MathDB

3

Part of 2015 Middle European Mathematical Olympiad

Problems(2)

Lots of cyclic quadrilaterals

Source: MEMO 2015, problem I-3.

8/27/2015
Let ABCDABCD be a cyclic quadrilateral. Let EE be the intersection of lines parallel to ACAC and BDBD passing through points BB and AA, respectively. The lines ECEC and EDED intersect the circumcircle of AEBAEB again at FF and GG, respectively. Prove that points CC, DD, FF, and GG lie on a circle.
geometrycyclic quadrilateralcircumcircleAngle Chasing
Maximal number of steps for students to permute

Source: MEMO 2015, problem T-3

8/28/2015
There are nn students standing in line positions 11 to nn. While the teacher looks away, some students change their positions. When the teacher looks back, they are standing in line again. If a student who was initially in position ii is now in position jj, we say the student moved for ∣i−j∣|i-j| steps. Determine the maximal sum of steps of all students that they can achieve.
combinatoricspermutationsoptimization