Problem: Prove that there exists a positive integer $n < 10^6$
such that $5^n$ has six consecutive zeros in its decimal representation.

Solution: We will prove that $\boxed{n = 20 + 2^{19} = 524308}$ fits the bill.

First, we claim that
\[ 5^n \equiv 5^{20} \pmod{5^{20}} \qquad\text{and}\qquad
  5^n \equiv 5^{20} \pmod{2^{20}}. \]
Indeed, the first equality holds since both sides are $0 \pmod{5^{20}}$,
and the second by $\varphi(2^{20}) = 2^{19}$ and Euler's theorem.
Hence \[ 5^n \equiv 5^{20} \pmod{10^{20}}. \]

In other words, the last $20$ digits of $5^n$
will match the decimal representation of $5^{20}$, with leading zeros.
However, we have
\[ 5^{20} = \frac{1}{2^{20}} \cdot 10^{20}
  < \frac{1}{1000^2} \cdot 10^{20} = 10^{-6} \cdot 10^{20} \]
and hence those first six of those $20$ digits will all be zero.
This completes the proof!
(To be concrete, it turns out that $5^{20} = 95367431640625$
and so the last $20$ digits of $5^n$ will be $00000095367431640625$.)

\begin{remark*}
  Many of the first posts in the JMO 2016 discussion thread
  (see \url{https://aops.com/community/c5h1230514})
  claimed that the problem was ``super easy''.
  In fact, the problem was solved by only about 10\% of contestants.
\end{remark*}

\paragraph{Authorship comments.}
This problem was inspired by the observation $5^8 \equiv 5^4 \pmod{10^4}$,
i.e.\ that $5^8$ ended with $0625$.

I noticed this one day back in November, when I was lying on my bed after
a long afternoon and was mindlessly computing powers of $5$ in my head
because I was too tired to do much else.
When I reached $5^8$ I noticed for the first time that the ending
$0625$ was actually induced by $5^4$.
(Given how much MathCounts I did, I really should have known this earlier!)

Thinking about this for a few more seconds,
I realized one could obtain arbitrarily long strings of $0$'s
by using a similar trick modulo larger powers of $10$.
This surprised me, because I would have thought that if this was true,
then I would have learned about it back in my contest days.
However, I could not find any references,
and I thought the result was quite nice,
so I submitted it as a proposal for the JMO,
where I thought it might be appreciated.

The joke about six consecutive zeros is due to Zuming Feng.