OSN 2019 Problem 4

Let us define a $\textit{triangle equivalence}$ a group of numbers that can be arranged as shown $a+b=c$ $d+e+f=g+h$ $i+j+k+l=m+n+o$ and so on...

Where at the $j$ -th row, the left hand side has $j+1$ terms and the right hand side has $j$ terms.

Now, we are given the first $N^2$ positive integers, where $N$ is a positive integer. Suppose we eliminate any one number that has the same parity with $N$ .

Prove that the remaining $N^2-1$ numbers can be formed into a $\textit{triangle equivalence}$ .

For example, if $10$ is eliminated from the first $16$ numbers, the remaining numbers can be arranged into a $\textit{triangle equivalence}$ as shown. $1+3=4$ $2+5+8=6+9$ $7+11+12+14=13+15+16$