IMO 2013 Shortlist N7
Let ν be an irrational positive number, and let m be a positive integer. A pair pa,bq of positive integers is called goo...
Category: Number Theory
Problem
Let ν be an irrational positive number, and let m be a positive integer. A pair pa,bq of positive integers is called good if arbνs ´ btaνu “ m. A good pair pa,bq is called excellent if neither of the pairs pa´b,bq and pa,b´aq is good. (As usual, by txu and rxs we denote the integer numbers such that x ´ 1 ă txu ď x and x ď rxs ă x ` 1.) Prove that the number of excellent pairs is equal to the sum of the positive divisors of m. (U.S.A.) 8 IMO 2013 Colombia