Daily math problems for August 4, 2026 — Calculus, Algebra, Absolute value & more

The limit $\displaystyle\lim_{n \to \infty} 2^n$ equals:

On $\mathbb{R}$, define $x * y = x + y - 2$. Then $3 * 5$

The sum of all real solutions of $|x - 3| = 5$ is:

The limit $\displaystyle\lim_{x \to 0} \dfrac{1 - \cos x}{x^2}$

Find $\left|\dfrac{(1+i\sqrt{3})^{2024}}{2^{2023}}\right|$

The value of $\binom{6}{0} - \binom{6}{1} + \binom{6}{2} - \cdots + \binom{6}{6}$

The value of $\arctan\sqrt{3}$ is:

At a meeting, every pair of people shakes hands exactly once. If there are $45$ handshakes total, how many people are there?

On Monday at 7:00 a.m. a monk begins climbing a winding mountain trail, arriving at the summit at 5:00 p.m. The next morning at 7:00 a.m. she begins descending the same trail and reaches the base at 5:00 p.m. There must exist a point on the trail and a clock time at which the monk was at the same place on both days. Which classical theorem most directly justifies this?

For $A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$, the entry $(A^n)_{12}$