Problem Bank

The parabola $C:y^2=2px$ ($p>0$) has focus $F$, and its directrix meets the $x$-axis at a point $D$. A line through $F$ meets $C$ at two points $A$ and $B$, with

Two right triangles $ABC$ and $ADC$ share the common hypotenuse $AC = 10$ and lie on opposite sides of it, so that $\…

Let $f(x) = \left(\mathrm{e}^{ax} - 1\right)\ln x$, where $a > 0$.

1. When $a = 1$, find the area of the triangle formed by the two coordinate axes and the tangent line to the cur…

Two distinct points $A$ and $B$ on the ellipse $\dfrac{x^2}2+y^2=1$ are symmetric about the line $y=mx+\dfrac 12$.

1. Find the range of $m$. 2. Find the maximum area of $\triangle…

In a cube $ABCD\text{-}A_1B_1C_1D_1$, let $P$ be a moving point on the four sides of the square $A_1B_1C_1D_1$, $O$ the center of the square $ABCD$, and $M,N$ the midpoints of $AB,…

Let $f(z) = z^2 - 19z$. There is a complex number $z$ such that the points corresponding to $z$, $f(z)$, and $f(f(z))$ in the complex plane are the vertices of a right triangle who…

Find all functions $f:\mathbb R\to\mathbb R$ such that for all $x,y\in\mathbb R$,

A triangle $ABC$ has vertices $A(-2,0)$ and $B(0,4)$, and its Euler line has equation $l:x+y-2=0$. (The Euler line joins the circumcenter $O$ and orthocenter $H$, with $2\overright…

Let $M(x_0, y_0)$ be a point in the first quadrant interior to the hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ ($a, b > 0$). Lines through $M$ meet the right branch at $A$ …

A regular dodecagon has area $144$. Find the area of the shaded region shown in the figure.

A sequence $\{a_n\}$ satisfies $a_1 = p$, $a_2 = p + 1$, and

for all positive integers $n$, where $p$ is a fixed real number.

For which val…

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

1. If $f$ has two extreme points $x_1, x_2$, find the range of $a$. 2. Under that condition, if $m > f(x_1) + f(x_2)$ always hol…