Difference between revisions of "1972 USAMO Problems/Problem 2"

Revision as of 17:51, 3 July 2013

Problem

A given tetrahedron $ABCD$ is isosceles, that is, $AB=CD, AC=BD, AD=BC$ . Show that the faces of the tetrahedron are acute-angled triangles.

Solutions

Solution 1

Suppose $\triangle ABD$ is fixed. By the equality conditions, it follows that the maximal possible value of $BC$ occurs when the four vertices are coplanar, with $C$ on the opposite side of $\overline{AD}$ as $B$ . In this case, the tetrahedron is not actually a tetrahedron, so this maximum isn't actually attainable.

For the sake of contradiction, suppose $\angle ABD$ is non-acute. Then, $(AD)^2\geq (AB)^2+(BD)^2$ . In our optimal case noted above, $ACDB$ is a parallelogram, so $\begin{align*} 2(BD)^2 + 2(AB)^2 &= (AD)^2 + (CB)^2 \\ &= 2(AD)^2 \\ &\geq 2(BD)^2+2(AB)^2. \end{align*}$ However, as stated, equality cannot be attained, so we get our desired contradiction.

Solution 2

It's not hard to see that the four faces are congruent from SSS Congruence. Without loss of generality, assume that $AB\leq BC \leq CA$ . Now assume, for the sake of contradiction, that each face is non-acute; that is, right or isosceles. Consider triangles $\triangle ABC$ and $\triangle ABD$ . They share side $AB$ . Let $k$ and $l$ be the planes passing through $A$ and $B$ , respectively, that are perpendicular to side $AB$ . We have that triangles $ABC$ and $ABD$ are non-acute, so $C$ and $D$ are not strictly between planes $k$ and $l$ . Therefore the length of $CD$ is at least the distance between the planes, which is $AB$ . However, if $CD=AB$ , then the four points $A$ , $B$ , $C$ , and $D$ are coplanar, and the volume of $ABCD$ would be zero. Therefore $CD>AB$ . However, we were given that $CD=AB$ in the problem, which leads to a contradiction. Therefore the faces of the tetrahedron must all be acute.

Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.

@@ Line 30: / Line 30: @@
 {{USAMO box|year=1972|num-b=1|num-a=3}}
+{{MAA Notice}}
 [[Category:Olympiad Geometry Problems]]

1972 USAMO (Problems • Resources)
Preceded by Problem 1	Followed by Problem 3
1 • 2 • 3 • 4 • 5
All USAMO Problems and Solutions

Page

Toolbox

Search