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max of x + y + z, if (x +cy<=36),(2x+ 3z>= 72), x,y,z >=0

Source: Vietnamese MO (VMO) 1966

August 22, 2018

algebrainequalitiesmaximum valueminimum value

Problem Statement

Let

x, y

and

z

be nonnegative real numbers satisfying the following conditions: (1)

x + cy \le 36

,(2)

2x+ 3z \le 72

, where

c

is a given positive number. Prove that if

c \ge 3

then the maximum of the sum

x + y + z

36

, while if

c < 3

, the maximum of the sum is

24 + \frac{36}{c}