MathDB

7

Part of 2006 IMO Shortlist

Problems(3)

common tangents and a superb similar triangle with ABC

Source: IMO Shortlist 2006, Geometry 7

6/28/2007
In a triangle ABC ABC, let Ma M_{a}, Mb M_{b}, Mc M_{c} be the midpoints of the sides BC BC, CA CA, AB AB, respectively, and Ta T_{a}, Tb T_{b}, Tc T_{c} be the midpoints of the arcs BC BC, CA CA, AB AB of the circumcircle of ABC ABC, not containing the vertices A A, B B, C C, respectively. For i∈{a,b,c} i \in \left\{a, b, c\right\}, let wi w_{i} be the circle with MiTi M_{i}T_{i} as diameter. Let pi p_{i} be the common external common tangent to the circles wj w_{j} and wk w_{k} (for all {i,j,k}={a,b,c} \left\{i, j, k\right\}= \left\{a, b, c\right\}) such that wi w_{i} lies on the opposite side of pi p_{i} than wj w_{j} and wk w_{k} do. Prove that the lines pa p_{a}, pb p_{b}, pc p_{c} form a triangle similar to ABC ABC and find the ratio of similitude.
Proposed by Tomas Jurik, Slovakia
geometrycircumcirclereflectionhomothetyIMO Shortlist
"antipodal" points of a polyhaedron

Source: IMO Shortlist 2006, Combinatorics 7

6/28/2007
Consider a convex polyhedron without parallel edges and without an edge parallel to any face other than the two faces adjacent to it. Call a pair of points of the polyhedron antipodal if there exist two parallel planes passing through these points and such that the polyhedron is contained between these planes. Let AA be the number of antipodal pairs of vertices, and let BB be the number of antipodal pairs of midpoint edges. Determine the difference A−BA-B in terms of the numbers of vertices, edges, and faces.
Proposed by Kei Irei, Japan
geometry3D geometryEulerpolyhedronIMO Shortlist
n divides 2^m+m

Source: IMO Shortlist 2006, N7, AIMO 2007, TST 7, P3

6/19/2007
For all positive integers nn, show that there exists a positive integer mm such that nn divides 2m+m2^{m} + m.
Proposed by Juhan Aru, Estonia
modular arithmeticnumber theoryDivisibilityexponentialIMO Shortlist