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Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam Team Selection Test
2016 Vietnam Team Selection Test
2016 Vietnam Team Selection Test
Part of
Vietnam Team Selection Test
Subcontests
(6)
3
1
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fixed chord given, fixed point and fixed circle wanted, Vietnam TST 2016 p3
Let
A
B
C
ABC
A
BC
be triangle with circumcircle
(
O
)
(O)
(
O
)
of fixed
B
C
BC
BC
,
A
B
≠
A
C
AB \ne AC
A
B
=
A
C
and
B
C
BC
BC
not a diameter. Let
I
I
I
be the incenter of the triangle
A
B
C
ABC
A
BC
and
D
=
A
I
∩
B
C
,
E
=
B
I
∩
C
A
,
F
=
C
I
∩
A
B
D = AI \cap BC, E = BI \cap CA, F = CI \cap AB
D
=
A
I
∩
BC
,
E
=
B
I
∩
C
A
,
F
=
C
I
∩
A
B
. The circle passing through
D
D
D
and tangent to
O
A
OA
O
A
cuts for second time
(
O
)
(O)
(
O
)
at
G
G
G
(
G
≠
A
G \ne A
G
=
A
).
G
E
,
G
F
GE, GF
GE
,
GF
cut
(
O
)
(O)
(
O
)
also at
M
,
N
M, N
M
,
N
respectively. a) Let
H
=
B
M
∩
C
N
H = BM \cap CN
H
=
BM
∩
CN
. Prove that
A
H
AH
A
H
goes through a fixed point. b) Suppose
B
E
,
C
F
BE, CF
BE
,
CF
cut
(
O
)
(O)
(
O
)
also at
L
,
K
L, K
L
,
K
respectively and
A
H
∩
K
L
=
P
AH \cap KL = P
A
H
∩
K
L
=
P
. On
E
F
EF
EF
take
Q
Q
Q
for
Q
P
=
Q
I
QP = QI
QP
=
Q
I
. Let
J
J
J
be a point of the circimcircle of triangle
I
B
C
IBC
I
BC
so that
I
J
⊥
I
Q
IJ \perp IQ
I
J
⊥
I
Q
. Prove that the midpoint of
I
J
IJ
I
J
belongs to a fixed circle.
5
1
Hide problems
A binary representation problem
Given
n
n
n
numbers
a
1
,
a
2
,
.
.
.
,
a
n
a_1,a_2,...,a_n
a
1
,
a
2
,
...
,
a
n
(
n
≥
3
n\geq 3
n
≥
3
) where
a
i
∈
{
0
,
1
}
a_i\in\{0,1\}
a
i
∈
{
0
,
1
}
for all
i
=
1
,
2.
,
,
,
.
n
i=1,2.,,,.n
i
=
1
,
2.
,,,
.
n
. Consider
n
n
n
following
n
n
n
-tuples
S
1
=
(
a
1
,
a
2
,
.
.
.
,
a
n
−
1
,
a
n
)
S
2
=
(
a
2
,
a
3
,
.
.
.
,
a
n
,
a
1
)
⋮
S
n
=
(
a
n
,
a
1
,
.
.
.
,
a
n
−
2
,
a
n
−
1
)
.
\begin{aligned} S_1 & =(a_1,a_2,...,a_{n-1},a_n)\\ S_2 & =(a_2,a_3,...,a_n,a_1)\\ & \vdots\\ S_n & =(a_n,a_1,...,a_{n-2},a_{n-1}).\end{aligned}
S
1
S
2
S
n
=
(
a
1
,
a
2
,
...
,
a
n
−
1
,
a
n
)
=
(
a
2
,
a
3
,
...
,
a
n
,
a
1
)
⋮
=
(
a
n
,
a
1
,
...
,
a
n
−
2
,
a
n
−
1
)
.
For each tuple
r
=
(
b
1
,
b
2
,
.
.
.
,
b
n
)
r=(b_1,b_2,...,b_n)
r
=
(
b
1
,
b
2
,
...
,
b
n
)
, let
ω
(
r
)
=
b
1
⋅
2
n
−
1
+
b
2
⋅
2
n
−
2
+
⋯
+
b
n
.
\omega (r)=b_1\cdot 2^{n-1}+b_2\cdot 2^{n-2}+\cdots+b_n.
ω
(
r
)
=
b
1
⋅
2
n
−
1
+
b
2
⋅
2
n
−
2
+
⋯
+
b
n
.
Assume that the numbers
ω
(
S
1
)
,
ω
(
S
2
)
,
.
.
.
,
ω
(
S
n
)
\omega (S_1),\omega (S_2),...,\omega (S_n)
ω
(
S
1
)
,
ω
(
S
2
)
,
...
,
ω
(
S
n
)
receive exactly
k
k
k
different values.a) Prove that
k
∣
n
k|n
k
∣
n
and \frac{2^n-1}{2^k-1}|\omega (S_i) \forall i=1,2,...,n.b) Let
M
=
max
i
=
1
,
n
‾
ω
(
S
i
)
m
=
min
i
=
1
,
n
‾
ω
(
S
i
)
.
\begin{aligned} M & =\max _{i=\overline{1,n}}\omega (S_i)\\ m & =\min _{i=\overline{1,n}}\omega (S_i). \end{aligned}
M
m
=
i
=
1
,
n
max
ω
(
S
i
)
=
i
=
1
,
n
min
ω
(
S
i
)
.
Prove that
M
−
m
≥
(
2
n
−
1
)
(
2
k
−
1
−
1
)
2
k
−
1
.
M-m\geq\frac{(2^n-1)(2^{k-1}-1)}{2^k-1}.
M
−
m
≥
2
k
−
1
(
2
n
−
1
)
(
2
k
−
1
−
1
)
.
6
1
Hide problems
Vietnam TST 2016 Problem 6
Given
16
16
16
distinct real numbers
α
1
,
α
2
,
.
.
.
,
α
16
\alpha_1,\alpha_2,...,\alpha_{16}
α
1
,
α
2
,
...
,
α
16
. For each polynomial
P
P
P
, denote
V
(
P
)
=
P
(
α
1
)
+
P
(
α
2
)
+
.
.
.
+
P
(
α
16
)
.
V(P)=P(\alpha_1)+P(\alpha_2)+...+P(\alpha_{16}).
V
(
P
)
=
P
(
α
1
)
+
P
(
α
2
)
+
...
+
P
(
α
16
)
.
Prove that there is a monic polynomial
Q
Q
Q
,
deg
Q
=
8
\deg Q=8
de
g
Q
=
8
satisfying:i)
V
(
Q
P
)
=
0
V(QP)=0
V
(
QP
)
=
0
for all polynomial
P
P
P
has
deg
P
<
8
\deg P<8
de
g
P
<
8
.ii)
Q
Q
Q
has
8
8
8
real roots (including multiplicity).
4
1
Hide problems
Vietnam TST 2016 Problem 4
Given an acute triangle
A
B
C
ABC
A
BC
satisfying
∠
A
C
B
<
∠
A
B
C
<
∠
A
C
B
+
∠
B
A
C
2
\angle ACB<\angle ABC<\angle ACB+\dfrac{\angle BAC}{2}
∠
A
CB
<
∠
A
BC
<
∠
A
CB
+
2
∠
B
A
C
. Let
D
D
D
be a point on
B
C
BC
BC
such that
∠
A
D
C
=
∠
A
C
B
+
∠
B
A
C
2
\angle ADC=\angle ACB+\dfrac{\angle BAC}{2}
∠
A
D
C
=
∠
A
CB
+
2
∠
B
A
C
. Tangent of circumcircle of
A
B
C
ABC
A
BC
at
A
A
A
hits
B
C
BC
BC
at
E
E
E
. Bisector of
∠
A
E
B
\angle AEB
∠
A
EB
intersects
A
D
AD
A
D
and
(
A
D
E
)
(ADE)
(
A
D
E
)
at
G
G
G
and
F
F
F
respectively,
D
F
DF
D
F
hits
A
E
AE
A
E
at
H
.
H.
H
.
a) Prove that circle with diameter
A
E
,
D
F
,
G
H
AE,DF,GH
A
E
,
D
F
,
G
H
go through one common point. b) On the exterior bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
and ray
A
C
AC
A
C
given point
K
K
K
and
M
M
M
respectively satisfying
K
B
=
K
D
=
K
M
KB=KD=KM
K
B
=
KD
=
K
M
, On the exterior bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
and ray
A
B
AB
A
B
given point
L
L
L
and
N
N
N
respectively satisfying
L
C
=
L
D
=
L
N
.
LC=LD=LN.
L
C
=
L
D
=
L
N
.
Circle throughs
M
,
N
M,N
M
,
N
and midpoint
I
I
I
of
B
C
BC
BC
hits
B
C
BC
BC
at
P
P
P
(
P
≠
I
P\neq I
P
=
I
). Prove that
B
M
,
C
N
,
A
P
BM,CN,AP
BM
,
CN
,
A
P
concurrent.
1
1
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Vietnam TST 2016 Problem 1
Find all
a
,
n
∈
Z
+
a,n\in\mathbb{Z}^+
a
,
n
∈
Z
+
(
a
>
2
a>2
a
>
2
) such that each prime divisor of
a
n
−
1
a^n-1
a
n
−
1
is also prime divisor of
a
3
2016
−
1
a^{3^{2016}}-1
a
3
2016
−
1
2
1
Hide problems
Vietnam TST 2016 Problem 2
Let
A
A
A
be a set contains
2000
2000
2000
distinct integers and
B
B
B
be a set contains
2016
2016
2016
distinct integers.
K
K
K
is the numbers of pairs
(
m
,
n
)
(m,n)
(
m
,
n
)
satisfying
{
m
∈
A
,
n
∈
B
∣
m
−
n
∣
≤
1000
\begin{cases} m\in A, n\in B\\ |m-n|\leq 1000 \end{cases}
{
m
∈
A
,
n
∈
B
∣
m
−
n
∣
≤
1000
Find the maximum value of
K
K
K