Problem Bank

Let $f(x) = |x - m|$ and $g(x) = x|x - m| + m^2 - 7m$.

1. If the equation $f(x) = |m|$ has two distinct real roots in $[-4, +\infty)$, find the range of possible values of $m$. 2.…

In $\triangle ABC$, let $D$ be the midpoint of $BC$, and suppose $\angle CAD=15^\circ$. Find the maximum value of $\angle ABC$.

You have $20$ grams of sugar water at $15\%$ sugar concentration and $15$ grams of sugar water at $40\%$ concentration, plus an unlimited supply of pure sugar and pure water. You w…

An acute triangle $ABC$ is inscribed in a circle $O$. The perpendicular from $A$ to $BC$ meets the circle $O$ again at $D$. A point $K$ lies on segment $BC$, and points $E,F$ lie o…

The figure shows the graph of the function

As the graph indicates, $f$ is con…

A line through $P(1,1)$ meets the left and right branches of the hyperbola $\dfrac{x^2}4-\dfrac{y^2}5=1$ at points $A$ and $B$ respectively, with

Let $A$ and $B$ be the left and right vertices of the ellipse $E:\dfrac{x^2}4+y^2=1$, and let $M(m,0)$ ($m>0$) be a point whose minimum distance to points of the ellipse equals $1$…

Let $S(x)$ be the digit sum of the natural number $x$. Solve

Let $f(x) = \dfrac{\ln x}{x}$ and $g(x) = x\mathrm{e}^{-x}$. Suppose there exist $x_1 \in (0, +\infty)$ and $x_2 \in \mathbb{R}$ with

Find …

Let $f(x) = (a - x)\mathrm{e}^x \ln x$, defined for $x > 0$.

1. If $f(x) \leqslant 0$ for all $x > 0$, find $a$. 2. For this value of $a$, prove that $f(x) < x^a$ for all $x > 0$.

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…

In a regular triangular prism $ABC\text{-}A_1B_1C_1$ with $AB=AA_1=1$, a point $P$ satisfies