MathDB

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Part of 2023 European Mathematical Cup

Problems(2)

Cute NT at EMC

Source: EMC 2023 Juniors P1

12/18/2023
Suppose a,b,ca,b,c are positive integers such that gcd⁡(a,b)+gcd⁡(a,c)+gcd⁡(b,c)=b+c+2023\gcd(a,b)+\gcd(a,c)+\gcd(b,c)=b+c+2023 Prove that gcd⁡(b,c)=2023\gcd(b,c)=2023.
Remark. For positive integers xx and yy, gcd⁡(x,y)\gcd(x,y) denotes their greatest common divisor.
Ivan Novak
nt2023emcnumber theoryGCD
Easy Algebra at EMC

Source: EMC 2023 Seniors P1

12/18/2023
Determine all sets of real numbers SS such that:
[*] 11 is the smallest element of SS, [*] for all x,y∈Sx,y\in S such that x>yx>y, x2−y2∈S\sqrt{x^2-y^2}\in S
Adian Anibal Santos Sepcic
algebraeasyemc2023