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Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
1978 Vietnam National Olympiad
1978 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(6)
6
1
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3 distances form a rectangular parallelepiped, are sidelengths of triangle
Given a rectangular parallelepiped
A
B
C
D
A
′
B
′
C
′
D
′
ABCDA'B'C'D'
A
BC
D
A
′
B
′
C
′
D
′
with the bases
A
B
C
D
,
A
′
B
′
C
′
D
′
ABCD, A'B'C'D'
A
BC
D
,
A
′
B
′
C
′
D
′
, the edges
A
A
′
,
B
B
′
,
C
C
′
,
D
D
′
AA',BB', CC',DD'
A
A
′
,
B
B
′
,
C
C
′
,
D
D
′
and
A
B
=
a
,
A
D
=
b
,
A
A
′
=
c
AB = a,AD = b,AA' = c
A
B
=
a
,
A
D
=
b
,
A
A
′
=
c
. Show that there exists a triangle with the sides equal to the distances from
A
,
A
′
,
D
A,A',D
A
,
A
′
,
D
to the diagonal
B
D
′
BD'
B
D
′
of the parallelepiped. Denote those distances by
m
1
,
m
2
,
m
3
m_1,m_2,m_3
m
1
,
m
2
,
m
3
. Find the relationship between
a
,
b
,
c
,
m
1
,
m
2
,
m
3
a, b, c,m_1,m_2,m_3
a
,
b
,
c
,
m
1
,
m
2
,
m
3
.
5
1
Hide problems
a river with a right-angle bend, longest boat of length c wanted
A river has a right-angle bend. Except at the bend, its banks are parallel lines of distance
a
a
a
apart. At the bend the river forms a square with the river flowing in across one side and out across an adjacent side. What is the longest boat of length
c
c
c
and negligible width which can pass through the bend?
4
1
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3 rational numbers in arithmetic progression, system
Find three rational numbers
a
d
,
b
d
,
c
d
\frac{a}{d}, \frac{b}{d}, \frac{c}{d}
d
a
,
d
b
,
d
c
in their lowest terms such that they form an arithmetic progression and
b
a
=
a
+
1
d
+
1
,
c
b
=
b
+
1
d
+
1
\frac{b}{a} =\frac{a + 1}{d + 1}, \frac{c}{b} = \frac{b + 1}{d + 1}
a
b
=
d
+
1
a
+
1
,
b
c
=
d
+
1
b
+
1
.
3
1
Hide problems
minimize 5 PA + 4 PB + 3 PC, for P interior point of triangle
The triangle
A
B
C
ABC
A
BC
has angle
A
=
3
0
o
A = 30^o
A
=
3
0
o
and
A
B
=
3
4
A
C
AB = \frac{3}{4} AC
A
B
=
4
3
A
C
. Find the point
P
P
P
inside the triangle which minimizes
5
P
A
+
4
P
B
+
3
P
C
5 PA + 4 PB + 3 PC
5
P
A
+
4
PB
+
3
PC
.
2
1
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non linear system with parameter
Find all values of the parameter
m
m
m
such that the equations
x
2
=
2
∣
x
∣
+
∣
x
∣
−
y
−
m
=
1
−
y
2
x^2 = 2^{|x|} + |x| - y - m = 1 - y^2
x
2
=
2
∣
x
∣
+
∣
x
∣
−
y
−
m
=
1
−
y
2
have only one root.
1
1
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find all three digit numbers abc such that 2(abc) = bca + cab
Find all three digit numbers
a
b
c
‾
\overline{abc}
ab
c
such that
2
⋅
a
b
c
‾
=
b
c
a
‾
+
c
a
b
‾
2 \cdot \overline{abc} = \overline{bca} + \overline{cab}
2
⋅
ab
c
=
b
c
a
+
c
ab
.