Problem:
A binary operation ⋄ has the properties that a⋄(b⋄c)=(a⋄b)⋅c and that a⋄a=1 for all nonzero real numbers a,b, and c. (Here the dot ⋅ represents the usual multiplication operation.) The solution to the equation 2016⋄(6⋄x)= 100 can be written as qp, where p and q are relatively prime positive integers. What is p+q?
Answer Choices:
A. 109
B. 201
C. 301
D. 3049
E. 33,601
Solution:
From the given properties, a⋄1=a⋄(a⋄a)=(a⋄a)⋅a= 1⋅a=a for all nonzero a. Then for nonzero a and b,a=a⋄1=a⋄(b⋄b)= (a⋄b)⋅b. It follows that a⋄b=ba. Thus
100=2016⋄(6⋄x)=2016⋄x6=x62016=336x
so x=336100=8425. The requested sum is 25+84=(A)109.
The problems on this page are the property of the MAA's American Mathematics Competitions