Apple's service document acknowledges some 27-inch iMacs shipped between December 2012 and July 2014 may be affected by an issue with the display hinge, resulting in the screen no longer adjusting and continuously tilting forward. The issue appears to be limited to late 2012 and late 2013 models in particular.
The hinge problem is blamed on the use of plastic washers in the hinge's spring mechanism. Over time the washers can break under the load and cause the screen to fall forward.
Check out the video below to see the issue demonstrated and contact Apple support if you have the problem or have paid to fix the problem in the past.