MathDB

Problem 3

Part of 2000 Croatia National Olympiad

Problems(4)

floor(x/m)=floor(x/(m-1)), # of solutions

Source: Croatia 2000 1st Grade P3

5/8/2021
Let m>1m>1 be an integer. Determine the number of positive integer solutions of the equation ⌊xm⌋=⌊xm−1⌋\left\lfloor\frac xm\right\rfloor=\left\lfloor\frac x{m-1}\right\rfloor.
algebrafloor function
floor inequality iff parameters equal

Source: Croatia 2000 2nd Grade P3

5/8/2021
Let jj and kk be integers. Prove that the inequality ⌊(j+k)α⌋+⌊(j+k)β⌋≥⌊jα⌋+⌊jβ⌋+⌊k(α+β)⌋\lfloor(j+k)\alpha\rfloor+\lfloor(j+k)\beta\rfloor\ge\lfloor j\alpha\rfloor+\lfloor j\beta\rfloor+\lfloor k(\alpha+\beta)\rfloorholds for all real numbers α,β\alpha,\beta if and only if j=kj=k.
inequalitiesfloor function
if rectangular parallelepiped has a hexagon cross-section, it is a cube

Source: Croatia 2000 3rd Grade P3

5/9/2021
A plane intersects a rectangular parallelepiped in a regular hexagon. Prove that the rectangular parallelepiped is a cube.
geometry3D geometry
maxmization with divisibility conditions

Source: Croatia 2000 4th Grade P3

5/9/2021
Let n≥3n\ge3 positive integers a1,…,ana_1,\ldots,a_n be written on a circle so that each of them divides the sum of its two neighbors. Let us denote Sn=an+a2a1+a1+a3a2+…+an−2+anan−1+…+an−1+a1an.S_n=\frac{a_n+a_2}{a_1}+\frac{a_1+a_3}{a_2}+\ldots+\frac{a_{n-2}+a_n}{a_{n-1}}+\ldots+\frac{a_{n-1}+a_1}{a_n}.Determine the minimum and maximum values of SnS_n.
number theoryinequalities