MathDB

2

Part of 2006 Switzerland Team Selection Test

Problems(4)

Problem with n-gone

Source: Swiss Imo Selection 2006

5/25/2006
Let n≥5n\ge5 be an integer. Find the biggest integer kk such that there always exists a nn-gon with exactly kk interior right angles. (Find kk in terms of nn).
geometryalgebra proposedalgebra
Two circumscribed circles

Source: Swiss Imo Selection 2006

5/25/2006
Let DD be inside △ABC\triangle ABC and EE on ADAD different to DD. Let ω1\omega_1 and ω2\omega_2 be the circumscribed circles of △BDE\triangle BDE and △CDE\triangle CDE respectively. ω1\omega_1 and ω2\omega_2 intersect BCBC in the interior points FF and GG respectively. Let XX be the intersection between DGDG and ABAB and YY the intersection between DFDF and ACAC. Show that XYXY is ∥\| to BCBC.
geometrycircumcirclepower of a pointradical axisgeometry proposed
2006 numbers around a circle

Source: Swiss Imo Selection 2006

5/25/2006
We place randomly the numbers 1,2,…,20061,2, \dots ,2006 around a circle. A move consists of changing two neighbouring numbers. After a limited numbers of moves all the numbers are diametrically opposite to their starting position. Show that we changed at least once two numbers which had the sum 20072007.
combinatorics proposedcombinatorics
3^k+5^k is the power of a natural number

Source: Swiss Imo Selection 2006

5/25/2006
Find all naturals kk such that 3k+5k3^k+5^k is the power of a natural number with the exponent ≥2\ge 2.
number theory proposednumber theory