Here is a problem from the UKMT Senior (17-18 year-old) Mathematics Challenge for 2009:

“Four positive integers *a, b, c*, and *d* are such that*_________abcd + abc + bcd + cda + dab + ab + bc + cd + da + ac + bd + a + b + c + d* = 2009.

What is the value of *a + b + c + d*?

_________A 73_________B 75_________C 77_________D 79_________E 81”

See the Challenging Sum

**(Update 4/17/2019)** I forgot to mention in my write-up that the UKMT solution only included the last few steps and did not provide any motivation for *why* one would think of it.