Problem of the Week

Mathematics & Statistics

Problem of the Week

Problem #7: A Square Integral

Fall 2026 · Due Monday, November 9 at 11:59 pm (Arizona time)

Find all continuous functions \(f \colon \mathbb{R} \to \mathbb{R}\) such that

\[\bigl(f(x)\bigr)^2 = \int_0^x f(t)\,dt \qquad \text{for every real number } x.\]

Printable PDF of Problem #7 (same content as this page)

How to enter

The contest is open to all undergraduates at Northern Arizona University. You don't need to be a math major, and you don't need to solve every problem to be on the ladder.

  1. Join the Problem of the Week Canvas course (one time only; sign in with your NAU account).
  2. Write up your solution as a single PDF, either typed or neatly handwritten and scanned, with your name at the top. If you don't have a scanner, use a phone app like Microsoft Lens or Adobe Scan, or scan at the library.
  3. Upload it to that week's assignment in Canvas by Monday at 11:59 pm Arizona time. Late submissions are not accepted. If you submit more than once, only your last submission before the deadline is graded.

If your instructor gives credit for Problem of the Week, add their name and course as a submission comment in Canvas (e.g., Swift, MAT 239).

Rules

  1. Your own work only. You may not discuss the problem with anyone (classmates, friends, tutors, or instructors) until after the deadline.
  2. No AI, no internet, no calculators. This means no ChatGPT, Gemini, Claude, Copilot, Photomath, Wolfram Alpha, or similar tools; no web searches or online forums; and no calculators, computer algebra systems, or code. You may use your own textbooks and class notes to look up standard definitions and theorems.
  3. Explain your reasoning. The goal is a clear, convincing mathematical argument, not an answer with a box around it. A correct answer without justification earns at most 1 point.
  4. Be ready to explain it. The coordinator may ask any participant to talk through their solution in a short conversation (in person or on Teams). A submission you can't explain will receive a score of 0.
  5. Don't share solutions or post them online before the deadline.
  6. Breaking these rules removes you from the ladder for the semester, and your instructor will not receive credit for your participation.

By submitting, you certify that your solution is entirely your own work and follows these rules.

Scoring and the ladder

Each solution is scored from 0 to 4:

  • 4 – complete, correct, and clearly explained
  • 3 – correct approach with a minor gap or error
  • 2 – substantial progress toward a solution
  • 1 – some relevant progress, or a correct answer without justification
  • 0 – no meaningful progress, or a rule violation

A ladder of total points is posted in the lobby of the Adel Math Building (across from the MAP room). Names appear on the posted ladder but are never published on this website. If you'd prefer a nickname on the ladder, or to be left off, send a Canvas message. A model solution is posted in Canvas after each deadline.

Questions

Send questions through Canvas Inbox or email ian.williams@nau.edu with "POTW" in the subject line. Questions asking to clarify what a problem means are welcome; hints are not given.

Past problems

Previous problems and their model solutions are posted in the Problem of the Week Canvas course. Only the current problem appears on this page.