Problem Bank

Let $f(x)$ satisfy

For $x>0$, determine, with justification, whether $f(x)$ has a local maximum and/or a lo…

The Riemann zeta function $\zeta(s)$ is closely tied to the distribution of primes. Writing $\operatorname{Re}(s)$ for the real part of a complex number $s$, define

In $\triangle ABC$, $\angle C=90^\circ$, $AC=1$, $BC=\sqrt{3}$, and $D$ is the midpoint of side $AB$. A point $M$ lies o…

As shown in the figure, in triangle $ABC$ we have $\angle ABC = 90^\circ$, $AB = \sqrt{3}$, and $BC = 1$. The poi…

Find the minimum distance from a lattice point (a point with integer coordinates) to the line

Points $A$ and $B$ lie on skew lines $a$ and $b$ respectively, and the segment $AB$ is perpendicular to both. Moving points $P\in a$ and $Q\in b$ satisfy that $PA+QB$ is constant. …

The ellipse $C:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a>b>0$) has upper vertex $A(0,1)$ and eccentricity $\dfrac{\sqrt 3}2$.

1. Find the equation of the ellipse $C$. 2. Let $B$ be…

The inequality

holds for all $x > -1$. Find the minimum value of $\dfrac{b}{a}$.

A moving point $Q$ in the coordinate plane satisfies $|QE| - |QF| = 2\sqrt{2}$, where $E(-\sqrt{3}, 0)$ and $F(\sqrt{3}, 0)$.

1. Find the equation of the locus $C$ of $Q$. 2. A li…

A function $f(x)$ is differentiable on $\mathbb{R}$, with

If

Let $f(x) = |1 - x^2|$. A number $a$ is chosen uniformly from $[0, 1]$ and a number $b$ uniformly from $[1, 2]$, independently. Find the probability that $f(a) \leqslant f(b)$.

Real numbers $a, b, c$ satisfy

Prove that $a + b + c = 0$.