Hint: The small symbols all stand for distinct integers and the big symbols for functions or series.
Maybe it's still a little too hard. This might help you on the way.
From the solutions to the upper left-most equations, you could deduce that symbols standing next to each other have to be added, since 19 is a prime number. Then if you choose the following values for the small square, triangle and diamond, square = 1, triangle = 10, diamond = 6, the upper left-most equation is solved. Moving on to the equation below that one. If you choose the Fibonacci function for the big circles, you get Fib(6) + 10 + Fib(1) = 19, which is correct as well, since Fib(6) yields 8 and Fib(1) = 1.
Of course this isn't the correct solution, I'm just showing you how it might be done.
The big symbols could stand for all kinds of functions, e.g. squares, cubes, square roots, prime numbers, triangular numbers, factorials, etc.
Nobody seems to have been able to solve this puzzle yet. Maybe I should give another clue.
It is the most efficient to start with the equations with the numerical values:
s+s+s+t+d = 19, C(s)+C(d)+c = 21, C(d)+t+C(s) = 19.
c = cirlce, s = square, t = triangle, d = diamond.
The small symbols all stand for distinct non-negative integers.
Now, if you can find out which of the previously mentioned functions the C() stands for, you might also be able to find out what the S(), T() and D() stand for and solve the final equation.
If anyone needs more help, please feel free to ask.