MathDB

Problem 1

Part of 2022 Kyiv City MO Round 2

Problems(4)

Easy NT on LCMs

Source: Kyiv City MO 2022 Round 2, Problem 7.1

1/30/2022
a) Do there exist positive integers aa and dd such that [a,a+d]=[a,a+2d][a, a+d] = [a, a+2d]?
b) Do there exist positive integers aa and dd such that [a,a+d]=[a,a+4d][a, a+d] = [a, a+4d]?
Here [a,b][a, b] denotes the least common multiple of integers a,ba, b.
number theoryLCM
LCM genius problem from our favorite author

Source: Kyiv City MO 2022 Round 2, Problem 8.1

1/30/2022
Find all triples (a,b,c)(a, b, c) of positive integers for which a+[a,b]=b+[b,c]=c+[c,a]a + [a, b] = b + [b, c] = c + [c, a].
Here [a,b][a, b] denotes the least common multiple of integers a,ba, b.
(Proposed by Mykhailo Shtandenko)
number theoryleast common multiple
GCD genius problem from our favorite author

Source: Kyiv City MO 2022 Round 2, Problem 9.1

1/30/2022
Find all triples (a,b,c)(a, b, c) of positive integers for which a+(a,b)=b+(b,c)=c+(c,a)a + (a, b) = b + (b, c) = c + (c, a).
Here (a,b)(a, b) denotes the greatest common divisor of integers a,ba, b.
(Proposed by Mykhailo Shtandenko)
number theorygreatest common divisor
Easy algebra

Source: Kyiv City MO 2022 Round 2, Problem 10.1

1/30/2022
Positive reals x,y,zx, y, z satisfy xy+1x+1=yz+1y+1=zx+1z+1\frac{xy+1}{x+1} = \frac{yz+1}{y+1} = \frac{zx+1}{z+1}
Do they all have to be equal?
(Proposed by Oleksii Masalitin)
algebrainequalitiesUkraine