MathDB

1

Part of 1988 National High School Mathematics League

Problems(2)

Three functions

Source: 1988 National High School Mathematics League, Exam One, Problem 1

2/25/2020
We have three functions. The first one is y=ϕ(x)y=\phi(x). The second one is the inverse function of the first one. The figure of the third funcion is symmetrical to the second one about line x+y=0x+y=0. Then, the third function is (A)y=−ϕ(x)(B)y=−ϕ(−x)(C)y=−ϕ−1(x)(D)y=−ϕ−1(x)\text{(A)}y=-\phi(x)\qquad\text{(B)}y=-\phi(-x)\qquad\text{(C)}y=-\phi^{-1}(x)\qquad\text{(D)}y=-\phi^{-1}(x)
function
Recurrence Sequence

Source: 1988 National High School Mathematics League, Exam Two, Problem 1

2/25/2020
Define sequence (an):a1=1,a2=2,an+2={5an+1−3an,if an⋅an+1 is evenan+1−an,if an⋅an+1 is odd(a_n):a_1=1,a_2=2,a_{n+2}=\begin{cases} 5a_{n+1}-3a_n,\text{if }a_n\cdot a_{n+1}\text{ is even}\\ a_{n+1}-a_n,\text{if }a_n\cdot a_{n+1}\text{ is odd} \end{cases} Prove that for all n∈Z+n\in\mathbb{Z}_+, an≠0a_n\neq0.