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Problem archive

Featured Math Problems.

A growing archive of favourite problems from Think-Nth educators. Sometimes a calculus curio, sometimes a number-theory gem, sometimes a classic puzzle.

To submit a solution, email your working (a photo of handwritten work is fine) with the problem title in the subject line. We read every submission and publish the clearest one with the student's first name, only where permission is given.

Featured problem

The unfinished sum

k=1nk2=n(n+1)(2n+1)6\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}
Induction
From the archive

A friendly integral

0π/2sinxsinx+cosxdx\int_{0}^{\pi/2} \frac{\sin x}{\sin x + \cos x} dx
Calculus
From the archive

Rank in disguise

rank(123246369)=?\text{rank}\begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{pmatrix} = ?
Linear Algebra
From the archive

Small gaps

pn+1pn246p_{n+1} - p_n \le 246
Number Theory
From the archive

The medical test paradox

P(sick+)=?P(\text{sick} \mid +) = ?
Probability
From the archive

Power of a point

PAPB=PCPDPA \cdot PB = PC \cdot PD
Geometry
From the archive

Diagonalise

Av=λvAv = \lambda v
Linear Algebra
From the archive

Lagrange's Theorem

H    G|H| \; \big| \; |G|
Abstract Algebra
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