IMO Shortlist 1969 problem 69
Dodao/la:
arhiva2. travnja 2012. ![(YUG 1)](/media/m/d/9/e/d9ef8ca09ae4e6eb077193e729116c4d.png)
Suppose that positive real numbers
![x_1, x_2, x_3](/media/m/9/d/6/9d6ad78bcb42c492dcb58a82bdde98e8.png)
satisfy
![x_1x_2x_3 > 1, x_1 + x_2 + x_3 <\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}](/media/m/6/1/5/61537bb93d72b690ccb4427b69c11263.png)
Prove that:
![(a)](/media/m/a/7/f/a7fedf50ce0b917a00dd07d5233906f1.png)
None of
![x_1, x_2, x_3](/media/m/9/d/6/9d6ad78bcb42c492dcb58a82bdde98e8.png)
equals
![1](/media/m/a/9/1/a913f49384c0227c8ea296a725bfc987.png)
.
![(b)](/media/m/9/2/7/92773ef234467079b4efc86655fdc459.png)
Exactly one of these numbers is less than
%V0
$(YUG 1)$ Suppose that positive real numbers $x_1, x_2, x_3$ satisfy
$x_1x_2x_3 > 1, x_1 + x_2 + x_3 <\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}$
Prove that:
$(a)$ None of $x_1, x_2, x_3$ equals $1$.
$(b)$ Exactly one of these numbers is less than $1.$
Izvor: Međunarodna matematička olimpijada, shortlist 1969