$$$Result = Max(0, Y-X)$$$

def read():
    return int(input())

def read_list():
    return [int(i) for i in input().split()]


def solve():
    x, y = read_list()
    print(max(0, y-x))

for i in range(read()):
    solve()

$$$QueryResult = Min(Max(X, \sum_{i = 1}^N A_i), \sum_{i = 1}^M A_i + Y)$$$

we can optimized this by pre-calculating a prefix sums arrays of $$$A$$$.

def read():
    return int(input())

def read_list():
    return [int(i) for i in input().split()]

def solve():
    n = read()
    a = read_list()
    prefix_sum = [0]*(n+1)
    for i in range(1, n+1):
        prefix_sum[i] = prefix_sum[i-1] + a[i-1]
    q = read()
    for i in range(q):
        x, y, m = read_list()
        result = min(max(x, prefix_sum[n]), prefix_sum[m]+y)
        print(result)


solve()   

we can pre-caculate $$$next(x)$$$ for all possible $$$x$$$ using a modified version of the Sieve of Eratosthenes.

def read():
    return int(input())
def read_list():
    return [int(i) for i in input().split()]


m = int(1e7)
nxt = [1] * (m+1)
for i in range(2, m+1):
    if nxt[i] == 1:
        for j in range(i*i, m+1, i):
            nxt[j] = max(nxt[j], j//i)


def solve():
    n = read()
    a = read_list()
    b = read_list()
    result = 0
    for i in range(n):
        x = a[i]
        y = b[i]
        while x != y:
            if x > y:
                x = nxt[x]
            else:
                y = nxt[y]
            result += 1
    print(result)

for i in range(read()):
    solve()
    

using a mono stack we can find the intervals of all the possible beautiful decreases, then sort them by length.

def read():
    return int(input())

def read_list():
    return [int(i) for i in input().split()]


def solve():
    n, q = read_list()
    a = read_list()
    total = sum(a)
    b_decreases = [] # (length, width)
    stack = [(0, -1)]
    

    def mono_push(value, index):
        if value > stack[-1][0]:
            stack.append((value, index))
        elif value < stack[-1][0]:
            while value < stack[-1][0]:
                if len(stack) > 1:
                    b_decreases.append([index-stack[-2][1]-1, min(stack[-1][0]-value, stack[-1][0]-stack[-2][0])])
                    stack.pop()

            if value != stack[-1][0]:
                stack.append((value, index))


    for i in range(n):
        mono_push(a[i], i)  
    mono_push(0, n)
    
    
        
    b_decreases.sort()

    for i in range(q):
        k = read()
        while len(b_decreases) > 0 and k > 0: 
            decrease_count = min(k, b_decreases[-1][1])
            total -= b_decreases[-1][0] * decrease_count
            k -= decrease_count
            b_decreases[-1][1] -= decrease_count

            if b_decreases[-1][1] == 0:
                b_decreases.pop()

        print(total)


solve()

def read():
    return int(input())
def read_list():
    return [int(i) for i in input().split()]

def solve():
    n = read()
    count = [0] * (n+1)
    for i in range(n):
        for j in read_list()[1:]:
            count[j] += 1

    result = []
    for i in range(1, n+1):
        if count[i] > n//2:
            result.append(i)
    print(len(result))
    print(' '.join(map(str, result)))


for i in range(read()):
    solve()

```python

```

def read():
    return int(input())
def read_list():
    return [int(i) for i in input().split()]

def solve():
    n, x = read_list()
    s = input()
    t = input()
    result = 0
    for i in range(n):
        j = (i+n-x%n)%n
        if t[i] != s[j]:
            result += 1
    print(result)

solve()

def read():
    return int(input())
def read_list():
    return [int(i) for i in input().split()]

def solve():
    n, m, a = read_list()
    print(min(n-m, a))


for i in range(read()):
    solve()

def read():
    return int(input())
def read_list():
    return [int(i) for i in input().split()]

def solve():
    n, x = read_list()
    h = read_list()
    result = [0] * n
    for i in range(n):
        if h[i] < x:
            result[i] = 1
    print(' '.join(map(str, result)))   


for i in range(read()):
    solve() 

```python

```