Problem:
There are N balls in a two-dimensional plane. The i-th ball is at coordinates (x_i, y_i).  We will collect all of these balls, by choosing two integers p and q such that p \neq 0 or q \neq 0 and then repeating the following operation:    * Choose a ball remaining in the plane and collect it. Let (a, b) be the coordinates of this ball. If we collected a ball at coordinates (a - p, b - q) in the previous operation, the cost of this operation is 0. Otherwise, including when this is the first time to do this operation, the cost of this operation is 1.  Find the minimum total cost required to collect all the balls when we optimally choose p and q.

Incorrect submission (verdict: Wrong Answer):
from fractions import gcd

n = int(input())
l = []
for i in range(n):
    x,y = map(int,input().split())
    l.append([x, y])

res = 10 ** 100
for i in range(n):
    for j in range(i + 1, n):
        sa = []
        invalid = False
        for p,q in zip(l[i], l[j]):
            sa.append(p - q)

        if not invalid:
            g = gcd(sa[0], sa[1])
            sa[0] //= g
            sa[1] //= g
        
        cnt = 0
        flag = [False] * n
        for p in range(n):
            if flag[p]:
                continue
            cnt += 1
            flag[p] = True
            for q in range(p, n):
                sb = []
                for a,b in zip(l[p], l[q]):
                    sb.append(a - b)
                
                if sa[0] == 0:
                    if sb[0] == 0 and sb[1] % sa[1] == 0:
                        flag[q] = True
                elif sa[1] == 0:
                    if sb[1] == 0 and sb[0] % sa[0] == 0:
                        flag[q] = True
                else:
                    if sb[0] % sa[0] == 0 and sb[1] % sa[1] == 0:
                        if sb[0] // sa[0] == sb[1] // sa[1]:
                            flag[q] = True
        res = min(res, cnt)
print(res)

Fixed, accepted submission:
n = int(input())
if n == 1:
    print(1)
    exit()

l = []
for i in range(n):
    x,y = map(int,input().split())
    l.append([x, y])

res = 10 ** 100
for i in range(n):
    for j in range(i + 1, n):
        sa = []
        invalid = False
        for p,q in zip(l[i], l[j]):
            sa.append(p - q)

        cnt = 0
        flag = [False] * n
        for p in range(n):
            if not flag[p]:
                cnt += 1
            flag[p] = True
            nxt = [l[p][0] + sa[0], l[p][1] + sa[1]]
            for q in range(n):
                if l[q] == nxt:
                    if flag[q]:
                        cnt -= 1
                    else:
                        flag[q] = True
        
        # cnt = 0
        
        # for p in range(n):
        #     if flag[p]:
        #         continue
        #     cnt += 1
        #     flag[p] = True
        #     for q in range(p + 1, n):
        #         if flag[q]:
        #             continue
        #         sb = []
        
        #             sb.append(a - b)
                
        #         if sa[0] == 0:
        #             if sb[0] == 0 and sb[1] % sa[1] == 0:
        #                 flag[q] = True
        #         elif sa[1] == 0:
        #             if sb[1] == 0 and sb[0] % sa[0] == 0:
        #                 flag[q] = True
        #         else:
        #             if sb[0] % sa[0] == 0 and sb[1] % sa[1] == 0:
        #                 if sb[0] // sa[0] == sb[1] // sa[1]:
        #                     flag[q] = True
        res = min(res, cnt)
print(res)