MathDB

11

Part of 2023 MMATHS

Problems(2)

MMATHS 2023 Individual Problem 11: Knight on infinite chessboard

Source:

9/24/2024
A knight is on an infinite chessboard. After exactly 100100 legal moves, how many different possible squares can it end on? A knight can move to any of the 88 closest squares not on the same row, column, or diagonal, as illustrated in the figure below. https://cdn.artofproblemsolving.com/attachments/0/7/144226144fb3ead533e7b517f5f65d8a70da5a.png
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MMATHS 2023 Team Problem 11: Two sequences and a function

Source:

9/24/2024
Suppose we have sequences (an)n≥0(a_n)_{n \ge 0} and (bn)n≥0(b_n)_{n \ge 0} and the function f(x)=1xf(x)=\tfrac{1}{x} such that for all nn we have:
[*]an+1=f(f(an+bn)−f(f(an)+f(bn))a_{n+1} = f(f(a_n+b_n)-f(f(a_n)+f(b_n)) [*]an+2=f(1−an)−f(1+an)a_{n+2} = f(1-a_n) - f(1+a_n) [*]bn+2=f(1−bn)−f(1+bn)b_{n+2} = f(1-b_n) - f(1+b_n)
Given that a0=16a_0=\tfrac{1}{6} and b0=17,b_0=\tfrac{1}{7}, then b5=mn,b_5=\tfrac{m}{n}, where mm and nn are relatively prime positive integers. Find the sum of the prime factors of mn.mn.
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