See also [:SymbolicBenchmarks: this other page].
The Symbolic Benchmark Suite
The conditions for something to be listed here: (a) it must be resemble an actual computation somebody actually wanted to do in Sage, and (b) the question must be precisely formulated with Sage code that uses the Sage symbolics in a straightforward way (i.e., don't cleverly use number fields).
== PROBLEM 1 ==
SETUP: Define a function f(z) = \sqrt{1/3}\cdot z^2 + i/3. COMPUTATION: Compute the real part of f(f(f(...(f(i/2))...) iterated 10 times.
def f(z): return sqrt(1/3)*z^2 + i/3 timeit('real(f(f(f(f(f(f(f(f(f(f(i/2)))))))))))')