MathDB

3

Part of 2022 South East Mathematical Olympiad

Problems(2)

Bai lu zhou Academy Problem

Source: 2022 China Southeast Grade 10 P3 and 11 P4

8/2/2022
If xix_i is an integer greater than 1, let f(xi)f(x_i) be the greatest prime factor of xi,xi+1=xi−f(xi)x_i,x_{i+1} =x_i-f(x_i) (i≥0i\ge 0 and i is an integer). (1) Prove that for any integer x0x_0 greater than 1, there exists a natural numberk(x0)k(x_0), such that xk(x0)+1=0x_{k(x_0)+1}=0 Grade 10: (2) Let V(x0)V_{(x_0)} be the number of different numbers in f(x0),f(x1),⋯ ,f(xk(x0))f(x_0),f(x_1),\cdots,f(x_{k(x_0)}). Find the largest number in V(2),V(3),⋯ ,V(781)V(2),V(3),\cdots,V(781) and give reasons. Note: Bai Lu Zhou Academy was founded in 1241 and has a history of 781 years. Grade 11: (2) Let V(x0)V_{(x_0)} be the number of different numbers in f(x0),f(x1),⋯ ,f(xk(x0))f(x_0),f(x_1),\cdots,f(x_{k(x_0)}). Find the largest number in V(2),V(3),⋯ ,V(2022)V(2),V(3),\cdots,V(2022) and give reasons.
algebranumber theory
Counting numbers and killing people

Source: 2022 China Southeast Grade11 P3

8/2/2022
There are nn people in line, counting 1,2,⋯ ,n1,2,\cdots, n from left to right, those who count odd numbers quit the line, the remaining people press 1,2 from right to left, and count off again, those who count odd numbers quit the line, and then the remaining people count off again from left to right⋯\cdots Keep doing that until only one person is in the line. f(n)f(n) is the number of the last person left at the first count. Find the expression for f(n)f(n) and find the value of f(2022)f(2022)
combinatorics