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Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
1966 Vietnam National Olympiad
1966 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(2)
1
1
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max of x + y + z, if (x +cy<=36),(2x+ 3z>= 72), x,y,z >=0
Let
x
,
y
x, y
x
,
y
and
z
z
z
be nonnegative real numbers satisfying the following conditions: (1)
x
+
c
y
≤
36
x + cy \le 36
x
+
cy
≤
36
,(2)
2
x
+
3
z
≤
72
2x+ 3z \le 72
2
x
+
3
z
≤
72
, where
c
c
c
is a given positive number. Prove that if
c
≥
3
c \ge 3
c
≥
3
then the maximum of the sum
x
+
y
+
z
x + y + z
x
+
y
+
z
is
36
36
36
, while if
c
<
3
c < 3
c
<
3
, the maximum of the sum is
24
+
36
c
24 + \frac{36}{c}
24
+
c
36
.
2
1
Hide problems
VietNam MO 1966-Pr.2
a
,
b
a, b
a
,
b
are two fixed lines through
O
O
O
. Variable lines
x
,
y
x, y
x
,
y
are parallel.
x
x
x
intersects a at
A
A
A
and
b
b
b
at
C
C
C
,
y
y
y
intersects
a
a
a
at
B
B
B
and
b
b
b
at
D
D
D
. The lines
A
D
AD
A
D
and
B
C
BC
BC
meet at
M
M
M
. The line through
M
M
M
parallel to
x
x
x
meets
a
a
a
at
L
L
L
and
b
b
b
at
N
N
N
. What can you say about
L
,
M
,
N
L, M, N
L
,
M
,
N
? Find the locus
M
M
M
.