MathDB

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Part of 2022 Rioplatense Mathematical Olympiad

Problems(4)

Rioplatense 2022 - Level 3 - Problem 1

Source:

12/6/2022
Prove that there exists infinitely many positive integers nn for which the equationx2+y11−z2022!=nx^2+y^{11}-z^{2022!}=nhas no solution (x,y,z)(x,y,z) over the integers.
number theory
Rioplatense 2022 - Level 2 - Problem 1

Source:

12/6/2022
In how many ways can the numbers from 22 to 20222022 be arranged so that the first number is a multiple of 11, the second number is a multiple of 22, the third number is a multiple of 33, and so on untile the last number is a multiple of 20212021?
number theory
Rioplatense 2022 - Level 1 - Problem 1

Source:

12/6/2022
Find three consecutive odd numbers a,b,ca,b,c such that a2+b2+c2a^2+b^2+c^2 is a four digit number with four equal digits.
number theory
Rioplatense 2022 - Level A - Problem 1

Source:

12/6/2022
In the blackboard there are drawn 2525 points as shown in the figure. Gastón must choose 44 points that are vertices of a square. In how many different ways can he make this choice?∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙\begin{matrix}\bullet & \bullet & \bullet & \bullet & \bullet \\\bullet & \bullet & \bullet & \bullet & \bullet \\\bullet & \bullet & \bullet & \bullet & \bullet \\\bullet & \bullet & \bullet & \bullet & \bullet \\\bullet & \bullet & \bullet & \bullet & \bullet \end{matrix}
combinatorics