1984 IMO Problems/Problem 1
Let , , be nonnegative real numbers with . Show that
Note that this inequality is symmetric with x,y and z.
To prove note that implies that at most one of , , or is greater than . Suppose , WLOG. Then, since , implying all terms are positive.
To prove , suppose . Note that since at most one of x,y,z is . Suppose not all of them equals -otherwise, we would be done. This implies and . Thus, define , Then, , , and . After some simplification, since and . If we repeat the process, defining after similar reasoning, we see that .