1984 IMO Problems/Problem 1
Contents
Problem
Let ,
,
be nonnegative real numbers with
. Show that
Solution
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
.
Video Solution
See Also
1984 IMO (Problems) • Resources | ||
Preceded by First Question |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 2 |
All IMO Problems and Solutions |