Problem Bank

In a methane molecule $\mathrm{CH}_4$, the four hydrogen atoms sit at the vertices $H_1, H_2, H_3, H_4$ of a regular tetrahedron with edge length $1$, and the carbon atom sits at t…

Let $f(x) = a\ln x + \dfrac{a-1}{x} - x$ with $a < 2$.

1. Find the extreme values of $f(x)$, discussing all cases. 2. Let $m \in \mathbb{Z}$ and $a = 1$. If the inequality

A degree-$4$ integer-coefficient polynomial $f$ satisfies

Find $f(x)$.

Let

If the equation $f(f(x)) = m$ has exactly two real solutions $x_1, x_2$, find the mi…

Evaluate

Let $f(x)=\ln(x+1)+a(x^2-x)$, where $a\in\mathbb R$.

1. Discuss the number of extreme points of $f(x)$, with justification. 2. If $f(x)\geqslant 0$ for all $x>0$, find the range o…

Consider the ellipse $\dfrac{x^2}4+\dfrac{y^2}3=1$ with right vertex $M(2,0)$. A variable line through the fixed point $P(2,3)$ meets the ellipse at points $A$ and $B$. Prove that …

The function $f(x) = x^2 + ax + b$ (with $a, b \in \mathbb{R}$) has a zero $x_0$ in the interval $(0, 1]$. Find the maximum value of

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…

Find all positive reals $a, b, c$ with

The parabola $C:y^2=2px$ ($p>0$) has focus $F$ and directrix $l:x=-1$. Points $A(x_1,y_1)$ and $B(x_2,y_2)$ (with $y_2>y_1\geqslant 0$) lie on $C$, the line $AB$ meets $l$ at a poi…

The ellipse $\dfrac{x^2}{4} + y^2 = 1$ has left vertex $A$. Points $P$ and $Q$ on the ellipse are symmetric about the $y$-axis, $O$ is the origin, and $B$ is a point on the ellipse…