Problem:
Find the number of integers such that the equation
has distinct real solutions.
Solution:
The equation has the following graph.
On the interval , the graph reaches a maximum of 100 at . The solutions to the equation are the -coordinates of the points of intersection of this graph with the lines and . Each of these lines intersects the graph exactly 6 times when is in the range , and no two of these intersections have the same -coordinate. Thus in order for the given equation to have 12 real solutions, must satisfy . There are integers in this range.