Problem Bank

Let $f(x) = \mathrm{e}^x - 2ax - 1 + 2a$.

1. For $a \in \mathbb{R}$, discuss the monotonicity of $f(x)$. 2. Let $g(x) = (x - 1)\ln(x - 1)$. If $f(x) \geqslant g(x)$ holds for all …

Let $f(x) = x^2 + (a-2)x - a\ln x$ with $a \in \mathbb{R}$.

1. If $a = 1$, find the extreme values of $f(x)$. 2. Discuss the monotonicity of $f(x)$. 3. For $n \in \mathbb{N}^*$, p…

The ellipse $E \colon \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$) has major axis of length $4$, and $D$ is its right vertex. For any four non-vertex points $P, Q, M…

Let $a \in \mathbb{R}$ and

1. If $f(x)$ is even, find $a$. 2. Suppose the graph of $f(x)$ meets the line $y = 2x$ at exactly $2$ points in the f…

Let $F_1$, $F_2$ be the common foci of an ellipse and a hyperbola, $P$ a common point of the two, with $\angle F_1PF_2=\dfrac{\pi}3$. Find the maximum value of the sum of the recip…

If for every acute angle $x$,

find the range of possible values of the real number $m$.

As shown in the figure, $F$ is the focus of the parabola $y^2=2px$ ($p>0$), and $M$ is the intersection point of its di…

A sequence $\{a_n\}$ has partial sums $S_n$, with $a_1 = 15$ and

For positive integers $m \le n$, find the minimum value of $…

A fraction $\dfrac pq$ (with $p,q$ positive integers) has decimal expansion $\dfrac pq=0.198\ldots$ (that is, its decimal begins $0.198$). When $q$ is as small as possible, find $p…

A sequence has $a_1 = 1$ and

Evaluate

Let $0 < a < b < \dfrac{1}{\mathrm{e}}$. Arrange

in increasing order.

Find all complex numbers $\alpha$ such that for any distinct $z_1, z_2$ with $|z_1|, |z_2| < 1$,