Problem:
Score : 300 points

Problem Statement

You are given N real values A_1, A_2, \ldots, A_N.
Compute the number of pairs of indices (i, j)
such that i < j and the product A_i \cdot A_j is integer.

Constraints

2 \leq N \leq 200\,000

0 < A_i < 10^4

A_i is given with at most 9 digits after the decimal.

Input

Input is given from Standard Input in the following format.

N
A_1
A_2
\vdots
A_N

Output

Print the number of pairs with integer product A_i \cdot A_j (and i < j).

Sample Input 1

5
7.5
2.4
17.000000001
17
16.000000000

Sample Output 1

3

There are 3 pairs with integer product:

7.5 \cdot 2.4 = 18

7.5 \cdot 16 = 120

17 \cdot 16 = 272

Sample Input 2

11
0.9
1
1
1.25
2.30000
5
70
0.000000001
9999.999999999
0.999999999
1.000000001

Sample Output 2

8

Solution in Python (3.8.2) (judge verdict: Accepted)

from collections import defaultdict

N = int(input())

AS = [input().strip() for _ in range(N)]

nums = [int(a.replace('.', '')) for a in AS]
dens = [1 if '.' not in a else 10**(len(a) - a.index('.') - 1) for a in AS]

# print(nums, dens)

def gcd(a, b):
    if a < b:
        return gcd(b, a)
    if b == 0:
        return a
    return gcd(b, a%b)

gs = [gcd(num, den) for num, den in zip(nums, dens)]
nums = [num // g for num, g in zip(nums, gs)]
dens = [den // g for den, g in zip(dens, gs)]

def signature(n):
    e2, e5 = 0, 0
    while n % 2 == 0:
        n //= 2
        e2 += 1
    while n % 5 == 0:
        n //= 5
        e5 += 1
    return (e2, e5)

lookup = defaultdict(int)
for num, den in zip(nums, dens):
    n2, n5 = signature(num)
    d2, d5 = signature(den)
    lookup[(n2-d2, n5-d5)]+=1

ans = 0
for k1 in lookup:
    for k2 in lookup:
        v1, v2 = lookup[k1], lookup[k2]
        x1, y1 = k1
        x2, y2 = k2
        if x1 + x2 >= 0 and y1 + y2 >=0:
            if k1 == k2:
                ans += v1*(v1-1)
            else:
                ans += v1*v2

ans //= 2
print(ans)