MathDB

Problem 1

Part of 2000 Mongolian Mathematical Olympiad

Problems(2)

sequence of radical (nt)

Source: Mongolia MO 2000 Grade 10 P1

4/22/2021
Let rad⁡(k)\operatorname{rad}(k) denote the product of prime divisors of a natural number kk (define rad⁡(1)=1\operatorname{rad}(1)=1). A sequence (an)(a_n) is defined by setting a1a_1 arbitrarily, and an+1=an+rad⁡(an)a_{n+1}=a_n+\operatorname{rad}(a_n) for n≥1n\ge1. Prove that the sequence (an)(a_n) contains arithmetic progressions of arbitrary length.
number theorySequences
range of (x+y+z)^2/xyz, x,y,z in N

Source: Mongolia MO 2000 Teachers P1

4/22/2021
Find all integers that can be written in the form (x+y+z)2xyz\frac{(x+y+z)^2}{xyz}, where x,y,zx,y,z are positive integers.
number theory