Worthy Matrix KAVGMAT Solution

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Worthy Matrix KAVGMAT April Long Challenge 2021

Chef found a matrix A with N rows (numbered 1 through N) and M columns (numbered 1 through M), where for each row r and column c, the cell in row r and column c (denoted by (r,c)) contains an integer Ar,c.

This matrix has two interesting properties:

The integers in each row form a non-decreasing sequence, i.e. for each valid i, Ai,1≤Ai,2≤…≤Ai,M.
The integers in each column also form a non-decreasing sequence, i.e. for each valid j, A1,j≤A2,j≤…≤AN,j.
A K-worthy submatrix is a square submatrix of A, i.e. a submatrix with l rows and l columns, for any integer l, such that the average of all the integers in this submatrix is ≥K.

Chef wants you to find the number of K-worthy submatrices of A.

Input
The first line of the input contains a single integer T denoting the number of test cases. The description of T test cases follows.
The first line of each test case contains three space-separated integers N, M and K.
N lines follow. For each valid i, the i-th of these lines contains M space-separated integers Ai,1,Ai,2,Ai,3,…,Ai,M.
Output
For each test case, print a single line containing one integer ― the number of K-worthy submatrices of A.

Constraints
1≤T≤10
1≤N⋅M≤106
N≤M
0≤K≤109
0≤Ar,c≤109 for each valid r,c
the sum of N⋅M over all test cases does not exceed 106
Subtasks
Subtask #1 (15 points): the sum of N⋅M over all test cases does not exceed 103
Subtask #2 (25 points): the sum of N⋅M over all test cases does not exceed 4⋅105
Subtask #3 (60 points): original constraints

Example Input
1
3 3 4
2 2 3
3 4 5
4 5 5
Example Output
7
Explanation
Example case 1: The following are the seven 4-worthy submatrices:

[3445] with average 4; this matrix occurs only once
[4555] with average 4.75; this matrix also occurs only once
[4] with average 4; we find this matrix twice in A
[5] with average 5; we find this matrix 3 times in A

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