number of indices differences

Question

I have a counting problem below:

Let $n>2$ be integer and $p>0$ be a real number. For all $1\leq i<j\leq n$, suppose $a_{ij}$'s and $b_k$'s satisfy

1. $a_{ij}=b_{j-i}=p^{j-i}$
2. $b_{j-i}=b_{n-(j-i)}$.

Now define $A_k=\{(i,j):a_{ij}=p^k\}$ for every $k< \frac n2$. What is the cardinality of $A_k$? I have tried for small $n$ i.e., $n=4,5$ but I haven't obtained the rule for general $n$. Thanks for any help.

Answer 1

I must be missing something, but it looks like this question is trivial.

Since $b_{j-i}=p^{j-i}$, we have $b_k = p^k$. Since $b_k = b_{n-k}$, we must have $p=1$. Thus, $a_{i,j}=1$ for all $i,j$, so $|A_k|=n^2$.