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Croatia Mathematical Competitions

Geometry Difficulty 5.5 AIME, harder Prove it Croatia

Let $ABCD$ be a tetrahedron such that $\angle BAC = \angle CAD = \angle DAB = 90^\circ$ , $|AD| = 2\sqrt{2}$ and $|AB| = |AC| = 3$ holds. Determine the radius of the inscribed sphere of the tetrahedron. (Mea Bombardelli)

Solution

MATHEMATICAL COMPETITIONS IN CROATIA IN 2015
If $a_1$ and $z_1$ are connected by bus, let us put $A' = \{z_1, g, a_1, \dots, a_n\}$ and $Z' = \{z_2, \dots, z_m\}$ . Then $(A', Z')$ is a good pair and the number of elements of $A' \cup Z'$ is greater than the number of elements of $A \cup Z$ , which contradicts the assumption.
If $a_1$ and $z_1$ are connected by train, let us put $A'' = \{a_2, \dots, a_n\}$ and $Z'' = \{a_1, g, z_1, z_2, \dots, z_m\}$ . Then $(A'', Z'')$ is a good pair and the number of elements of $A'' \cup Z''$ is greater than the number of elements of $A \cup Z$ , which contradicts the assumption.
Since all cases lead to contradiction, we conclude that the assumption was wrong and that every city is either in the set $A$ or in the set $Z$ .
3.4. Let $F'$ be the intersection of the circumcircle of the triangle $ABD$ and the line $BG$ (different from $B$ ).

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