Problem Bank

Prove that for every positive integer $n$,

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

1. Discuss the monotonicity of $f(x)$. 2. If the equation $f(x) = a$ has two real solutions, find the largest possible integer value of $…

A line $l$ through the focus $F$ of the parabola $C \colon y^2 = 2px$ meets $C$ at $A$ and $B$, and meets the $y$-axis at $P$. If

Let $a, b, c$ be the side lengths of a triangle. Prove that

Let $f(x) = (x + t)\ln x$, and suppose the tangent line to $f(x)$ at $x = 1$ is parallel to the line $x - y = 0$.

1. Find $t$, and find the intervals on which $f(x)$ is monotonic.…

The ellipse $C:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a>b>0$) has a vertex $A(2,0)$ and eccentricity $\dfrac{\sqrt 2}2$. The line $y=k(x-1)$ meets $C$ at two distinct points $M$ an…

From an arbitrary point $P$ of the line $l \colon x + y = 2$, two tangent lines are drawn to the circle $C \colon x^2 + y^2 = 1$, touching it at $A$ and $B$. Let $Q$ be the midpoin…

If the inequality

holds for every $x \in \left[\dfrac{1}{2}, 1\right]$ (where $\mathrm{e}$ is the base of the natural logarithm), …

Let $z \in \mathbb{C}$ be such that $\dfrac{z-1}{z+1}$ is purely imaginary. Find the minimum of

Let $f(x) = \dfrac{1}{x} + k\ln x$ with $k \neq 0$.

1. When $k = 1$, find the intervals of monotonicity and the extreme values of $f(x)$. 2. If the equation $f(x) = k$ has a solut…

Functions $f(x)$ and $g(x)$ are defined on $\mathbb{R}$ and satisfy

Determine, with proof, wh…

Call a $2020$-digit number type $A$ if it can be written as the product of two $1010$-digit numbers, and type $B$ otherwise. Which type is more common: type $A$ or type $B$?