I'll the post the next problem. Undergraduates will probably find it pretty easy and it can be done with GCSE knowledge.
Problem 2*
Define a function f:R → R using the following rule:
"For any x,y∈R, f(x+y)=f(x)f(y)"
Show that, if f is non-zero:
(a) f(0)=1
(b) f(x)=0 has no solutions
(c) f(-x)=1/f(x)