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Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
1971 Vietnam National Olympiad
1971 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(2)
1
1
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m<n,p<q,(m,n)=1,(p,q)=1, m/n= tan \alpha p/q=tan\beta, \alpha+\beta=45^o
Consider positive integers
m
<
n
,
p
<
q
m <n,p < q
m
<
n
,
p
<
q
such that
(
m
,
n
)
=
1
,
(
p
,
q
)
=
1
(m, n) = 1, (p, q) = 1
(
m
,
n
)
=
1
,
(
p
,
q
)
=
1
and satisfy the condition that if
m
n
=
t
a
n
α
\frac{m}{n}= tan\alpha
n
m
=
t
an
α
and
p
q
=
t
a
n
β
\frac{p}{q} = tan\beta
q
p
=
t
an
β
, then
α
+
β
=
4
5
o
\alpha + \beta = 45^o
α
+
β
=
4
5
o
. i) Given
m
,
n
m, n
m
,
n
, find
p
,
q
p, q
p
,
q
. ii) Given
n
,
q
n, q
n
,
q
, find
m
,
p
m, p
m
,
p
. ii) Given
m
,
q
m, q
m
,
q
, find
n
,
p
n, p
n
,
p
.
2
1
Hide problems
6 questions in one geometry problem, starting with a cube, old Vietnamese
A
B
C
D
A
′
B
′
C
′
D
′
ABCDA'B'C'D'
A
BC
D
A
′
B
′
C
′
D
′
is a cube (with
A
B
C
D
ABCD
A
BC
D
and
A
′
B
′
C
′
D
′
A'B'C'D'
A
′
B
′
C
′
D
′
faces, and
A
A
′
,
B
B
′
,
C
C
′
,
D
D
′
AA', BB', CC', DD'
A
A
′
,
B
B
′
,
C
C
′
,
D
D
′
edges).
L
L
L
is a line which intersects or is parallel to the lines
A
A
′
,
B
C
AA', BC
A
A
′
,
BC
and
D
B
′
DB'
D
B
′
.
L
L
L
meets the line
B
C
BC
BC
at
M
M
M
(which may be the point at infinity). Let
m
=
∣
B
M
∣
m = |BM|
m
=
∣
BM
∣
. The plane
M
A
A
′
MAA'
M
A
A
′
meets the line
B
′
C
′
B'C'
B
′
C
′
at
E
E
E
. Show that
∣
B
′
E
∣
=
m
|B'E| = m
∣
B
′
E
∣
=
m
. The plane
M
D
B
′
MDB'
M
D
B
′
meets the line
A
′
D
′
A'D'
A
′
D
′
at
F
F
F
. Show that
∣
D
′
F
∣
=
m
|D'F| = m
∣
D
′
F
∣
=
m
. Hence or otherwise show how to construct the point
P
P
P
at the intersection of
L
L
L
and the plane
A
′
B
′
C
′
D
′
A'B'C'D'
A
′
B
′
C
′
D
′
. Find the distance between
P
P
P
and the line
A
′
B
′
A'B'
A
′
B
′
and the distance between
P
P
P
and the line
A
′
D
′
A'D'
A
′
D
′
in terms of
m
m
m
. Find a relation between these two distances that does not depend on
m
m
m
. Find the locus of
M
M
M
. Let
S
S
S
be the envelope of the line
L
L
L
as
M
M
M
varies. Find the intersection of
S
S
S
with the faces of the cube.