MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam Team Selection Test
2009 Vietnam Team Selection Test
2009 Vietnam Team Selection Test
Part of
Vietnam Team Selection Test
Subcontests
(3)
2
2
Hide problems
Sequence and coefficients of a 3-degree polynomial
Let a polynomial P(x) \equal{} rx^3 \plus{} qx^2 \plus{} px \plus{} 1
(
r
>
0
)
(r > 0)
(
r
>
0
)
such that the equation P(x) \equal{} 0 has only one real root. A sequence
(
a
n
)
(a_n)
(
a
n
)
is defined by a_0 \equal{} 1, a_1 \equal{} \minus{} p, a_2 \equal{} p^2 \minus{} q, a_{n \plus{} 3} \equal{} \minus{} pa_{n \plus{} 2} \minus{} qa_{n \plus{} 1} \minus{} ra_n. Prove that
(
a
n
)
(a_n)
(
a
n
)
contains an infinite number of nagetive real numbers.
Median pass over a fix point
Let a circle
(
O
)
(O)
(
O
)
with diameter
A
B
AB
A
B
. A point
M
M
M
move inside
(
O
)
(O)
(
O
)
. Internal bisector of
A
M
B
^
\widehat{AMB}
A
MB
cut
(
O
)
(O)
(
O
)
at
N
N
N
, external bisector of
A
M
B
^
\widehat{AMB}
A
MB
cut
N
A
,
N
B
NA,NB
N
A
,
NB
at
P
,
Q
P,Q
P
,
Q
.
A
M
,
B
M
AM,BM
A
M
,
BM
cut circle with diameter
N
Q
,
N
P
NQ,NP
NQ
,
NP
at
R
,
S
R,S
R
,
S
. Prove that: median from
N
N
N
of triangle
N
R
S
NRS
NRS
pass over a fix point.
3
1
Hide problems
Quadratic diophatine equation
Let a, b be positive integers. a, b and a.b are not perfect squares. Prove that at most one of following equations ax^2 \minus{} by^2 \equal{} 1 and ax^2 \minus{} by^2 \equal{} \minus{} 1 has solutions in positive integers.
1
2
Hide problems
Concurent!
Let an acute triangle
A
B
C
ABC
A
BC
with curcumcircle
(
O
)
(O)
(
O
)
. Call
A
1
,
B
1
,
C
1
A_1,B_1,C_1
A
1
,
B
1
,
C
1
are foots of perpendicular line from
A
,
B
,
C
A,B,C
A
,
B
,
C
to opposite side.
A
2
,
B
2
,
C
2
A_2,B_2,C_2
A
2
,
B
2
,
C
2
are reflect points of
A
1
,
B
1
,
C
1
A_1,B_1,C_1
A
1
,
B
1
,
C
1
over midpoints of
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
respectively. Circle
(
A
B
2
C
2
)
,
(
B
C
2
A
2
)
,
(
C
A
2
B
2
)
(AB_2C_2),(BC_2A_2),(CA_2B_2)
(
A
B
2
C
2
)
,
(
B
C
2
A
2
)
,
(
C
A
2
B
2
)
cut
(
O
)
(O)
(
O
)
at
A
3
,
B
3
,
C
3
A_3,B_3,C_3
A
3
,
B
3
,
C
3
respectively. Prove that:
A
1
A
3
,
B
1
B
3
,
C
1
C
3
A_1A_3,B_1B_3,C_1C_3
A
1
A
3
,
B
1
B
3
,
C
1
C
3
are concurent.
Vietnam TST 2009, Problem 4
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive numbers.Find
k
k
k
such that: (k \plus{} \frac {a}{b \plus{} c})(k \plus{} \frac {b}{c \plus{} a})(k \plus{} \frac {c}{a \plus{} b}) \ge (k \plus{} \frac {1}{2})^3