Due to symmetry, the answer of n m is equal to the answer of -n m, so work with the absolute value of n.

We are looking for the probability of starting at position 0 and ending at n positions to the right, using m moves.

A move goes to Right with probability and Left with probability , we can imagine we have a binary string of instructions of length m (executed one by one), where 0 means go left and 1 means go right. because of this discrete uniform distribution, every unique binary string of length m will have a probability of as we have 2^m different strings.

So the answer will be the number of such strings that x - y = n where x is the number of 1's and y is the number of 0's, with some work on paper you can find that . Now the probability will be where mCx means m Choose x. You can handle the output format using the multiplicative inverse.

If m - n is odd or negative then the answer is 0.

We need n numbers that sums up to n × a so that their average is equal to a.

To choose the numbers to be the most distinct, we want to include as many smaller distinct number as possible so we will take numbers: 1, 2, 3 ..., x so that their summation is n × a, we can find x using Binary Search.
Why is it better to choose smaller numbers? because if we replace a small number with a bigger number, then we get closer to n × a thus less chances to use more distinct numbers.

A problem arise when there is no summation from 1 to x that is equal to n × a, so we choose a summation less than n × a, and what's left will be equal to which will be distributed on n - x unused numbers (because we already used x numbers).
So at least we can give each of the n - x numbers a value of 1 since values must be positive, so must be at least equal to n - x which will be our Binary Search condition which makes x a valid answer.

The answer is the number of possible values for a multiplied by the number of possible values for b.

We can see that we can use any value for b from 1 to m - 1, thus m - 1 possible values for b, as for a, we can only use a value x that has a multiplicative inverse in respect to the module m which means x and m are co-primes, and the number of positive numbers co-primes with m less than m are calculated using the totient function. so the final answer is (m - 1) × Totient(m).

Since the number of nodes are so low, you can work on all different subsets of nodes.

One solution is a minimization Dynamic Programming, with state holding only a mask telling which nodes are already included in the spanning tree, and try to include any un-included node that can connect directly to one of the included nodes until all the important nodes are in the tree.

Another slower solution is to find all the subsets of nodes that include all the important nodes, and do an MST for each of them and take the minimum.

Minimize when the condition mentioned in the statement applies :)

Since values are small, as well as the grid size, we can make a frequency array for every cell in a grid and make a 2D cumulative summation on every frequency array, this way we can get the frequency of a number in the sub-rectangle in O(1).

For every query, loop over each number from 1 to 500 and find its frequency in that sub-rectangle, as we count the sum of the frequencies we know the median is at the element that makes the sum exceed half the size of the rectangle.

The solution above gets AC, but to make the solution faster, we can do a cumulative summation on the frequency arrays so then we can binary search the point where the sum exceeds half the size of the sub-rectangle instead of looping over them.

Notice that abcd is a cyclic quadrilateral, which means its vertices all lie on a single circle.

A property for cyclic quadrilateral is that its opposite angles are supplementary. https://en.wikipedia.org/wiki/Cyclic_quadrilateral#Characterizations

With some work on paper we can find that the 2 triangles which areas are given are similar, thus the similarity ratio can be calculated using the areas: , from which we can find and then x and y.

Keep track of the number of different opposite characters, and at the end of each query add 1 to the answer if the number of different opposite characters is zero.

Read problem statement carefully to handle the cases where both teams have the same number of goals overall.

You can solve this problem by brute forcing on the numbers from b down to a until you find a solution, this solution is valid and won't cause a Time Limit Exceeded.

Why no TLE? because the biggest gap between any 2 consecutive 9 digits primes (half of the given number) is less than 320, so at most you will loop 320 times and find the second half of the number to be a prime and the GCD between a prime and any other number is 1, unless the other number is a multiple of that prime. but even if it is, the next prime number below that will work because if the first half is a multiple of both primes then it can't only be 9 digits (must be 18 digits if both primes are 9 digits). https://en.wikipedia.org/wiki/Prime_gap#Numerical_results

Show me your implementation powers :P