Problem of the Week (POTW)

Over the summer, a problem will be posted each week for students to work on and submit their solutions. These problems are curated by the HMMT Problems staff — the same people who write the HMMT November and February tests — and are intended to increase in difficulty every week, resetting with the start of a new month. After the submission deadline closes, the problem writers will review submissions and post one of their favorites alongside the next POTW. Discussion of the POTW through any form of media is closed until the submission deadline passes.

This Week's Problem (September 20 to September 24)

This week's problem will be released on Monday, September 20 at 12 PM EDT (UTC-0400).

Submission Form

The link to the submission form will be available once the problem has been released.

Schedule

The current POTW and past week’s solution will be posted on Monday, September 20 at 12 PM EDT (UTC-0400).
The deadline for submission is Friday, September 24 at 06 PM EDT (UTC-0400).

Last Week's Problem

Find the number of positive integers $n$ for which \[ 2021^n + 2022^n + 2023^n + \cdots + 2029^n \] is prime.

The main idea of this problem is considering the sum modulo 3. Daniel B. writes:
Reducing modulo $3$, each base can thus also be reduced modulo $3$. This gives us $3$ copies of $2^n$, $3$ copies of $1^n$, and $3$ copies of $0^n$.
Indeed, we find that
\[2021^n \equiv 2024^n \equiv 2027^n \equiv 2^n \pmod 3\] \[2022^n \equiv 2025^n \equiv 2028^n \equiv 0^n \pmod 3\] \[2023^n \equiv 2026^n \equiv 2029^n \equiv 1^n \pmod 3\]
So that \[2021^n + \cdots + 2029^n \equiv 3(2^n + 0^n + 1^n) \equiv 0 \pmod 3\]
So the sum is always divisible by 3, and no integers $n$ satisfy the requirement.
Aakash H. rigorously shows that $n \equiv (n + 3)^k \pmod 3$ for all positive integers $k$.
\[(n + 3)^k = \binom n 0 n^k + \binom n 1 3n^{k - 1} + \binom n 2 3^2n^{k - 2} + \cdots + \binom n n 3^k\] Every term after the first contains 3 as a factor, so by taking modulo 3, we arrive at $x^n \equiv (x + 3)^n \pmod 3$
In general, many well-known properties of modular arithmetic can be invoked without proof on contests. However, it is important to understand where these properties come from to use them effectively, and so deriving them for yourself can be a good exercise. For example, the property that
\begin{equation} a \equiv b \pmod n \implies a^k \equiv b^k \pmod n \end{equation} is a consequence of a more general fact
\begin{equation} a \equiv b \pmod n \text{ and } c \equiv d \pmod n \implies ac \equiv bd \pmod n. \end{equation} Can you prove $(2)$?

Archive

Past POTWs and their solutions can be found in the archive.