Difference between revisions of "1970 IMO Problems/Problem 5"
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Latest revision as of 22:35, 18 July 2016
Problem
In the tetrahedron 
, angle 
is a right angle. Suppose that the foot 
of the perpendicular from 
to the plane 
in the tetrahedron is the intersection of the altitudes of 
. Prove that
.
For what tetrahedra does equality hold?
Solution
Let us show first that angles 
and 
are also right. Let be the intersection of the altitudes
of and let 
meet 
at 
. Planes 
and are perpendicular and is perpendicular to
the line of intersection 
. Hence is perpendicular to the plane 
and hence to 
. So 
Also 
Therefore 
But 
so 
, so angle 
. But angle 
, so 
is perpendicular to
the plane 
, and hence angle 
= 
. Similarly, angle 
.
Hence 
. But now we are done, because Cauchy's
inequality gives 
We have equality if and only if
we have equality in Cauchy's inequality, which means 
| 1970 IMO (Problems) • Resources | ||
| Preceded by Problem 4 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 6 |
| All IMO Problems and Solutions | ||
