MathDB

8

Part of 2019 South East Mathematical Olympiad

Problems(2)

Long Problem I

Source: 2019 CSMO Grade 10 P8

7/31/2019
For positive integer x>1x>1, define set SxS_x as Sx={pα∣p is one of the prime divisor of x,α∈N,pα∣x,α≡vp(x)(mod2)},S_x=\{p^\alpha|p \textup{ is one of the prime divisor of }x,\alpha \in \mathbb{N},p^\alpha|x,\alpha \equiv v_p(x)(\textup{mod} 2)\}, where vp(n)v_p(n) is the power of prime divisor pp in positive integer n.n. Let f(x)f(x) be the sum of all the elements of SxS_x when x>1,x>1, and f(1)=1.f(1)=1. Let mm be a given positive integer, and the sequence a1,a2,⋯ ,an,⋯a_1,a_2,\cdots,a_n,\cdots satisfy that for any positive integer n>m,n>m, an+1=max⁡{f(an),f(an−1+1),⋯ ,f(an−m+m)}.a_{n+1}=\max\{ f(a_n),f(a_{n-1}+1),\cdots,f(a_{n-m}+m)\}. Prove that (1)there exists constant A,B(0<A<1),A,B(0<A<1), such that when positive integer xx has at least two different prime divisors, f(x)<Ax+Bf(x)<Ax+B holds; (2)there exists positive integer QQ, such that for any positive integer n,an<Q.n,a_n<Q.
number theory
Long Problem II

Source: 2019 CSMO Grade 11 P8

7/31/2019
For positive integer x>1x>1, define set SxS_x as Sx={pα∣p is one of the prime divisor of x,α∈N,pα∣x,α≡vp(x)(mod2)},S_x=\{p^\alpha|p \textup{ is one of the prime divisor of }x,\alpha \in \mathbb{N},p^\alpha|x,\alpha \equiv v_p(x)(\textup{mod} 2)\}, where vp(n)v_p(n) is the power of prime divisor pp in positive integer n.n. Let f(x)f(x) be the sum of all the elements of SxS_x when x>1,x>1, and f(1)=1.f(1)=1. Let mm be a given positive integer, and the sequence a1,a2,⋯ ,an,⋯a_1,a_2,\cdots,a_n,\cdots satisfy that for any positive integer n>m,n>m, an+1=max⁡{f(an),f(an−1+1),⋯ ,f(an−m+m)}.a_{n+1}=\max\{ f(a_n),f(a_{n-1}+1),\cdots,f(a_{n-m}+m)\}. Prove that (1)there exists constant A,B(0<A<1),A,B(0<A<1), such that when positive integer xx has at least two different prime divisors, f(x)<Ax+Bf(x)<Ax+B holds; (2)there exists positive integer N,lN,l, such that for any positive integer n≥N,an+l=ann\geq N ,a_{n+l}=a_n holds.
number theory