Problem Bank

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

1. If $a = 2$, prove that $f(x) > \dfrac{1}{6}$. 2. If $a = \dfrac{9}{4}$, prove that $f(x) > 0$.

In $\triangle ABC$, $AM$ is the median to side $BC$ (with $M$ on $BC$), satisfying $AM=AB-AC$, and $BC=4$. Find the maximum distance from $A$ to the line $BC$.

Let $f(x) = 2x|x|$. If

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

In the coordinate plane, a point $P$ satisfies: the distance from $P$ to the $x$-axis equals the distance from $P$ to the point $\left(0, \dfrac{1}{2}\right)$. Let $W$ be the locus…

The parabola $C \colon y^2 = 2px$ ($p > 0$) has focus $F$, and the point $P(4, t)$ on $C$ (with $t > 0$) is at distance $5$ from $F$.

1. Find $p$ and $t$. 2. Let $A, B$ be points …

An equilateral triangle $ABC$ has vertices $A, B$ on the parabola $y^2 = 4x$ and third vertex $C(4, 0)$. How many such equilateral triangles are there?

Suppose

Find the maximum value of the real number $a$.

Let $f(x) = \ln x - \mathrm{e}^{1-x}$ and $g(x) = a\left(x^2 - 1\right) - \dfrac{1}{x}$.

1. Determine the number of zeros of $y = f(x)$, with justification. 2. Let

The function

has exactly $2$ zeros. Find the range of possible values of the real number $a$.

Points $A, B, C$ lie on the surface of a sphere $O$, with $\angle AOB = \angle AOC = 45^\circ$. If the maximum possible volume of the tetrahedron $O\text{-}ABC$ is $\dfrac{2}{3}$, …

Let $a\in\mathbb R$. Suppose there exists a function $f(x)$ with domain $\mathbb R$ such that

1. for every $x_0\in\mathbb R$, the value $f(x_0)$ equals either $x_0$ or $x_0^2$; an…

As shown, $\angle B=\angle CDE=90^{\circ}$, $BC=6$, and $AB=8$. Point $D$ lies on segment $AB$, and point $E$ lies on segmen…