The well known William Lowell Putnam Mathematical Competition (76th edition) took place this month.
Here is a Maple approach for two of the problems.
1. For each real number x, 0 <= x < 1, let f(x) be the sum of 1/2^n where n runs through all positive integers for which floor(n*x) is even.
Find the infimum of f.
(Putnam 2015, A4 problem)
local n, s:=0;
for n to N do
if type(floor(n*x),even) then s:=s+2^(-n) fi;
#if floor(n*x) mod 2 = 0 then s:=s+2^(-n) fi;
min([seq(f(t), t=0.. 0.998,0.0001)]);
So, the infimum is 4/7.
Of course, this is not a rigorous solution, even if the result is correct. But it is a valuable hint.
I am not sure if in the near future, a CAS will be able to provide acceptable solutions for such problems.
2. If the function f is three times differentiable and has at least five distinct real zeros,
then f + 6f' + 12f'' + 8f''' has at least two distinct real zeros.
(Putnam 2015, B1 problem)
F := f + 6*D(f) + 12*(D@@2)(f) + 8*(D@@3)(f);
We are sugested to consider
So, F(x) = k(x) * g3 = k(x) * g'''
g has 5 distinct zeros implies g''' and hence F have 5-3=2 distinct zeros, q.e.d.