Daily math problems for October 31, 2026 — Fundamental Algebra, Descriptive Statistics & Sampling, Euclidean Geometry & more

$y=x+5\Rightarrow x=y-5$.

${\displaystyle \frac{\pi }{3}}={60}^{\circ }$, and $\cos {60}^{\circ }={\displaystyle \frac{1}{2}}$.

Congruent shapes are exactly the same shape and the same size.

$v(t)=3t^2-12t+9$. At $t=2$: $v=12-24+9=-3$ m/s (the particle is moving backward).

$\widehat{2},\widehat{4},\widehat{6}=\widehat{0}$ — three additions, so the order is $3$.

$f^{\mathrm{\prime }}(x)=2x\ln x+x^2\cdot \frac{1}{x}=2x\ln x+x=x(2\ln x+1)$.

The contrapositive of "$n^2$ even $\Rightarrow$ $n$ even" is "$n$ odd $\Rightarrow$ $n^2$ odd". A statement and its contrapositive always have the same truth value.

$3x+4y=25$.

The units are the residues coprime to $8$: $\{1,3,5,7\}$, four elements.

The polynomial $z^n-1$ has zero coefficient for $z^{n-1}$, so by Viète's formula the sum of its roots is $0$. Equivalently, the geometric sum ${\sum }_{k=0}^{n-1}{e}^{2\pi ik\mathrm{/}n}={\displaystyle \frac{1-1}{1-e^{2\pi i\mathrm{/}n}}}=0$ (for $n\ge 2$).