Problem of the Week
MathLumen expects complete mathematical proofs, not just answers. This guide explains how to write, verify, and format your submission using MLTeX — our built-in LaTeX editor.
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A valid submission must be a complete mathematical proof — not a numerical answer, not a sketch, and not a reference to a known result without derivation. The standard is the same as a written homework solution in a graduate analysis or algebra course.
Every logical step must be present. Do not write 'by standard arguments' or 'it is easy to see' for non-trivial steps. If you invoke a theorem, name it and verify its hypotheses apply.
Use standard mathematical notation throughout. Define every symbol when it first appears. If your proof introduces a function, state its domain and codomain. Ambiguous notation is grounds for rejection.
Submit your proof as LaTeX source. All mathematical expressions — including inline variables like $n$ or $f(x)$ — must be wrapped in LaTeX delimiters. Do not mix Unicode math symbols with LaTeX; use \int, \sum, \to instead.
State your final answer clearly and unambiguously at the end of your proof, set apart from the derivation. For closed-form answers, verify numerically where possible and include that check.
Before writing anything, read the problem statement twice. Identify what is given, what is to be proved, and which techniques are likely relevant. For series problems, determine whether the series converges absolutely — this affects which manipulations are valid.
Write a rough proof by hand before opening any editor. Identify the key steps, the theorems you will invoke, and the structure of the argument. A proof written directly in LaTeX without prior thought is almost always poorly structured.
Go to MLTeX at /tools/mltex (opens in a new tab). Type your proof in the left pane. The right pane renders your LaTeX live using KaTeX. Use display math for key equations:
$$
\begin{align*}
S &= \sum_{n=1}^{\infty} \frac{H_{n+\frac{1}{2}}}{(2n-1)^2(2n+1)} \\
&= \frac{\pi^2}{8}\ln 2 - \frac{3}{16}\zeta(3) + \cdots
\end{align*}
$$Check that every equation in the right pane of MLTeX looks exactly as intended. Pay particular attention to: fraction nesting (\frac inside \frac), subscript and superscript grouping, align environment column alignment, and that all \left( \right) pairs are balanced. Fix any rendering errors before proceeding.
Once your proof is correct, copy the raw LaTeX from MLTeX's left pane and paste it into the solution textarea on the submission page. Fill in your name and email, then submit. We render and review every submission — you will not see a live preview on the submission page itself.
The constructs you will most likely need for POW submissions.
| What you need | LaTeX | Notes |
|---|---|---|
| Inline math | $f(x) = x^2$ | Single dollar signs. Use for variables in prose. |
| Display equation | $$...$$ | On its own line. For any equation worth visual space. |
| Aligned derivation | \begin{align*} ... \end{align*} | Use & before = to align. \\ to break lines. |
| Fraction | \frac{a}{b} | Never use a/b for display fractions. |
| Sum / Integral | \sum_{n=1}^{\infty} \int_a^b | Limits go in {} after _ and ^ |
| Digamma function | \psi(x) | Standard notation. Define on first use. |
| Euler–Mascheroni | \gamma | No subscript needed; \gamma alone is conventional. |
| Zeta function | \zeta(3) | Riemann zeta. Spell out \zeta, not z. |
| Natural logarithm | \ln x | Never \log unless base is ambiguous and stated. |
| Real / integer sets | \mathbb{R} \mathbb{N} \mathbb{Z} | Requires no extra package — KaTeX supports \mathbb. |
| Therefore / QED | \therefore \blacksquare | Put \blacksquare at the end of the final line. |
| Cases | \begin{cases} ... \end{cases} | Use & to separate case from condition. |
MLTeX uses KaTeX for rendering. All standard KaTeX commands are supported. For a full reference see katex.org/docs/support_table.
Writing ∑ or using copy-pasted Unicode symbols makes your proof unrenderable. Always use \sum_{n=1}^{\infty} and proper LaTeX delimiters.
Writing 'Let n be an integer' instead of 'Let $n$ be an integer' means n will appear as plain text. Every mathematical symbol, including single-letter variables, must be inside $ $.
Writing 'by the dominated convergence theorem' without verifying domination, or 'by symmetry' without explaining the symmetry, are incomplete arguments. Name and verify every theorem you apply.
Interchanging \sum and \int, or two limits, requires explicit justification — uniform convergence, dominated convergence, Fubini's theorem, or Tonelli's theorem as appropriate. State which theorem applies and why.
A proof that ends mid-derivation without isolating the final closed-form value is incomplete. State the answer explicitly on its own display line at the end, for example: $$S = \frac{\pi^2}{8}\ln 2 - \frac{3}{16}\zeta(3)$$
Every submission is stored and associated with the problem number. You will not receive an automated confirmation email — the success message on screen confirms receipt.
After the submission deadline, we read every proof submitted. We look for correctness, completeness, and clarity. Partially correct proofs are noted — elegant or novel approaches to sub-problems may be highlighted in the published solution.
A full solution article is published on MathLumen the following week. If your proof is selected as exemplary, it will be credited by name in the solution article.
Draft your proof in MLTeX, then come back to submit.