MathDB

3

Part of 2007 Middle European Mathematical Olympiad

Problems(2)

5 Circles and a Rectangle (nice but well known)

Source: MEMO Individual Competition, Question 3

9/24/2007
Let k k be a circle and k1,k2,k3,k4 k_{1},k_{2},k_{3},k_{4} four smaller circles with their centres O1,O2,O3,O4 O_{1},O_{2},O_{3},O_{4} respectively, on k k. For i \equal{} 1,2,3,4 and k_{5}\equal{} k_{1} the circles ki k_{i} and k_{i\plus{}1} meet at Ai A_{i} and Bi B_{i} such that Ai A_{i} lies on k k. The points O1,A1,O2,A2,O3,A3,O4,A4 O_{1},A_{1},O_{2},A_{2},O_{3},A_{3},O_{4},A_{4} lie in that order on k k and are pairwise different. Prove that B1B2B3B4 B_{1}B_{2}B_{3}B_{4} is a rectangle.
geometryrectanglegeometry proposed
Tetrahedron with Integer Sidelengths

Source: MEMO Team Competition, Quesiton 7

9/24/2007
A tetrahedron is called a MEMO-tetrahedron if all six sidelengths are different positive integers where one of them is 2 2 and one of them is 3 3. Let l(T) l(T) be the sum of the sidelengths of the tetrahedron T T. (a) Find all positive integers n n so that there exists a MEMO-Tetrahedron T T with l(T)\equal{}n. (b) How many pairwise non-congruent MEMO-tetrahedrons T T satisfying l(T)\equal{}2007 exist? Two tetrahedrons are said to be non-congruent if one cannot be obtained from the other by a composition of reflections in planes, translations and rotations. (It is not neccessary to prove that the tetrahedrons are not degenerate, i.e. that they have a positive volume).
geometry3D geometrytetrahedrongeometric transformationreflectiongeometry proposed