Problem Bank

A sequence $\{a_n\}$ satisfies $a_1 = 1$ and

Prove that for every $n \geqslant 2$,

is a positive in…

Let $x, y \in (0, 1)$. Prove that

Let $a, b, c > 0$ with $a + b + c = 1$. Prove that

For a bounded planar region, the largest distance between any two of its points is called the region's *diameter…

Six students sit in a row of connected desks and can leave only from the two ends of the row. They finish their exams in a uniformly random order, and a student who wants to leave …

Let $O$ be the origin, and let $A, B, C$ be three points on the ellipse $E \colon \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$) satisfying

Let $a, b \in \mathbb{R}$ be such that

When $ab$ attains its maximum value, find $\dfrac{1}{a} + 2\l…

Four players each start with $4$ coins. A bag contains one green ball, one red ball, and two white balls. In each round, the four players draw balls from the bag one at a time with…

A moving line $l$ through $P(3, 1)$ meets the left and right branches of the hyperbola $C \colon \dfrac{x^2}{3} - y^2 = 1$ at $A$ and $B$ respectively. A point $Q$ on segment $AB$,…

A line through the left focus $F$ of the ellipse $\dfrac{x^2}{9} + \dfrac{y^2}{5} = 1$ meets the ellipse at $M$ and $N$, and $A(1, 0)$ is a fixed point on the major axis. Lines $MA…

Real numbers $a, b$ satisfy $a^2 + ab + b^2 = 3$, and $M = a^2 + b^2 - ab$. Find the minimum and maximum values of $M$.

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…