Canadian Computing Olympiad: 2018 Day 1, Problem 2

Troy made the following problem (titled WA) for a programming contest:

There is a game with $N$ levels numbered from $1$ to $N$. There are two characters, both are initially at level $1$. For $i<j$, it costs $A_{i,j}$ coins to move a character from level $i$ to level $j$. It is not allowed to move a character from level $i$ to level $j$ if $i>j$. To win the game, every level (except level $1$) must be visited by exactly one character. What is the minimum number of coins needed to win?

JP is a contestant and submitted the following Python solution.

def Solve(N, A):
    # A[i][j] is cost of moving from level i to level j
    # N is the number of levels
    x, y, sx, sy = 1, 1, 0, 0 # Initialize x and y to 1, sx and sy to 0
    for i in range(2, N + 1): # loop from 2 to N
        if sx + A[x][i] < sy + A[y][i]:
            sx += A[x][i]
            x = i
        else:
            sy += A[y][i]
            y = i
    return sx + sy

Troy is certain that JP's solution is wrong. Suppose for an input to WA, JP's solution returns $X$ but the minimum number of coins needed is $Y$. To show how wrong JP's solution is, help Troy find an input $N$ and $A_{i,j}$ such that ${\displaystyle \frac{X}{Y}}$ is maximized.

Input Specification

There is no input.

Output Specification

Print an input to WA in the following format:

On the first line, print one integer $N$ $(2\le N\le 100)$.

Then print $N-1$ lines; the $i$-th line should contain $N-i$ integers $A_{i,i+1},\dots ,A_{i,N}$ $(1\le A_{i,j}\le 100)$.

If your output is not in the correct format, it will get an incorrect verdict on the sample test in the grader and score 0 points.

Otherwise, suppose that for your input, JP's solution returns $X$ but the minimum number of coins needed is $Y$. Then you will receive $\lceil min(25,{\displaystyle \frac{X}{4Y}})\rceil$ points where $\lceil Z\rceil$ is the smallest integer that is not less than $Z$.

Sample Output

5
1 2 3 4
10 9 8
7 6
5

Explanation for Sample Output

The optimal way to win the game is for one character to visit level 2 and the other character to visit levels 3, 4 and 5. This costs $(1)+(2+7+5)=15$ coins. JP's solution returns 18. Thus ${\displaystyle \frac{X}{4Y}}={\displaystyle \frac{18}{4\times 15}}=0.3$, so this output will receive $\lceil 0.3\rceil =1$ point.