Problem Bank

In $\triangle ABC$, let $D$ be the midpoint of the side $BC$, and let $E$ be a point on $BC$ with $AE=AC=BE$ and $DE=1$. Given $\cos C=\dfrac 13$, find the area of $\triangle ABC$.

Let

If $m \neq n$ and $f(m) + f(n) = 4$, find the minimum value of $m + n$.

The lines $y = k_1x + b_1$ and $y = k_2x + b_2$ (with $k_1 \neq k_2$) are both tangent to the curve $y = \ln x$ and both tangent to the curve $y = \mathrm{e}^x$. Find the value of …

Let $A, B$ be the left and right vertices of the hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ ($a > 0$, $b > 0$), and let $P_1, P_2, \ldots, P_n$ be distinct points on the h…

Given

find $\sin!\left(\alpha + \dfrac{\pi}{6…

Let $z,w\in\mathbb C$, and suppose the equation $w^2+zw+z\mathrm i=0$ (in $w$) always has a real root. Let $\Gamma$ be the locus of the point $Z$ in the complex plane $xOy$ corresp…

Prove that for all $x > 0$,

For curves $C_1, C_2$, define their distance $d(C_1, C_2)$ as the minimum of $|PQ|$ over $P \in C_1$, $Q \in C_2$.

1. Find $d(C_1, C_2)$ for

For $x \in \left(0, \dfrac{\pi}{2}\right)$, find the minimum value of

The ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a>b>0$) has left and right vertices $A_1$, $A_2$ and right focus $F$, with $|A_1F|=3$ and $|A_2F|=1$.

1. Find the equation of t…

The ellipse $E \colon \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$) passes through $(\sqrt{6}, 0)$ and has eccentricity $\dfrac{\sqrt{2}}{2}$. A line meets $E$ at poi…

Let $f(x) = \dfrac{x + a}{x + 1}$ for $x > 0$, and suppose the tangent line to $y = f(x)$ at $(1, f(1))$ has $y$-intercept $\dfrac{11}{2}$.

1. Find $a$. 2. Discuss the monotonicit…