Subcontests
(16)Infinitely many solutions to floor equation
Prove that there exist infinitely many triples of positive integers (a,b,c) so that a,b,c are pairwise coprime and
⌊2021a2⌋+⌊2021b2⌋=⌊2021c2⌋. Weird length conditions
Five points A,B,C,P,Q are chosen so that A,B,C aren't collinear. The following length conditions hold: BPAP=BQAQ=2021 and CPBP=CQBQ=1920. Prove that line PQ goes through the circumcentre of △ABC. Easy process problem
Initially on the blackboard eight zeros are written. In one step, it is allowed to choose numbers a,b,c,d, erase them and replace them with the numbers a+1, b+2, c+3, d+3. Determine:
a) the minimum number of steps required to achieve 8 consecutive integers on the board
b) whether it is possible to achieve that sum of the numbers is 2021
c) whether it is possible to achieve that product of the numbers is 2145