Given some integers, find triplets where the product of two numbers equals the third number
Your combinations(…, 3) loop makes your algorithm O(N3).

It's easy to improve it to be O(N2).  The question is, essentially: for every pair of entries, how many occurrences of their product are in the list?  So, make an indexes data structure to help you find, in O(1) time, where the product might be located.

from collections import defaultdict
from itertools import combinations

a = [int(ai) for ai in input('Input: ').split()]

indexes = defaultdict(set)
for i, ai in enumerate(a):
    indexes[ai].add(i)

triplets = set()
for x, y in combinations(range(len(a)), 2):
    for z in indexes[a[x] * a[y]].difference([x, y]):
        triplets.add(tuple(sorted((x, y, z))))

print(len(triplets))


Here, I've chosen to stick closer to the notation used in the challenge itself, with a being the list, and x, y, z as the indexes of entries (but 0-based rather than 1-based).