MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
1972 Vietnam National Olympiad
1972 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(3)
3
1
Hide problems
midpoints, parallels, intersections, collinear in the end
A
B
C
ABC
A
BC
is a triangle.
U
U
U
is a point on the line
B
C
BC
BC
.
I
I
I
is the midpoint of
B
C
BC
BC
. The line through
C
C
C
parallel to
A
I
AI
A
I
meets the line
A
U
AU
A
U
at
E
E
E
. The line through
E
E
E
parallel to
B
C
BC
BC
meets the line
A
B
AB
A
B
at
F
F
F
. The line through
E
E
E
parallel to
A
B
AB
A
B
meets the line
B
C
BC
BC
at
H
H
H
. The line through
H
H
H
parallel to
A
U
AU
A
U
meets the line
A
B
AB
A
B
at
K
K
K
. The lines
H
K
HK
HK
and
F
G
FG
FG
meet at
T
.
V
T. V
T
.
V
is the point on the line
A
U
AU
A
U
such that
A
A
A
is the midpoint of
U
V
UV
U
V
. Show that
V
,
T
V, T
V
,
T
and
I
I
I
are collinear.
4
1
Hide problems
Compute the volume of EFGE'F'G', starting with a regular tetrahedron
Let
A
B
C
D
ABCD
A
BC
D
be a regular tetrahedron with side
a
a
a
. Take
E
,
E
′
E,E'
E
,
E
′
on the edge
A
B
,
F
,
F
′
AB, F, F'
A
B
,
F
,
F
′
on the edge
A
C
AC
A
C
and
G
,
G
′
G,G'
G
,
G
′
on the edge AD so that
A
E
=
a
/
6
,
A
E
′
=
5
a
/
6
,
A
F
=
a
/
4
,
A
F
′
=
3
a
/
4
,
A
G
=
a
/
3
,
A
G
′
=
2
a
/
3
AE =a/6,AE' = 5a/6,AF= a/4,AF'= 3a/4,AG = a/3,AG'= 2a/3
A
E
=
a
/6
,
A
E
′
=
5
a
/6
,
A
F
=
a
/4
,
A
F
′
=
3
a
/4
,
A
G
=
a
/3
,
A
G
′
=
2
a
/3
. Compute the volume of
E
F
G
E
′
F
′
G
′
EFGE'F'G'
EFG
E
′
F
′
G
′
in term of
a
a
a
and find the angles between the lines
A
B
,
A
C
,
A
D
AB,AC,AD
A
B
,
A
C
,
A
D
and the plane
E
F
G
EFG
EFG
.
1
1
Hide problems
T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x), polynomial, cosn\alpha=T_n(x)
Let
α
\alpha
α
be an arbitrary angle and let
x
=
c
o
s
α
,
y
=
c
o
s
n
α
x = cos\alpha, y = cosn\alpha
x
=
cos
α
,
y
=
cos
n
α
(
n
∈
Z
n \in Z
n
∈
Z
). i) Prove that to each value
x
∈
[
−
1
,
1
]
x \in [-1, 1]
x
∈
[
−
1
,
1
]
corresponds one and only one value of
y
y
y
. Thus we can write
y
y
y
as a function of
x
,
y
=
T
n
(
x
)
x, y = T_n(x)
x
,
y
=
T
n
(
x
)
. Compute
T
1
(
x
)
,
T
2
(
x
)
T_1(x), T_2(x)
T
1
(
x
)
,
T
2
(
x
)
and prove that
T
n
+
1
(
x
)
=
2
x
T
n
(
x
)
−
T
n
−
1
(
x
)
T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x)
T
n
+
1
(
x
)
=
2
x
T
n
(
x
)
−
T
n
−
1
(
x
)
. From this it follows that
T
n
(
x
)
T_n(x)
T
n
(
x
)
is a polynomial of degree
n
n
n
. ii) Prove that the polynomial
T
n
(
x
T_n(x
T
n
(
x
) has
n
n
n
distinct roots in
[
−
1
,
1
]
[-1, 1]
[
−
1
,
1
]
.