Problem Bank

Let $a > 0$ with $a \neq 1$, and let $f(x) = x^a - a^x$ for $x > 0$.

1. When $a = 2$, find the intervals on which the derivative $f'(x)$ is monotonic. 2. Suppose $y = f(x)$ has ex…

The sides $a, b, c$ of a right triangle satisfy

Find the maximum possible area of the triangle.

Real numbers $a, b$ satisfy $a^2 + ab + b^2 = 3$, and $M = a^2 + b^2 - ab$. Find the minimum and maximum values of $M$.

The ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$) has eccentricity $\dfrac{\sqrt{3}}{2}$. The line $y = \dfrac{1}{2}x + 1$ meets the ellipse at $A$ and $B$, …

A function $f(x)$ is differentiable on $\mathbb{R}$, with

If

A sequence $\{a_n\}$ satisfies $a_1 = 3$ and

1. Prove that $2 < a_{n+1} < a_n$ for all $n$. 2. Let $S_n = a_1…

Let $f(x) = \mathrm{e}^x - \mathrm{e}x$ and

1. Discuss the monotonicity of $y = f(ax)$. 2. If for all $x \ge 0$…

Let $a, b > 0$ with

Find the minimum value of $\dfrac{3}{a} + \dfrac{1}{b}$.

Positive reals $a, b$ satisfy

Find the range of possible values of $a + b$.

Let $a > 0$ and $f(x) = ax^2 - x + 1$. Suppose that on every closed interval of length $1$ there exist two points at which the values of $f$ differ by at least $1$ in absolute valu…

Prove that for every positive integer $n$,

Call $\triangle ABC$ harmonious if, after folding along its three midsegments, the four small triangles can be assembled into a (non-degenerate) tetrahedron. With $A, B, C$ the a…