Problem Bank

Consider the circle $O \colon x^2 + y^2 = 4$ and the line $l \colon y = kx + 5$.

1. If there exist a point $A$ on $l$ and a point $B$ on the circle $O$ with $\angle OAB = \dfrac{\…

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…

Let $A, B, C$ be the interior angles of a triangle. Find the maximum value of

Let $f(x) = x\ln x - 3x$.

1. Find the extreme values of $f(x)$. 2. If the inequality

holds for all $x$ in the domain, find the minimum…

Triangle $ABC$ has $AB = AC = 1$. A moving point $P$ in its plane satisfies

A sequence satisfies

and the sum of its first $16$ terms is $540$. Find $a_1$.

Suppose that for some positive integer $n$, using an unlimited supply of stamps worth $5$, $n$, and $n+1$ cents, the greatest postage that cannot be formed is exactly $91$ cents. F…

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 …

The hyperbola $C:\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ ($a>0$, $b>0$) has left and right foci $F_1$, $F_2$ and left and right vertices $A_1$, $A_2$. The circle with diameter $F_1F_2…

In acute triangle $ABC$, the sides opposite angles $A, B, C$ are $a, b, c$, with

Find the range of possible areas of triangle $ABC$.

Real numbers $x, y$ satisfy

Find the minimum value of

Let $f(x) = \dfrac{1}{x-a} + \dfrac{1}{x-b}$ with $a, b \in \mathbb{R}$.

1. If $a = b = 1$, solve the inequality $f(x) > 1$. 2. Define the length of each of the intervals $(m, n)$…