Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.5 AIME Find the answer

3. Find conditions on the positive real number $a$ such that there exists a tetrahedron $k$ of whose edges $(k=1,2,3,4,5)$ have length $a$ , and the other $6-k$ edges have length 1. Second Day (July 11)

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

3. We have several cases: $1^{\circ} k=1$ . W.l.o.g. let $AB=a$ and the remaining segments have length 1. Let $M$ be the midpoint of $CD$ . Then $AM=BM=\sqrt{3} / 2$ (triangles $CDA$ and $CDB$ are equilateral) and $01$ . Assume $AB=AC=AD=a$ . Varying $A$ along the line perpendicular to the plane $BCD$ and through the center of $\triangle BCD$ we achieve all values of $a>1 / \sqrt{3}$ . For $a\sqrt{2-\sqrt{3}}$ . $5^{\circ} k=5$ . We reduce to $k=1$ and get $a>1 / \sqrt{3}$ .

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.