## CCC '19 S5 - Triangle: The Data Structure

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Points: 20 (partial)
Time limit: 1.0s
Memory limit: 512M

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##### Canadian Computing Competition: 2019 Stage 1, Senior #5

In a parallel universe, the most important data structure in computer science is the triangle. A triangle of size consists of rows, with the row containing elements. Furthermore, these rows must be arranged to form the shape of an equilateral triangle. That is, each row is centred around a vertical line of symmetry through the middle of the triangle. For example, the diagram below shows a triangle of size 4:

A triangle contains sub-triangles. For example, the triangle above contains ten sub-triangles of size 1, six sub-triangles of size 2 (two of which are the triangle containing and the triangle containing ), three sub-triangles of size 3 (one of which contains ). Note that every triangle is a sub-triangle of itself.

You are given a triangle of size and must find the sum of the maximum elements of every sub-triangle of size .

#### Input Specification

The first line contains two space-separated integers and ().

Following this are lines describing the triangle. The of these lines contains space-separated integers (), representing the row of the triangle.

For 4 of the 15 available marks, .

#### Output Specification

Output the integer sum of the maximum elements of every sub-triangle of size .

#### Sample Input

4 2
3
1 2
4 2 1
6 1 4 2

#### Output for Sample Input

23