MathDB

3

Part of 2023 European Mathematical Cup

Problems(2)

Configuration with median and right angles

Source: EMC 2023 Junior P3

12/18/2023
Consider an acute-angled triangle ABCABC with AB<ACAB < AC. Let MM and NN be the midpoints of segments BCBC and ABAB, respectively. The circle with diameter ABAB intersects the lines BC,AMBC, AM and ACAC at D,ED, E, and FF, respectively. Let GG be the midpoint of FCFC. Prove that the lines NF,DENF, DE and GMGM are concurrent.
Michal Pecho
geometry
You cannot have common descendants!

Source: EMC 2023 Seniors P3

12/18/2023
Let nn be a positive integer. Let BnB_n be the set of all binary strings of length nn. For a binary string s_1\hdots s_n, we define it's twist in the following way. First, we count how many blocks of consecutive digits it has. Denote this number by bb. Then, we replace sbs_b with 1−sb1-s_b. A string aa is said to be a descendant of bb if aa can be obtained from bb through a finite number of twists. A subset of BnB_n is called divided if no two of its members have a common descendant. Find the largest possible cardinality of a divided subset of BnB_n.
Remark. Here is an example of a twist: 101100→101000101100 \rightarrow 101000 because 1∣0∣11∣001\mid 0\mid 11\mid 00 has 44 blocks of consecutive digits.
Viktor Simjanoski
emc2023combinatorics