| 1. ABC is an equilateral triangle. D and D' are points on opposite sides of the plane ABC such that the two tetrahedra ABCD and ABCD' are congruent (but not necessarily with the vertices in that order). If the polyhedron with the five vertices A, B, C, D, D' is such that the angle between any two adjacent faces is the same, find DD'/AB. | |
| 2. At a party every girl danced with at least one boy, but not with all of them. Similarly, every boy danced with at least one girl, but not with all of them. Show that there were two girls G and G' and two boys B and B', such that each of B and G danced, B' and G' danced, but B and G' did not dance, and B' and G did not dance. | |
| 3. Show that for any positive reals w, x, y, z we have ( (w2 + x2 + y2 + z2)/4)1/2 ≥ ( (wxy + wxz + wyz + xyz)/4)1/3. |
The original problems are in Hungarian. Many thanks to Péter Dombi for the translation.
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(C) John Scholes
jscholes@kalva.demon.co.uk
19 Apr 2003
Last corrected/updated 19 Apr 2003