MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam Team Selection Test
2015 Vietnam Team selection test
2015 Vietnam Team selection test
Part of
Vietnam Team Selection Test
Subcontests
(4)
Problem 5
1
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7 circles, Vietnam TST 2015 (VNTST) P5
Let
A
B
C
ABC
A
BC
be a triangle with an interior point
P
P
P
such that
∠
A
P
B
=
∠
A
P
C
=
α
\angle APB = \angle APC = \alpha
∠
A
PB
=
∠
A
PC
=
α
and
α
>
18
0
o
−
∠
B
A
C
\alpha > 180^o-\angle BAC
α
>
18
0
o
−
∠
B
A
C
. The circumcircle of triangle
A
P
B
APB
A
PB
cuts
A
C
AC
A
C
at
E
E
E
, the circumcircle of triangle
A
P
C
APC
A
PC
cuts
A
B
AB
A
B
at
F
F
F
. Let
Q
Q
Q
be the point in the triangle
A
E
F
AEF
A
EF
such that
∠
A
Q
E
=
∠
A
Q
F
=
α
\angle AQE = \angle AQF =\alpha
∠
A
QE
=
∠
A
QF
=
α
. Let
D
D
D
be the symmetric point of
Q
Q
Q
wrt
E
F
EF
EF
. Angle bisector of
∠
E
D
F
\angle EDF
∠
E
D
F
cuts
A
P
AP
A
P
at
T
T
T
. a) Prove that
∠
D
E
T
=
∠
A
B
C
,
∠
D
F
T
=
∠
A
C
B
\angle DET = \angle ABC, \angle DFT = \angle ACB
∠
D
ET
=
∠
A
BC
,
∠
D
FT
=
∠
A
CB
. b) Straight line
P
A
PA
P
A
cuts straight lines
D
E
,
D
F
DE, DF
D
E
,
D
F
at
M
,
N
M, N
M
,
N
respectively. Denote
I
,
J
I, J
I
,
J
the incenters of the triangles
P
E
M
,
P
F
N
PEM, PFN
PEM
,
PFN
, and
K
K
K
the circumcenter of the triangle
D
I
J
DIJ
D
I
J
. Straight line
D
T
DT
D
T
cut
(
K
)
(K)
(
K
)
at
H
H
H
. Prove that
H
K
HK
HK
passes through the incenter of the triangle
D
M
N
DMN
D
MN
.
Problem 2
1
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fixed circle and fixed point
sorry if this has been posted before .given a fixed circle
(
O
)
(O)
(
O
)
and two fixed point
B
,
C
B,C
B
,
C
on it.point A varies on circle
(
O
)
(O)
(
O
)
. let
I
I
I
be the midpoint of
B
C
BC
BC
and
H
H
H
be the orthocenter of
△
A
B
C
\triangle ABC
△
A
BC
. ray
I
H
IH
I
H
meet
(
O
)
(O)
(
O
)
at
K
K
K
,
A
H
AH
A
H
meet
B
C
BC
BC
at
D
D
D
,
K
D
KD
KD
meet
(
O
)
(O)
(
O
)
at
M
M
M
.a line pass
M
M
M
and perpendicular to
B
C
BC
BC
meet
A
I
AI
A
I
at
N
N
N
. a) prove that
N
N
N
varies on a fixed circle. b) a circle pass
N
N
N
and tangent to
A
K
AK
A
K
at
A
A
A
cut
A
B
,
A
C
AB,AC
A
B
,
A
C
at
P
,
Q
P,Q
P
,
Q
. let
J
J
J
be the midpoint of
P
Q
PQ
PQ
.prove that
A
J
AJ
A
J
pass through a fixed point.
Problem 6
1
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VietNam TST 2015, day 2, problem 6
Find the smallest positive interger number
n
n
n
such that there exists
n
n
n
real numbers
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\ldots,a_n
a
1
,
a
2
,
…
,
a
n
satisfied three conditions as follow: a.
a
1
+
a
2
+
⋯
+
a
n
>
0
a_1+a_2+\cdots+a_n>0
a
1
+
a
2
+
⋯
+
a
n
>
0
; b.
a
1
3
+
a
2
3
+
⋯
+
a
n
3
<
0
a_1^3+a_2^3+\cdots+a_n^3<0
a
1
3
+
a
2
3
+
⋯
+
a
n
3
<
0
; c.
a
1
5
+
a
2
5
+
⋯
+
a
n
5
>
0
a_1^5+a_2^5+\cdots+a_n^5>0
a
1
5
+
a
2
5
+
⋯
+
a
n
5
>
0
.
Problem 1
1
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VietNam TST 2015, day 1, problem 1
Let
α
\alpha
α
be the positive root of the equation
x
2
+
x
=
5
x^2+x=5
x
2
+
x
=
5
. Let
n
n
n
be a positive integer number, and let
c
0
,
c
1
,
…
,
c
n
∈
N
c_0,c_1,\ldots,c_n\in \mathbb{N}
c
0
,
c
1
,
…
,
c
n
∈
N
be such that
c
0
+
c
1
α
+
c
2
α
2
+
⋯
+
c
n
α
n
=
2015.
c_0+c_1\alpha+c_2\alpha^2+\cdots+c_n\alpha^n=2015.
c
0
+
c
1
α
+
c
2
α
2
+
⋯
+
c
n
α
n
=
2015.
a. Prove that
c
0
+
c
1
+
c
2
+
⋯
+
c
n
≡
2
(
m
o
d
3
)
c_0+c_1+c_2+\cdots+c_n\equiv 2 \pmod{3}
c
0
+
c
1
+
c
2
+
⋯
+
c
n
≡
2
(
mod
3
)
. b. Find the minimum value of the sum
c
0
+
c
1
+
c
2
+
⋯
+
c
n
c_0+c_1+c_2+\cdots+c_n
c
0
+
c
1
+
c
2
+
⋯
+
c
n
.