MathDB

4

Part of 2020 Bundeswettbewerb Mathematik

Problems(2)

Find the prime factorization of recursive sequence involving maximum

Source: Bundeswettbewerb Mathematik 2020, Round 1 - Problem 4

11/17/2020
Define a sequence (an)(a_n) recursively by a1=0,a2=2,a3=3a_1=0, a_2=2, a_3=3 and an=max⁡0<d<nad⋅an−da_n=\max_{0<d<n} a_d \cdot a_{n-d} for n≥4n \ge 4. Determine the prime factorization of a19702020a_{19702020}.
number theoryprime factorizationnumber theory proposed
Non-negative numbers in a table, row sum larger than column sum

Source: Bundeswettbewerb Mathematik 2020, Round 2 - Problem 4

11/17/2020
In each cell of a table with mm rows and nn columns, where m<nm<n, we put a non-negative real number such that each column contains at least one positive number.
Show that there is a cell with a positive number such that the sum of the numbers in its row is larger than the sum of the numbers in its column.
combinatoricsinequalitiesnonnegativeSum