IOQM 2022 Part 3 #ioqm2022
IOQM 2022 Questions 18 to 22 πΏππ‘ π, π ππ πππ‘π’πππ ππ’πππππ π π’πβ π‘βππ‘ π + 3π β 5 =2πΏπΆπ(π, π) β 11πΊπΆπ·(π, π). πΉπππ π‘βπ πππ₯πππ’π πππ π ππππ π£πππ’π ππ π + π. Consider a string of n 1βs. 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. πΉππ ππ πππ‘ππππ πβ₯ 3 πππ π πππππ’π‘ππ‘πππ πΎ = (π1, π2, β¦, ππ)ππ {1, 2, β¦, π}, π€π π ππ¦ π1 ππ π ππππππππ πππππ‘ ππ 2 less than π less than π β 1 πππ (ππ β1 β ππ)(ππ + 1 β ππ) greater than 0. πΉππ ππ₯πππππ, πππ π = 7, π‘βπ πππππ’π‘ππ‘πππ (2, 7, 6, 4, 5, 1, 3)βππ πππ’π ππππππππ πππππ‘π : π2 = 7, π4 = 4, π5 = 5 πππ π6 = 1. πΉππ π πππ£ππ πβ₯ 3, πππ‘ πΏ(π)πππππ‘π π‘βπ ππ’ππππ ππ πππππ’π‘ππ‘πππ ππ {1, 2, β¦, π}π€ππ‘β ππ₯πππ‘ππ¦ ππππ¦ πππ ππππππππ πππππ‘. πΉπππ π‘βπ πππ₯πππ’π π greater than 3 πππ π€βππβ πΏ(π) ππ π πππππππ‘ π ππ’πππ. 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 π΅πππππ¦ π πππ’ππππ ππ π π πππ’ππππ ππ π€βππβ πππβ π‘πππ ππ πππ’ππ π‘π 0 ππ 1. π΄ ππππππ¦ π πππ’ππππ ππ ππππππ ππππππππ¦ ππ πππβ π‘πππ ππ ππππππππ‘ π‘π ππ‘ ππππ π‘ πππ π‘πππ π‘βππ‘ ππ πππ’ππ π‘π 1. πΉππ ππ₯πππππ, π‘βπ π πππ’ππππ 0, 1, 1, 0, 0, 1, 1, 1 ππ ππππππππ¦. πΏππ‘ πΉ(π) πππππ‘π π‘βπ ππ’ππππ ππ ππππππππ¦ ππππππ¦ π πππ’πππππ π€ππ‘β π π‘ππππ . πΉπππ π‘βπ π ππππππ π‘ πππ ππ‘ππ£π πππ‘ππππ π greater than or equal to 2 π π’πβ π‘βππ‘ πΉ(π) greater than100.
