MathDB

6

Part of 2022 Rioplatense Mathematical Olympiad

Problems(4)

Covering with domino tiles

Source: Rioplatense L-2 2022 #6

12/13/2022
Let N(a,b)N(a,b) be the number of ways to cover a table a×ba \times b with domino tiles. Let M(a,2b+1)M(a,2b+1) be the number of ways to cover a table a×2b+1a \times 2b+1 with domino tiles, such that there are no vertical tile in the central column. Prove that M(2m,2n+1)=2m⋅N(2m,n)⋅N(2m,n−1)M(2m,2n+1)=2^m \cdot N(2m,n)\cdot N(2m,n-1)
combinatorics
Switching the order of the digits

Source: Rioplatense L-3 2022 #6

12/13/2022
In a board, the positive integer NN is written. In each round, Olive can realize any one of the following operations: I - Switch the current number by a positive multiple of the current number. II - Switch the current number by a number with the same digits of the current number, but the digits are written in another order(leading zeros are allowed). For instance, if the current number is 20222022, Olive can write any of the following numbers 222,2202,2220222,2202,2220. Determine all the positive integers NN, such that, Olive can write the number 11 after a finite quantity of rounds.
combinatoricsnumber theory
fake coins are always odd

Source: Rioplatense L-1 2022 #6

12/13/2022
In Vila Par, all the truth coins weigh an even quantity of grams and the false coins weigh an odd quantity of grams. The eletronic device only gives the parity of the weight of a set of coins. If there are 20202020 truth coins and 22 false coins, detemine the least kk, such that, there exists a strategy that allows to identify the two false coins using the eletronic device, at most, kk times.
combinatorics
Rioplatense L-A 2022 #6

Source:

12/13/2022
A sequence of numbers is platense if the first number is greater than 11, and an+1=anpna_{n+1}=\frac{a_n}{p_n} which pnp_n is the least prime divisor of ana_n, and the sequence ends if an=1a_n=1. For instance, the sequences 864,432,216,108,54,27,9,3,1864, 432,216,108,54,27,9,3,1 and 2022,1011,337,12022,1011,337,1 are both sequence platense. A sequence platense is cuboso if some term is a perfect cube greater than 11. For instance, the sequence 864864 is cuboso, because 27=3327=3^3, and the sequence 20222022 is not cuboso, because there is no perfect cube. Determine the number of sequences cuboso which the initial term is less than 20222022.
number theory