MathDB

5

Part of 2021 Argentina National Olympiad

Problems(3)

Argentina MO 2021 National Level 1 P5

Source:

4/23/2022
Mica wrote a list of numbers using the following procedure. The first number is 11, and then, at each step, he wrote the result of adding the previous number plus 33. The first numbers on Mica's list are 1,4,7,10,13,16,… .1, 4, 7, 10, 13, 16,\dots. Next, Facu underlined all the numbers in Mica's list that are greater than 1010 and less than 100000,100000, and that have all their digits the same. What are the numbers that Facu underlined?
combinatoricscontests
Argentina MO 2021 National Level 2 P5

Source:

4/23/2022
Determine all positive integers nn such that n⋅2n−1+1n\cdot 2^{n-1}+1 is a perfect square.
number theoryPerfect Squarescontests
Argentina MO 2021 National Level 3 P5

Source:

4/24/2022
The sequence an(n≥1)a_n (n\geq 1) of natural numbers is defined as an+1=an+bn,a_{n+1}=a_n+b_n, where bnb_n is the number that has the same digits as ana_n but in the opposite order (bnb_n can start with 00). For example, if a1=180,a_1=180, then a2=261,a3=423.a_2=261, a_3=423.
a) Decide if a1a_1 can be chosen so that a7a_7 is prime. b) Decide if a1a_1 can be chosen so that a5a_5 is prime.
number theoryprime numberscontestsArgentina