Problem Bank

A contest has three problems $A, B, C$. It is known that:

1. $30$ students solved at least one problem; 2. among students who solved exactly one problem, half solved $C$; 3. the n…

A sequence has $a_0 = \dfrac12$ and $a_n = a_{n-1} + \dfrac{1}{n^2}a_{n-1}^2$.

1. Prove $\dfrac{1}{a_{n-1}} - \dfrac{1}{a_n} < \dfrac{1}{n^2}$. 2. Prove $a_n < n$ for $n \ge 1$. 3…

Let

1. When $a = \sqrt{3}$, find the intervals of monotonicity of $f(x)$. 2. Suppose $f(x)$ has two zeros.

1. Find the range of possible…

Let $f(x) = \dfrac{1}{2}x^2 - 2x + a\ln x$ with $a > 0$.

1. Discuss the monotonicity of $f(x)$. 2. If $f(x)$ has two extreme points $x_1, x_2$, prove that

A positive sequence $\{a_n\}$ satisfies $a_1 = \dfrac{3}{2}$ and

Let $S_n$ be the sum of the first $n$ terms. Find

Recall that the graph of an inverse-proportion function $y=\dfrac kx$ ($k\neq 0$) is a hyperbola whose two asymptotes are the coordinate axes.

1. Find the length of the transverse…

Five people $A, B, C, D, E$ stand in a row. $A$ is adjacent to neither $B$ nor $C$, and $D$ is not adjacent to $E$. How many arrangements are there?

Let $AB$ be a chord of a circle $\omega$, and let $P$ be a point on the chord $AB$. A circle $\omega_1$ passes through $P$ and is internally tangent to $\omega$ at $A$; a circle $\…

The ellipse $M \colon \dfrac{x^2}{a^2} + \dfrac{y^2}{3} = 1$ (with $a > \sqrt{3}$) has right focus $F$, and the line $y = \dfrac{3}{\sqrt{7}}$ meets $M$ at $P$ and $Q$ with $PF \pe…

A triangle $ABC$ has perimeter $12$ and inradius $1$. Determine, with proof, whether $\triangle ABC$ must be right-angled, must be acute, must be right or acute, or none of these.

The ellipse $C:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a>b>0$) has right focus $F$. A line $l$ through $F$ meets $C$ at points $A$, $B$; the line through $F$ perpendicular to $l$ me…

Let $f(x) = e^x - ax + a$ (with $a \in \mathbb{R}$), and suppose its graph meets the $x$-axis at $A(x_1, 0)$ and $B(x_2, 0)$ with $x_1 < x_2$.

1. Find the range of possible values…