MathDB

3

Part of 2022 Middle European Mathematical Olympiad

Problems(2)

Parallelogram configuration with equal segments

Source: MEMO 2022 I3

9/2/2022
Let ABCDABCD be a parallelogram with ∠DAB<90\angle DAB < 90 Let EE be the point on the line BCBC such that AE=ABAE = AB and let FF be the point on the line CDCD such that AF=ADAF = AD. The circumcircle of the triangle CEFCEF intersects the line AEAE again in PP and the line AFAF again in QQ. Let XX be the reflection of PP over the line DEDE and YY the reflection of QQ over the line BFBF. Prove that A,X,YA, X, Y lie on the same line.
geometry
Swapping blocks of equal number of elements

Source: MEMO 2022 T3

9/2/2022
Let nn be a positive integer. There are nn purple and nn white cows queuing in a line in some order. Tim wishes to sort the cows by colour, such that all purple cows are at the front of the line. At each step, he is only allowed to swap two adjacent groups of equally many consecutive cows. What is the minimal number of steps Tim needs to be able to fulfill his wish, regardless of the initial alignment of the cows?
combinatorics