Mathematicians have just created three new operation symbols, @, $,and _, each doing a different thing to a pair of numbers (All of these symbols can be derived from already-known operations and notations). The following equations using the new symbols are correct:

5@5= 2/5

2@2= 1

2@3= 5/9

3@4= 7/16

2@4= 3/8

1$0= undefined

1$4= 1/10

4$1= 1/10

10$10= 1/50

(1@1) $ 1= (√2)/10

2_2= 6

2_3= 9

6_2= 42

3_5= 30

5_6= 90

If the previous equations are correct, compute the following equation.

1 @ (1 $ (1_1))=

I enjoyed the problem, Victor, and was able to resolve it without
reading the postings (I'm the type that will fail to solve something
for days rather than look at the solution, if I think the problem was
well stated, as was the case here!). I rated it a 4 because it
was fun to solve, but I am new here and am leary of issuing a 5 before
I've got a better grip on how I should rate things.

I will offer this problem to the University of Houston Math
Enthusiasts, a happy group that will be happier for having had the
opportunity to work on your question.

Robert