Problem Bank

The function

satisfies $f(x) \ge 0$ for all $x > 0$. Find the range of $m$.

The function $f(x) = \ln x + \dfrac{1}{2}x^2 + ax$ has two extreme points $x_1, x_2$, and

Find the range of possible values of the real number $…

In a cube $ABCD\text{-}A_1B_1C_1D_1$ of edge length $1$, let $E$ and $F$ be the midpoints of the edges $BC$ and $CC_1$ respectively. A point $P$ moves on the face $ADD_1A_1$ so tha…

Let $M,N$ be two points on the ellipse $\dfrac{x^2}4+y^2=1$ with the line $MN$ of slope $1$. Let $P$ be the reflection of $M$ across the $x$-axis. The circle with diameter $PN$ mee…

Find all functions $f \colon \mathbb{R} \to \mathbb{R}$ satisfying

Let $f(x) = \ln x$ and $g(x) = \mathrm{e}^x$.

1. Find the intervals on which $y = f(x) - x$ is monotonic. 2. Prove that on the common domain of $f$ and $g$,

…

Points $A, B, C$ lie on the surface of a sphere $O$, with $\angle AOB = \angle AOC = 45^\circ$. If the maximum possible volume of the tetrahedron $O\text{-}ABC$ is $\dfrac{2}{3}$, …

How many planes are at distances from the four vertices of a regular tetrahedron in the ratio $1 : 1 : 1 : \sqrt2$?

In $\triangle ABC$, $\angle A=60^\circ$, and $\angle BAP=\angle CAP$ with $P$ inside the triangle. The extension of $BP$ meets $AC$ at $Q$, and

In the right trapezoid $ABCD$, $BC\parallel AD$, $AB\perp AD$, $BC=8$, $AD=9$, and $AB=2\sqrt{3}$. Point $E$ is an interi…

$F_1, F_2$ are the foci of the ellipse $C: \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ ($a > b > 0$). The point $P\left(\dfrac{2\sqrt6}{3}, 1\right)$ lies on $C$, and the orthocenter …

A sequence has $a_0 = \dfrac1k$ ($k$ a positive integer) and

Find all $k$ such that $a_n < 1$ for every $n \in \mathbb{N}^*$.