Putnam Problem, Pigeonhole Principle

Call the sequence $a_1,\ldots,a_m$ and the $n$ distinct terms $t_1,\ldots,t_n$. For each $k$ from $1$ to $m$ define the set $$S(k)=\{j\mid \hbox{$t_j$ occurs an odd number of times among $a_1,\ldots,a_k$}\}\ .$$ Now consider two cases.

1. For some $k$ we have $S(k)=\varnothing$.

2. $S(k)$ is never empty. Then there are fewer than $m$ distinct $S(k)$ so at some point we have $S(k_1)=S(k_2)$ and then...

Since you asked for hints rather than a solution I'll leave it there....