Problem Bank

Visitors to a landmark are surveyed one at a time about whether they plan to visit a second attraction. A visitor who declines scores $1$ point and a visitor who accepts scores $2$…

As shown, $\angle B=\angle CDE=90^{\circ}$, $BC=6$, and $AB=8$. Point $D$ lies on segment $AB$, and point $E$ lies on segmen…

A sequence $a_1, a_2, \ldots, a_{21}$ satisfies $|a_{n+1} - a_n| = 1$ for $n = 1, \ldots, 20$, and $a_1, a_7, a_{21}$ form a geometric sequence. If $a_1 = 1$ and $a_{21} = 9$, find…

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

1. For $a \in \mathbb{R}$, discuss the monotonicity of $f(x)$. 2. Let $g(x) = (x - 1)\ln(x - 1)$. If $f(x) \geqslant g(x)$ holds for all …

You may use the fact that for $x \in \left(0, \dfrac{\pi}{2}\right)$,

1. Prove that $\dfrac{\sin x}{x} > \dfrac{1}{2}$ for $x \in \left(0, \d…

Let $f(x) = \dfrac{1 + \ln x}{x - 1}$ and $g(x) = \dfrac{k}{x}$ with $k \in \mathbb{N}^*$. Suppose that for every $c > 1$ there exist $a, b$ with $0 < a < b < c$ such that

For a finite sequence $E \colon a_1, a_2, \dots, a_M$ (with $M \geqslant 2$), an index $n \geqslant 2$ is called a low point of $E$ if $a_i > a_n$ for every $i = 1, 2, \dots, n-1…

Plane vectors $\vec a, \vec b, \vec c$ satisfy $|\vec b|\cdot|\vec c| = 1$ and

Prove that there are infinitely many triples of positive integers $x, y, z$ with

Positive reals $x, y, z$ satisfy

Prove that

Let

Call a positive integer $m$ ($m \geqslant 4$) a related number of $A_{2n}$ if every $m$-el…

A sequence $\{a_n\}$ satisfies $a_{n+1}=\dfrac 14(a_n-6)^3+6$ ($n\in\mathbb N^{*}$). Determine, with proof, which of the following claims are true.

1. When $a_1=3$, $\{a_n\}$ is d…