MathDB

5

Part of 2022 New Zealand MO

Problems(2)

5 times in round-robin tournament

Source: 2022 NZMO - New Zealand Maths Olympiad Round 1 p5

10/8/2022
A round-robin tournament is one where each team plays every other team exactly once. Five teams take part in such a tournament getting: 33 points for a win, 11 point for a draw and 00 points for a loss. At the end of the tournament the teams are ranked from first to last according to the number of points. (a) Is it possible that at the end of the tournament, each team has a different number of points, and each team except for the team ranked last has exactly two more points than the next-ranked team? (b) Is this possible if there are six teams in the tournament instead?
combinatorics
max m, x_n(x_n - 1)(x_n - 2) . . . (x_n - m + 1)/{m!} never multiple of 7

Source: 2022 NZMO - New Zealand Maths Olympiad Round 2 p5

10/8/2022
The sequence x1,x2,x3,...x_1, x_2, x_3, . . . is defined by x1=2022x_1 = 2022 and xn+1=7xn+5x_{n+1}= 7x_n + 5 for all positive integers nn. Determine the maximum positive integer mm such that xn(xn−1)(xn−2)...(xn−m+1)m!\frac{x_n(x_n - 1)(x_n - 2) . . . (x_n - m + 1)}{m!} is never a multiple of 77 for any positive integer nn.
number theorymultipledividesdivisiblerecurrence relation