Let ABC be $a$ triangle let D be $a$ point on the segment BC such that AD = BC. Suppose ∠CAD = x°, ∠ABC = y° and ∠ACB = z° and x, y, z are in an arithmetic progression in that order where the first term and the common difference are positive integers. Find the largest possible value of ∠ABC in degrees.

Step 1: sin $y$ sin  $x$  $y$  z 

Step 2: sin z sin x

Step 3: sin  $x$  $y$  z  sin x

Step 4:  1

Step 5: sin $y$ sin z

Step 6: sin  3 $y$  sin  $y$ – $d$ 

Step 7:  1

Step 8: sin $y$ sin  $y$  $d$ 

Step 9: sin  $y$ – $d$ 

Step 10:  – 2cos 2 y

Step 11: sin  $y$  $d$ 

Step 12: sin  3 $y$  $d$   0 => 3 $y$  $d$  180°, 360°

Step 13: y & $d$ are integers => $d$ is multiple of 3

Step 14: If 3y + $d$ = 180° => ymax = 59°

Step 15: 3y + $d$ = 360° => ymax = 119° (if $y$ is obtuse, then z must be acute)

Step 16: Only 1st case is possible ymax = 59°