IOQM 2022 Part 3

Описание: IOQM 2022 Questions 18 to 22
𝐿𝑒𝑡 𝑚, 𝑛 𝑏𝑒 𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑚 + 3𝑛 – 5 =2𝐿𝐶𝑀(𝑚, 𝑛) – 11𝐺𝐶𝐷(𝑚, 𝑛). 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑚 + 𝑛.

Consider a string of n 1’s. We wish to place some + signs in between so that the sum is 1000. For instance, if n = 190, one may put + signs so as to get 11 ninety times and 1 ten times and get the sum 1000. If a is the number of positive integers n for which it is possible to place + signs so as to get the sum 1000, then find the sum of the digits of a.

𝐹𝑜𝑟 𝑎𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑛≥ 3 𝑎𝑛𝑑 𝑎 𝑝𝑒𝑟𝑚𝑢𝑡𝑎𝑡𝑖𝑜𝑛 𝐾 = (𝑝1, 𝑝2, …, 𝑝𝑛)𝑜𝑓 {1, 2, …, 𝑛},
𝑤𝑒 𝑠𝑎𝑦 𝑝1 𝑖𝑠 𝑎 𝑙𝑎𝑛𝑑𝑚𝑎𝑟𝑘 𝑝𝑜𝑖𝑛𝑡 𝑖𝑓 2 less than 𝑙 less than 𝑛 – 1 𝑎𝑛𝑑 (𝑝𝑙 –1 – 𝑝𝑙)(𝑝𝑙 + 1 – 𝑝𝑙) greater than 0.
𝐹𝑜𝑟 𝑒𝑥𝑎𝑚𝑝𝑙𝑒, 𝑓𝑜𝑟 𝑛 = 7, 𝑡ℎ𝑒 𝑝𝑒𝑟𝑚𝑢𝑡𝑎𝑡𝑖𝑜𝑛 (2, 7, 6, 4, 5, 1, 3)ℎ𝑎𝑠 𝑓𝑜𝑢𝑟 𝑙𝑎𝑛𝑑𝑚𝑎𝑟𝑘 𝑝𝑜𝑖𝑛𝑡𝑠: 𝑝2 = 7, 𝑝4 = 4, 𝑝5 = 5 𝑎𝑛𝑑 𝑝6 = 1. 𝐹𝑜𝑟 𝑎 𝑔𝑖𝑣𝑒𝑛 𝑛≥ 3, 𝑙𝑒𝑡 𝐿(𝑛)𝑑𝑒𝑛𝑜𝑡𝑒 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑒𝑟𝑚𝑢𝑡𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 {1, 2, …, 𝑛}𝑤𝑖𝑡ℎ 𝑒𝑥𝑎𝑐𝑡𝑙𝑦 𝑜𝑛𝑙𝑦 𝑜𝑛𝑒 𝑙𝑎𝑛𝑑𝑚𝑎𝑟𝑘 𝑝𝑜𝑖𝑛𝑡. 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑛 greater than 3 𝑓𝑜𝑟 𝑤ℎ𝑖𝑐ℎ 𝐿(𝑛) 𝑖𝑠 𝑎 𝑝𝑒𝑟𝑓𝑒𝑐𝑡 𝑠𝑞𝑢𝑎𝑟𝑒.

An ant is at a vertex of a cube. Every 10 minutes it moves to an adjacent vertex along an edge. If N is the number of one hour journeys that end at the starting vertex, find the sum of the squares of the digits of N

𝐵𝑖𝑛𝑎𝑟𝑦 𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒 𝑖𝑠 𝑎 𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒 𝑖𝑛 𝑤ℎ𝑖𝑐ℎ 𝑒𝑎𝑐ℎ 𝑡𝑒𝑟𝑚 𝑖𝑠 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 0 𝑜𝑟 1. 𝐴 𝑏𝑖𝑛𝑎𝑟𝑦 𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒 𝑖𝑠 𝑐𝑎𝑙𝑙𝑒𝑑 𝑓𝑟𝑖𝑒𝑛𝑑𝑙𝑦 𝑖𝑓 𝑒𝑎𝑐ℎ 𝑡𝑒𝑟𝑚 𝑖𝑠 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 𝑎𝑡 𝑙𝑒𝑎𝑠𝑡 𝑜𝑛𝑒 𝑡𝑒𝑟𝑚 𝑡ℎ𝑎𝑡 𝑖𝑠 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 1. 𝐹𝑜𝑟 𝑒𝑥𝑎𝑚𝑝𝑙𝑒, 𝑡ℎ𝑒 𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒 0, 1, 1, 0, 0, 1, 1, 1 𝑖𝑠 𝑓𝑟𝑖𝑒𝑛𝑑𝑙𝑦. 𝐿𝑒𝑡 𝐹(𝑛) 𝑑𝑒𝑛𝑜𝑡𝑒 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑟𝑖𝑒𝑛𝑑𝑙𝑦 𝑏𝑖𝑛𝑎𝑟𝑦 𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒𝑠 𝑤𝑖𝑡ℎ 𝑛 𝑡𝑒𝑟𝑚𝑠. 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑠𝑚𝑎𝑙𝑙𝑒𝑠𝑡 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑛 greater than or equal to 2 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝐹(𝑛) greater than100.