2

Part of 2024 Czech-Polish-Slovak Junior Match

Problems(2)

Counting sets with product a perfect cube

Source: Czech-Polish-Slovak Junior Match 2024, I-2

How many non-empty subsets of

\{1,2,\dots,11\}

are there with the property that the product of its elements is the cube of an integer?

number theorynumber theory proposedProductperfect cubegeometry3D geometry

Factoring 2024 in Z[\sqrt{3}]

Source: Czech-Polish-Slovak Junior Match 2024, T-2

Among all triples

(a,b,c)

of natural numbers satisfying

(a+14\sqrt{3})(b-14c\sqrt{3})=2024,

determine the one with the maximal value of

a

number theorynumber theory proposedalgebraalgebra proposed