MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
1970 Vietnam National Olympiad
1970 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(5)
4
1
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construction, locus, starting with perpendicular diameteres and tangents
A
B
AB
A
B
and
C
D
CD
C
D
are perpendicular diameters of a circle.
L
L
L
is the tangent to the circle at
A
A
A
.
M
M
M
is a variable point on the minor arc
A
C
AC
A
C
. The ray
B
M
,
D
M
BM, DM
BM
,
D
M
meet the line
L
L
L
at
P
P
P
and
Q
Q
Q
respectively. Show that
A
P
⋅
A
Q
=
A
B
⋅
P
Q
AP\cdot AQ = AB\cdot PQ
A
P
⋅
A
Q
=
A
B
⋅
PQ
. Show how to construct the point
M
M
M
which gives
B
Q
BQ
BQ
parallel to
D
P
DP
D
P
. If the lines
O
P
OP
OP
and
B
Q
BQ
BQ
meet at
N
N
N
find the locus of
N
N
N
. The lines
B
P
BP
BP
and
B
Q
BQ
BQ
meet the tangent at
D
D
D
at
P
′
P'
P
′
and
Q
′
Q'
Q
′
respectively. Find the relation between
P
′
P'
P
′
and
Q
Q
Q
'. The lines
D
D
D
P and
D
Q
DQ
D
Q
meet the line
B
C
BC
BC
at
P
"
P"
P
"
and
Q
"
Q"
Q
"
respectively. Find the relation between
P
"
P"
P
"
and
Q
"
Q"
Q
"
.
3
1
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f(x,y), f(x,0)=ax, f(c,d) = f(h,k) is constant on the line through (c,d),(h,k)
The function
f
(
x
,
y
)
f(x, y)
f
(
x
,
y
)
is defined for all real numbers
x
,
y
x, y
x
,
y
. It satisfies
f
(
x
,
0
)
=
a
x
f(x,0) = ax
f
(
x
,
0
)
=
a
x
(where
a
a
a
is a non-zero constant) and if
(
c
,
d
)
(c, d)
(
c
,
d
)
and
(
h
,
k
)
(h, k)
(
h
,
k
)
are distinct points such that
f
(
c
,
d
)
=
f
(
h
,
k
)
f(c, d) = f(h, k)
f
(
c
,
d
)
=
f
(
h
,
k
)
, then
f
(
x
,
y
)
f(x, y)
f
(
x
,
y
)
is constant on the line through
(
c
,
d
)
(c, d)
(
c
,
d
)
and
(
h
,
k
)
(h, k)
(
h
,
k
)
. Show that for any real
b
b
b
, the set of points such that
f
(
x
,
y
)
=
b
f(x, y) = b
f
(
x
,
y
)
=
b
is a straight line and that all such lines are parallel. Show that
f
(
x
,
y
)
=
a
x
+
b
y
f(x, y) = ax + by
f
(
x
,
y
)
=
a
x
+
b
y
, for some constant
b
b
b
.
1
1
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sin A/2 sin B /2 sin C/ 2 < 1/ 4.
Prove that for an arbitrary triangle
A
B
C
ABC
A
BC
:
s
i
n
A
2
s
i
n
B
2
s
i
n
C
2
<
1
4
sin \frac{A}{2} sin \frac{B}{2} sin \frac{C}{2} < \frac{1}{4}
s
in
2
A
s
in
2
B
s
in
2
C
<
4
1
.
2
1
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Divisors
Let
N
=
1890
∗
1930
∗
1970
N=1890*1930*1970
N
=
1890
∗
1930
∗
1970
, find the number of divisors of N which are not divisors of
45
45
45
5
1
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VietNam MO 1970-Pr.B2
A plane
p
p
p
passes through a vertex of a cube so that the three edges at the vertex make equal angles with
p
p
p
. Find the cosine of this angle. Find the positions of the feet of the perpendiculars from the vertices of the cube onto
p
p
p
. There are 28 lines through two vertices of the cube and 20 planes through three vertices of the cube. Find some relationship between these lines and planes and the plane
p
p
p
.