Problem Bank

The base of the tetrahedron $P\text{-}ABC$ is an equilateral triangle of side $2$. Given $PA = PB = \sqrt{2}$ and that the dihedral angle $P\text{-}BA\text{-}C$ measures $60^\circ$…

Consider the ellipse $\dfrac{x^2}4+\dfrac{y^2}3=1$ with left vertex $A$ and upper vertex $B$. Let $P$ and $Q$ be two points on the ellipse symmetric about the origin $O$. The line …

In the figure, square $ABCD$ has side length $6$. Points $E$ and $F$ lie on sides $AD$ and $BC$ respectively, with $DE …

Let $f(x) = (x^2 + ax + b)e^x$ with $b < 1$, and suppose $f(x)$ is increasing on both $(-\infty, -2)$ and $(1, +\infty)$. Find the range of possible values of $\dfrac{a+b}{a-2}$.

The $2n$ consecutive integers $1, 2, \ldots, 2n$ (where $n \in \mathbb{N}^*$, $n \geqslant 2$) are randomly split into two groups $A$ and $B$ of $n$ numbers each. Let $a$ be the la…

A sequence $\{a_n\}$ satisfies $a_1 = 1$ and

and $b_n = \dfrac{a_n}{n(n+1)}$.

1. Find $b_n$ and $a_n$. 2. Prove that

Arrange in increasing order:

Let $x, y, z \ge 0$ with at most one of them zero. Find the minimum of

Call $f(x)$ convex on $(a, b)$ if it is differentiable there and $f'(x)$ is increasing.

1. Decide whether $y = x^3$ and $y = \ln\dfrac{1}{x}$ are convex on their domains. 2. Let…

Let $f(x) = \mathrm{e}^{mx} + x - x\ln x$ with $m \geqslant 0$.

1. When $m = 1$, find the range of $f(x)$ on $[1, \mathrm{e}]$. 2. Let $f'(x)$ be the derivative of $f(x)$. Determi…

Let $f(x) = x|x + a| + m|x - 1|$.

1. For $a = 0$, $m = 1$, describe the monotonicity of $f$. 2. If the maximum of $f$ on $[0, 2]$ equals $a + 1$, find the range of $m$ (in terms o…

Let $a, b, c$ be positive real numbers with $a + b + c = 3$. Prove that