Problem Bank

In the coordinate plane $xOy$, let $A,B,C$ be distinct points on the circle $x^2+y^2=1$. If there exist reals $\lambda,\mu$ with $\overrightarrow{OC}=\lambda\overrightarrow{OA}+\mu…

Let $\lfloor x \rfloor$ denote the greatest integer not exceeding $x$. How many real solutions does the equation

have?

Let $A,B,C,D$ be four points on the hyperbola $xy=k$. Let $AD$ meet $BC$ at $P$, let $BD$ meet $AC$ at $Q$, and let $CD$ meet $AB$ at $R$. Let $O$ be the origin. Prove that $O,P,Q,…

Positive reals $x, y, z$ satisfy $x + y + z = 1$. Prove that for every positive integer $n$,

Let $I$ be the incenter of $\triangle ABC$ with $AB = 2$ and $AC = 3$, and write

Find t…

Let $S_n = 1 + \dfrac{1}{2} + \cdots + \dfrac{1}{n}$ for positive integers $n$. Prove that for any real numbers $a, b$ with $0 \leqslant a < b \leqslant 1$, infinitely many terms o…

The three medians of $\triangle ABC$ have lengths $6$, $9$, $12$. Find the sum of the longest and shortest sides of $\triangle ABC$ (as a numerical value).

Let $f(x) = (x^2 + ax + b)e^x$ with $b < 1$, and suppose $f(x)$ is increasing on both $(-\infty, -2)$ and $(1, +\infty)$. Find the range of possible values of $\dfrac{a+b}{a-2}$.

Let $f(x) = ax^2 - \mathrm{e}^{x-1}$.

1. When $a = \dfrac{1}{2}$, prove that $f(x)$ is decreasing on $\mathbb{R}$. 2. If $f(x) \leqslant a\cos x$ for all $x \in \left[0, \dfrac{\p…

In a sequence $\{a_n\}$, $a_1=3$ and $a_{n+1}a_n+\lambda a_{n+1}+\mu a_n^2=0$ for $n\in\mathbb N^{*}$.

1. If $\lambda=0$ and $\mu=-2$, find the general term of $\{a_n\}$. 2. If $\…

The hyperbola $E \colon \dfrac{x^2}{4} - y^2 = 1$ meets the line $l \colon y = kx - 3$ at two points $A$ and $B$, and $M$ is the midpoint of segment $AB$.

1. As $k$ varies, find t…

Sequences $\{a_n\}$ and $\{b_n\}$ satisfy $b_1 = 2a_1 = 4$ and

1. Find…