Problem Bank

Let $P$ be a moving point on the ellipse $C:\dfrac{x^2}3+\dfrac{y^2}2=1$. The two tangent lines from $P$ to the circle $x^2+y^2=1$ meet the ellipse again at points $A$ and $B$; fro…

Let $f(x) = x\ln x + \dfrac{1}{x}$.

1. Find the intervals on which $f(x)$ is monotonic. 2. If the curve $y = f(x)$ and the line $y = kx - k$ (with $k > 0$) have exactly one point …

A positive sequence has $a_1 = 1$ and

Prove that

The function $f(x) = ax^2 + x + 1 - \mathrm{e}^x$ has a unique zero in the interval $(0, 1)$. Find the range of possible values of the real number $a$.

For a positive integer $n$, the expansion of

after combining like terms in $x^i y^j$ ($i, j = 0, 1, \dots, n$), has at least $2021$ terms. Find the min…

Find the range of the function

Let $f(x) = e^x - ax$ and $g(x) = \ln x - ax$. Suppose that for every $x_1 \in (0, +\infty)$ there exists $x_2 \in (0, +\infty)$ such that

Find the …

A sequence $\{a_n\}$ satisfies $a_1 = -\dfrac{1}{2}$ and

Determine, with proof, which of the following hold:

1. $a_n > a_{n+1}$; 2. $-\dfr…

Define

1. For $a > 0$, let $f(x) = \rho(x, a)$. Find the number of…

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

1. If $a = 2$, prove that $f(x) > \dfrac{1}{6}$. 2. If $a = \dfrac{9}{4}$, prove that $f(x) > 0$.

A particle starts at $(4, 4)$ and stops upon first touching a coordinate axis. From $(a, b)$ it moves to $(a-1, b)$, $(a, b-1)$, or $(a-1, b-1)$, each with probability $\tfrac13$, …

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