MathDB

3

Part of 2019 Middle European Mathematical Olympiad

Problems(2)

4 concyclic points

Source: 2019 MEMO Problem I-3

8/29/2019
Let ABCABC be an acute-angled triangle with AC>BCAC>BC and circumcircle ω\omega. Suppose that PP is a point on ω\omega such that AP=ACAP=AC and that PP is an interior point on the shorter arc BCBC of ω\omega. Let QQ be the intersection point of the lines APAP and BCBC. Furthermore, suppose that RR is a point on ω\omega such that QA=QRQA=QR and RR is an interior point of the shorter arc ACAC of ω\omega. Finally, let SS be the point of intersection of the line BCBC with the perpendicular bisector of the side ABAB. Prove that the points P,Q,RP, Q, R and SS are concyclic.
Proposed by Patrik Bak, Slovakia
geometrycircumcircleperpendicular bisectorMEMO 2019memomoving points
n boys and n girls

Source: 2019 MEMO Problem T-3

8/30/2019
There are nn boys and nn girls in a school class, where nn is a positive integer. The heights of all the children in this class are distinct. Every girl determines the number of boys that are taller than her, subtracts the number of girls that are taller than her, and writes the result on a piece of paper. Every boy determines the number of girls that are shorter than him, subtracts the number of boys that are shorter than him, and writes the result on a piece of paper. Prove that the numbers written down by the girls are the same as the numbers written down by the boys (up to a permutation).
Proposed by Stephan Wagner, Austria
combinatoricsmemoMEMO 2019induction