Faster algorithm to tailor given mathematical expression
To start, let's fix the PEP8 style errors using any PEP8 checker:

counts = 0
ans = 0
while(ans == 0):
    s = 0
    for i in range(counts, len(arr)):
        s += int(arr[i]/(i+1-counts))
        if(s &gt; M):
            break
    if((i+1) == len(arr) and s &lt;= M):
        print(counts)
        ans = 1
    counts += 1




The parentheses around the conditions are also not usual Python style:

counts = 0
ans = 0
while ans == 0:
    s = 0
    for i in range(counts, len(arr)):
        s += int(arr[i]/(i+1-counts))
        if s &gt; M:
            break
    if (i+1) == len(arr) and s &lt;= M:
        print(counts)
        ans = 1
    counts += 1




The inner loop uses break; there's no reason that the outer loop can't also, and that would simplify things slightly:

counts = 0
while True:
    s = 0
    for i in range(counts, len(arr)):
        s += int(arr[i]/(i+1-counts))
        if s &gt; M:
            break
    if (i+1) == len(arr) and s &lt;= M:
        print(counts)
        break
    counts += 1




When we get to the line

    if (i+1) == len(arr) and s &lt;= M:


the first part of the condition can only fail because we called break in the inner loop, and in that case the second part of the condition also fails, so we can simplify that line to

    if s &lt;= M:




Python has integer division: it's just that you need to use // instead of /. So

        s += int(arr[i]/(i+1-counts))


can be changed to

        s += arr[i] // (i+1-counts)


for readability.



It would be more Pythonic to operate on a slice rather than a range of indices. I.e. instead of

    for i in range(counts, len(arr)):
        s += arr[i] // (i+1-counts)


we would have

    for i, z_i in enumerate(arr[counts:]):
        s += z_i // (i+1)




The outer loop, on the other hand, could perfectly well be a range: it starts at 0 and is incremented once each time round the loop.

for counts in range(len(arr)+1):




Finally, there's the question of names. arr isn't great, but in context it passes. M comes from the problem statement. counts is, in my opinion, not at all helpful. I would call it offset, start, or something similar. And s is really partial_sum or weighted_sum.

This gives us tidied code:

for offset in range(len(arr)+1):
    partial_sum = 0
    for i, z_i in enumerate(arr[offset:]):
        partial_sum += z_i // (i+1)
        if partial_sum &gt; M:
            break
    if partial_sum &lt;= M:
        print(offset)
        break




As for speed: flooring discards information, so it's not easy to calculate the partial sum for a given offset any faster by using information from other offsets. Moreover, you can't use binary chop or some similar search because the partial sum is not monotonic in the offset: consider input array

[1000, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1000]


So I don't think there's much you can do to speed this up.