Problem Bank

For the ellipse $E \colon \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$), the quadrilateral whose vertices are the foci and the endpoints of the minor axis is a square…

A regular quadrilateral pyramid $P\text{-}ABCD$ has base edge $1$ and height $h$, and its apex $P$ lies inside (or on the surface of) the cube $ABCD\text{-}A_1B_1C_1D_1$. Determine…

A sequence $\{a_n\}$ satisfies $a_1=1$, $a_{n+3}\leqslant a_n+3$, and $a_{n+2}\geqslant a_n+2$ for all $n$. Find $\{a_n\}$.

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

1. If $a = 1$, prove that $f(x) < \ln(x+1)$ for $x > 0$. 2. If $h(x) = \ln(x+1) - f(x)$ has a local minimum point $…

Let $f(x)=x(x-3)^2$, and suppose $f(a)=f(b)=f(c)$ with $a<b<c$. Determine, with proof, which of the following claims are true.

1. $1<a<2$. 2. $a+b+c=6$. 3. $a+b>2$. 4. The range o…

A round table has twelve chairs numbered $1$ through $12$ in order. Four ambassadors, each accompanied by one advisor, are to be seated. Each ambassador must sit in an even-numbere…

Let $f(x)$ be a continuous even function that is strictly monotonic for $x>0$. Find the sum of all $x$ satisfying $f(x)=f\left(\dfrac{x+3}{x+4}\right)$.

As shown in the figure, rectangle $ABCD$ is folded along its diagonal $BD$, carrying triangle $ABD$ to triangle $A'BD$, where $…

A line $l$ meets the ellipse $C \colon \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ at points $P$ and $Q$, the triangle $POQ$ (with $O$ the origin) has area $\dfrac{ab}{2}$, and $M$ is…

As shown in the figure, the quadrilaterals $ABCD$ and $ADPQ$ are both squares, and the planes containing them are per…

Let $a, b, c, d$ be real numbers with $abcd = 1$. Prove that

Let $A$ be the right vertex of the ellipse $E \colon \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$). Two mutually perpendicular lines $AP$ and $AQ$ through $A$ meet th…