MathDB

4

Part of 2024 Dutch IMO TST

Problems(2)

Replacing a number by multiples or digit permutations, can we reach 1?

Source: Dutch TST 2024, 3.4

6/28/2024
Initially, a positive integer NN is written on a blackboard. We repeatedly replace the number according to the following rules: 1) replace the number by a positive multiple of itself 2) replace the number by a number with the same digits in a different order. (The new number is allowed to have leading digits, which are then deleted.) A possible sequence of moves is given by 5→20→140→041=415 \to 20 \to 140 \to 041=41. Determine for which values of NN it is possible to obtain 11 after a finite number of such moves.
number theorynumber theory proposedblackboardDigits
Fire on n islands

Source: 2023 Singapore MO Round 2 Senior Q3

6/24/2023
Let nn be a positive integer. There are nn islands with n−1n-1 bridges connecting them such that one can travel from any island to another. One afternoon, a fire breaks out in one of the islands. Every morning, it spreads to all neighbouring islands. (Two islands are neighbours if they are connected by a bridge.) To control the spread, one bridge is destroyed every night until the fire has nowhere to spread the next day. Let XX be the minimum possible number of bridges one has to destroy before the fire stops spreading. Find the maximum possible value of XX over all possible configurations of bridges and island where the fire starts at.
combinatorics