Problem Bank

The quadratic $f(x) = ax^2 + 2bx + c$ with $c > b > a$ passes through $(1, 0)$ and meets the line $y = -a$. Prove that

In right triangle $ABC$, $\angle C = 90^\circ$ and $AC : CB = 1 : 3$. Points $E$ and $D$ are the two trisection poin…

Suppose

where $m,n\in\mathbb N^{*}$ with $m,n\geqslant 2$. Find $m+n$. (Hint: use $\sin 3x=4\sin x\sin(60^\circ-x)\sin(60^\circ+x)$.)

Consider the hyperbola $E \colon \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ and a point $M(x_0, y_0)$ not on $E$ with $x_0y_0 \neq 0$. Let $N(\lambda x_0, \lambda y_0)$, where

A function $f(x)$ is strictly decreasing on $(0, +\infty)$, and for every $x \in (0, +\infty)$,

Let $g(x) = f(x) …

Let $n \geqslant 2$ and $a_1, a_2, \ldots, a_n \geqslant 0$ with $\displaystyle\sum_{i=1}^n a_i^2 = S$. Find, in terms of $S$ and $n$, the maximum and minimum values of

For $a, b \in [0, 1]$, find the minimum value of

In $\triangle ABC$, $AC=6$, $BC=7$, $\cos A=\dfrac 15$, and $I$ is the incenter. A point $P$ satisfies $\overrightarrow{IP}=x\overrightarrow{IA}+y\overrightarrow{IB}$ with $x,y\in[…

Let $\omega = \cos\dfrac{\pi}{5} + \mathrm{i}\sin\dfrac{\pi}{5}$. Expand

as a polynomial in $x$ with integer coefficien…

In the coordinate plane $xOy$, let $A(4,0)$ and $B(0,3)$, and let $P$ be a moving point on the circle $x^2+y^2=4$. Find the minimum value of

The quadratic $f(x) = ax^2 - 4x + c$ (for $x \in \mathbb{R}$) has range $[0, +\infty)$. Find the maximum value of

Let $f(x) = x^2 + \dfrac{a}{x}$ for $x > 0$. If $f(f(x))$ has a unique zero, find the range of possible values of $a$.