MathDB

5

Part of 2018 Czech-Polish-Slovak Junior Match

Problems(2)

perpendicularity for Czech, Polish and Slovak juniors

Source: Czech-Polish-Slovak Junior Match 2018, individual p5 CPSJ

2/1/2020
An acute triangle ABCABC is given in which AB<ACAB <AC. Point EE lies on the ACAC side of the triangle, with AB=AEAB = AE. The segment ADAD is the diameter of the circumcircle of the triangle ABCABC, and point SS is the center of this arc BCBC of this circle to which point AA does not belong. Point FF is symmetric of point DD wrt SS. Prove that lines FEF E and ACAC are perpendicular.
geometryperpendicularcircumcirclearc midpointSymmetric
2n people sit, knowing both neighbors in a round table

Source: Czech-Polish-Slovak Junior Match 2018, Team p5 CPSJ

3/7/2020
There are 2n2n people (n≥2n \ge 2) sitting around the round table, with each person getting to know both with his neighbors and exactly opposite him sits a person he does not know. Prove that people can rearrange in such a way that everyone knows one of their two neighbors.
combinatorics