A round table is pushed into a corner of a rectangular room so that the tabletop touches both walls. Suppose that there is a point on the perimeter of the table that is 12 inches from one wall and 6 inches from the other. What is the diameter of the table?
Solution by Steven (Hyungmin) Jin , Ann Arbor Huron High School
If the circle was thought to be on the grid and its tangent to the x-axis and y-axis, then the equation of the circle is 
. However, the distance of the center away from the axes is equal to the radius r; so the equation is 
. Because x = 12 and y = 6 is a point on the circle, we know that 
. Solving this equaton, we get:
So, r = 30 inches or r = 6 inches, which means there are two possible diameters fot the table: d = 60 inches or d = 12 inches.
Also solved by: Trevor Barker, Ryan Cantrell, Cody Nyp (Central Montcalm H.S.); Mu-Wei Huang, Matthew Nicholas (Detroit Catholic Central H.S.); Galina Gagin (East Grand Rapids H.S.); Kyle Helzer (East Kentwood H.S.); Michelle DeWitt, Sonya Kavalam, Kristin Kopec, Mary Maier, Paige O'Dowd, Marc Walker, Sheila Waslawski (Forest Hills Eastern H.S.); Robert Harrison, Alex Robinson (Lansing Catholic H.S.); Britt Gabel (Linden H.S.); Luke Tillotson (Homeschool, Petoskey); Sachin Padmanabhan (Novi H.S.); David Maniez (St. Louis H.S.); Chirag Pathre (Troy H.S.); Louisa Avellar, Gio Campagna, Johnny Gragg (Alamo Heights H.S., San Antonio, TX).
Those finding only one solution: Kedao Wang (Ann Arbor Huron H.S.); Matthew Bates, Maxwell Mikelic (East Grand Rapids H.S.); Brooke Boer, Shelby Eagle, Eric Graverson, Ryan Klock, Kailene Schabes, Christopher Yang, Katee Yang (Forest Hills Eastern H.S.); Persis Sopariwala (Forest Hills Northern H.S.); Darren Brown (Lansing Catholic H.S.); Spencer Frost, Rachel Juengel (Linden H.S.) ; Dalton Allan, Xiao Xiang (Saginaw Arts and Sciences Academy); Allison Liao (Troy H.S.); Andrew Dubinski (Alamo Heights H.S., San Antonio, TX).