Problem Bank

Let $O$ be the origin and $F$ the focus of the parabola $C \colon y^2 = x$. Points $A, B$ lie on $C$ with $OA \perp AB$.

1. When $\angle AOF = 45^\circ$, find $|AB|$. 2. Find the …

Let $A = \{1, 2, m\}$ where $m$ is a real number. Let $B = \{a^2 \mid a \in A\}$ and $C = A \cup B$. If the sum of all elements of $C$ is $6$, find the product of all elements of $…

The function $f(x) = \mathrm{e}^{2x} - ax^2 - 1$ (for $x \in \mathbb{R}$) has two extreme points $x_1, x_2$.

1. Find the range of possible values of the real number $a$. 2. Prove …

In the prism $ABCD\text{-}A_1B_1C_1D_1$, the edge $AA_1$ is perpendicular to the base plane $ABCD$, and the …

In the coordinate plane, consider the ellipse $\Gamma \colon \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$). Let $A$ be an endpoint of the major axis, $B$ an endpoint …

In $\triangle ABC$, $A = 120^\circ$ and $D$ is the midpoint of side $BC$. Points $E, F$ move along sides $AB, AC$ respectively with $EF \parallel BC$. Prove that

In $\triangle ABC$, $B = \dfrac{2\pi}{3}$ and $BC > AB$. A point $D$ satisfies $\overrightarrow{AD} = 2\overrightarrow{DC}$ and $BD = 1$. Find the range of possible values of $AC$.

Let $f(x)=2+3\sin x$ for $x\in\left[0,\dfrac{\pi}2\right]$. Suppose $\theta$ is such that for every $x_1\in\left[0,\dfrac{\pi}2\right]$ there exists $x_2\in\left[0,\dfrac{\pi}2\rig…

Joey, Chloe, and their daughter Zoe share a birthday. Joey is one year older than Chloe, and Zoe turns exactly $1$ today. Today Chloe's age is an integer multiple of Zoe's age, and…

Suppose $x_1\ln x_1 = x_2\ln x_2$ with $x_1 < x_2$, and the integer

Find $k$.

Consider the ellipse $C \colon \dfrac{x^2}{a^2} + \dfrac{3y^2}{a^2} = 1$ (with $a > 0$), and points $P, Q, R$ on it such that the distances from $R$ to the lines $OP$ and $OQ$ both…

Let $a > 0$, and suppose

Determine, with proof, the size order of $a, b, c$.