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Featured math problems

Each problem is chosen because it rewards thinking, not recall. Hints and solutions stay hidden until you ask for them.

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Published problems

New problems are added by our educators. Nothing here is auto-generated.

IntermediatePrecalculus / first-year university25 min

The unfinished sum

∑k=1nk2  =  n(n+1)(2n+1)6\sum_{k=1}^{n} k^2 \;=\; \frac{n(n+1)(2n+1)}{6}

Prove the identity by induction, then find a closed form for the sum of the first n cubes.

Mathematical inductionSeriesFactoring
AdvancedCalculus II / A-Level Further Maths15 min

A friendly integral

∫0π/2sin⁡xsin⁡x+cos⁡x dx\int_{0}^{\pi/2} \frac{\sin x}{\sin x + \cos x}\, dx

Evaluate the integral exactly.

Definite integralsSymmetrySubstitution
IntermediateLinear Algebra10 min

Rank in disguise

A=(123246369)A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{pmatrix}

State the rank of A, then describe its null space and column space.

RankNull spaceRank–nullity theorem