MathDB

3

Part of 1998 National High School Mathematics League

Problems(2)

Geometric Series

Source: 1998 National High School Mathematics League, Exam One, Problem 3

3/9/2020
For geometric series (an)(a_n) with all items real, if S10=10,S30=70S_{10}=10,S_{30}=70, then S40=S_{40}= (A)150(B)−200(C)150 or −200(D)−50 or 400\text{(A)}150\qquad\text{(B)}-200\qquad\text{(C)}150\text{ or }-200\qquad\text{(D)}-50\text{ or }400 Note: Sn=∑i=1naiS_n=\sum_{i=1}^{n}a_i.
geometric series
Number Theory

Source: 1998 National High School Mathematics League, Exam Two, Problem 3

3/9/2020
For positive integers a,na,n, define Fn(a)=q+rF_n(a)=q+r, where a=qn+ra=qn+r (q,rq,r are nonnegative integers, 0≤q<n0\leq q<n). Find the largest integer AA, there are positive integers n1,n2,n3,n4,n5,n6n_1,n_2,n_3,n_4,n_5,n_6, for all positive integer a≤Aa\leq A, Fn6(Fn5(Fn4(Fn3(Fn2(Fn1(a))))))=1F_{n_6}(F_{n_5}(F_{n_4}(F_{n_3}(F_{n_2}(F_{n_1}(a))))))=1.
number theory