) (suppose that
a
2
n
+
1
=
a
1
a_{2n+1}=a_1
a
2
n
+
1
=
a
1
). a. For
n
=
6
n=6
n
=
6
, point out a beautiful permutation. b. Prove that there exists a beautiful permutation for every
n
n
n
.
2
2
Hide problems
Vietnam TST 2017 problem 2
For each positive integer
n
n
n
, set
x
n
=
(
2
n
n
)
x_n=\binom{2n}{n}
x
n
=
(
n
2
n
)
. a. Prove that if
201
7
k
2
<
n
<
201
7
k
\frac{2017^k}{2}<n<2017^k
2
201
7
k
<
n
<
201
7
k
for some positive integer
k
k
k
then
2017
2017
2017
divides
x
n
x_n
x
n
. b. Find all positive integer
h
>
1
h>1
h
>
1
such that there exists positive integers
N
,
T
N,T
N
,
T
such that
(
x
n
)
n
>
N
(x_n)_{n>N}
(
x
n
)
n
>
N
is periodic mod
h
h
h
with period
T
T
T
.
Vietnam TST 2017 problem 5
Given
2017
2017
2017
positive real numbers
a
1
,
a
2
,
…
,
a
2017
a_1,a_2,\dots ,a_{2017}
a
1
,
a
2
,
…
,
a
2017
. For each
n
>
2017
n>2017
n
>
2017
, set
a
n
=
max
{
a
i
1
a
i
2
a
i
3
∣
i
1
+
i
2
+
i
3
=
n
,
1
≤
i
1
≤
i
2
≤
i
3
≤
n
−
1
}
.
a_n=\max\{ a_{i_1}a_{i_2}a_{i_3}|i_1+i_2+i_3=n, 1\leq i_1\leq i_2\leq i_3\leq n-1\}.
a
n
=
max
{
a
i
1
a
i
2
a
i
3
∣
i
1
+
i
2
+
i
3
=
n
,
1
≤
i
1
≤
i
2
≤
i
3
≤
n
−
1
}
.
Prove that there exists a positive integer
m
≤
2017
m\leq 2017
m
≤
2017
and a positive integer
N
>
4
m
N>4m
N
>
4
m
such that
a
n
a
n
−
4
m
=
a
n
−
2
m
2
a_na_{n-4m}=a_{n-2m}^2
a
n
a
n
−
4
m
=
a
n
−
2
m
2
for every
n
>
N
n>N
n
>
N
.
1
2
Hide problems
Vietnam TST 2017 Problem 1
There are
44
44
44
distinct holes in a line and
2017
2017
2017
ants. Each ant comes out of a hole and crawls along the line with a constant speed into another hole, then comes in. Let
T
T
T
be the set of moments for which the ant comes in or out of the holes. Given that
∣
T
∣
≤
45
|T|\leq 45
∣
T
∣
≤
45
and the speeds of the ants are distinct. Prove that there exists two ants that don't collide.
Vietnam TST 2017 problem 4
Triangle
A
B
C
ABC
A
BC
is inscribed in circle
(
O
)
(O)
(
O
)
.
A
A
A
varies on
(
O
)
(O)
(
O
)
such that
A
B
>
B
C
AB>BC
A
B
>
BC
.
M
M
M
is the midpoint of
A
C
AC
A
C
. The circle with diameter
B
M
BM
BM
intersects
(
O
)
(O)
(
O
)
at
R
R
R
.
R
M
RM
RM
intersects
(
O
)
(O)
(
O
)
at
Q
Q
Q
and intersects
B
C
BC
BC
at
P
P
P
. The circle with diameter
B
P
BP
BP
intersects
A
B
,
B
O
AB, BO
A
B
,
BO
at
K
,
S
K,S
K
,
S
in this order. a. Prove that
S
R
SR
SR
passes through the midpoint of
K
P
KP
K
P
. b. Let
N
N
N
be the midpoint of
B
C
BC
BC
. The radical axis of circles with diameters
A
N
,
B
M
AN, BM
A
N
,
BM
intersects
S
R
SR
SR
at
E
E
E
. Prove that
M
E
ME
ME
always passes through a fixed point.