MathDB

6

Part of 2023 Caucasus Mathematical Olympiad

Problems(2)

\gcd(a, b) + lcm(a, b) = \gcd(a, c) + lcm(a, c)

Source: Caucasus MO 2023

7/16/2023
Let a,b,ca, b, c be positive integers such that gcd⁡(a,b)+lcm(a,b)=gcd⁡(a,c)+lcm(a,c).\gcd(a, b) + \text{lcm}(a, b) = \gcd(a, c) + \text{lcm}(a, c). Does it follow from this that b=cb = c?
number theory
Find the greatest $n$

Source: Caucasus MO 2023

7/16/2023
Let n≤100n \leq 100 be an integer. Hare puts real numbers in the cells of a 100×100100 \times 100 table. By asking Hare one question, Wolf can find out the sum of all numbers of a square n×nn \times n, or the sum of all numbers of a rectangle 1×(n−1)1 \times (n - 1) (or (n−1)×1(n - 1) \times 1). Find the greatest nn{} such that, after several questions, Wolf can find the numbers in all cells, with guarantee.