Problem Bank

Three moving points $A,B,C$ are arranged clockwise, with $A$ on the $x$-axis and $B$ on the $y$-axis, and

Find the equation of the locus of the po…

Let $f(x) = a - \dfrac{1}{x} - \ln x$ with $a \in \mathbb{R}$.

1. If $a = 2$, find the number of zeros of $f(x)$ on $\left(1, e^2\right)$. 2. If $f(x)$ has exactly one zero, find …

A sequence $\{a_n\}$ is defined for $n \in \mathbb{N}^*$ by

so every term of the sequence is…

Let $n$ be an even integer with $n \geqslant 6$, and let $S_n$ denote the area of a regular $n$-gon inscribed in the unit circle.

1. Prove that

A planar quadrilateral $ABCD$ has side lengths $AB = a$, $BC = b$, $CD = c$, $DA = d$, and

Find the area $S$ of the quadrilateral in terms of $…

In a regular triangular pyramid, the cosine of the dihedral angle between a lateral face and the base is $\dfrac 16$. Find the ratio of the height $h$ of the pyramid to the radius …

Let

and let $g(x)=f(x)-|kx^2-2x|$, where $k\in\mathbb R$. Suppose $g(x)$ has exactly $4$ zeros. Find the range of $k$…

Let $a \neq 0$, and suppose $x = a$ is a local maximum point of

Determine the relationship between $ab$ and $a^2$, with proof.

Let $f(x) = a\mathrm{e}^x - \sin x$, where $a$ is a real number.

1. If $f(x)$ has an extreme value on the interval $\left(0, \dfrac{\pi}{2}\right)$, find the range of possible val…

The parabola $y=ax^2$ passes through the point $P(-1,1)$. Through the point $Q\left(-\dfrac 12,0\right)$, a line $l$ of positive slope meets the parabola at two points $M$ and $N$,…

The function $f(x) = ax - \ln x$ has two zeros $x_1, x_2$.

1. Find the range of possible values of $a$. 2. Prove that $x_1 x_2 > \mathrm{e}^2$.

In the plane, the ellipse $C: \dfrac{x^2}{3} + y^2 = 1$ has top vertex $A$. A line $\ell$ not through $A$ meets $C$ at $P$ and $Q$ with $\overrightarrow{AP} \cdot \overrightarrow{A…