2013 Mock AIME I Problems/Problem 6
Find the number of integer values can have such that the equation has a solution.
is a continuous function, so every value between its minimum and maximum is attainable. By Cauchy-Schwarz, Giving a maximum of , which is achievable when . Note that a minimum of can be attained at . Thus the values of that work are the integers from to , inclusive, giving a total of .