Problem Bank

The hyperbola $C \colon \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ (with $a > 0$, $b > 0$) passes through the point $A\left(4\sqrt{2}, 3\right)$ and has focal distance $10$.

1. Find…

Let $a$ and $b$ be real numbers, and let $f(x)=|a^2x^2-1|+ax$. Suppose that

Find the minimum value of

The function $f(x) = \mathrm{e}^x - ax$ has two zeros $x_1, x_2$. Prove that

A quadratic function $f(x)=ax^2+bx+c$ satisfies the following three conditions:

1. $2$ is a zero of $f(x)$; 2. the maximum value of $f(x)$ is $1$; 3. $f(x+1)=f(1-x)$ for every rea…

A sequence has $a_1 = 2$ and

1. Find a formula for $a_n$. 2. Let $T_n$ be the sum of the first $n$ terms of $\left\…

The function

has exactly one extreme point. Find the range of possible values of the real number $a$.

Two rays $l_1, l_2$ emanate from a point $O$. A moving line $l$ meets $l_1$ at $A$ and $l_2$ at $B$ so that the area of $\triangle OAB$ is a constant $c$. Let $D$ be the midpoint o…

Let

where $a\leqslant b\leqslant c$. Determine, with proof, which of the following claims are true.

1. $f(b)\leqslant 0$. 2. If $b=\dfra…

Let $f(x)=-\ln x+ax^2+bx-a-2b$ have two extreme points $x_1,x_2$, where $-\dfrac 12<a<0<b$ and $f(x_2)=x_2>x_1$. Find the number of real roots of the equation

Vectors $\vec{a}, \vec{b}$ make an angle of $\dfrac{\pi}{3}$ and $\left|\vec{a} - \vec{b}\right| = 5$; the vectors $\vec{c} - \vec{a}$ and $\vec{c} - \vec{b}$ make an angle of $\df…

In $\triangle ABC$, $AD$ is the altitude to side $BC$. If $BC+AD=AB+AC$, find the range of $\angle BAC$.

A sequence $\{a_n\}$ satisfies $a_n>0$ and

Determine, with proof, which of the following claims are true.

1. If $a_1=\dfrac …