Problem Bank

Suppose $|2x-a|\geqslant x-1$ for all real $x\in[0,2]$. Find the range of $a$.

On the curve $C: xy = 1$, starting from $A_1(x_1, y_1)$ with $x_1 = 1$, each point $A_n(x_n, y_n)$ is joined to the next intersection $A_{n+1}$ of $C$ with the line through $A_n$ o…

Let $f(x) = \ln x + x^2 - kx + 1$ with $k \in \mathbb{R}$, and $g(x) = x^2 - 3x + x\mathrm{e}^x$.

1. Find the intervals on which $f(x)$ is monotonic, discussing all cases. 2. If t…

Let $a,b$ be distinct real numbers. If the quadratic function $f(x)=x^2+ax+b$ satisfies $f(a)=f(b)$, find $f(2)$.

Let $ABCDEF$ be a hexagon whose interior angles are all equal, with $AB=6$, $BC=8$, $CD=10$, $DE=12$. Let $d$ be the diameter of the largest circle contained in the hexagon. Find $…

An ellipse $E$ centered at the origin, with axes of symmetry the coordinate axes, passes through $A(0,-2)$ and $B\left(\dfrac 32,-1\right)$.

1. Find the equation of $E$. 2. A line…

Let $f(x)=\sin x$. Suppose there exist $x_1,x_2,\dots,x_m$ with $0\leqslant x_1<x_2<\cdots<x_m\leqslant 6\pi$ such that

Real numbers $a, b$ satisfy

Find the range of possible values of $a^2 + b^2$.

Positive $x, y$ satisfy $xy(x+y) = 4$. Find the maximum of $xy$ and the minimum of $2x + y$.

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

1. If the graph of $f(x)$ has a tangent line of slope zero at some point with $x$-coordinate in $[0, 1]$, find the range of possible valu…

Suppose there exists $x \in \mathbb{R}$ such that

where $a \in \mathbb{Z}$. Find the sum of all integers $a$ satisfying this condition.

Suppose the equation

has two distinct real roots in the interval $[k-1,k+1]$. Find the range of the real number $k$.