Problem Bank

Let $a$ and $b$ be coprime positive integers with $a+b=2005$. Let $[x]$ denote the integer part of $x$, and set

In the equilateral triangle $ABC$ with side length $2$ shown in the figure, $D$ is the midpoint of $BC$, and $E$, $F…

Let $f(x) = \mathrm{e}^x - a\ln(ax - a) + a$ with $a > 0$. If $f(x) > 0$ holds for all $x$ in the domain, find the range of possible values of the real number $a$.

Find the number of zeros of the function $f(x)=2^x|\log_{0.5}x|-1$.

Let $a > 0$, $A(2a, 0)$, $B(0, 2)$, and let $O$ be the origin.

1. Find the standard equation of the circle that has $OA$ as a chord and is tangent to line $AB$ at $A$. 2. This cir…

A convex quadrilateral $ABCD$ is circumscribed about a circle $O$; that is, each of its four sides is tangent to the circle. A line through $O$ meets the sides $AB$ and $CD$ at $K$…

In a rectangle $ABCD$ with $AB = 2\sqrt{3}$ and $BC = 6$, points $E, F$ move along $AD, BC$ respectively with $CF = 2AE$. An equilateral triangle $EFG$ is erected on segment $EF$. …

The inequality

holds for all $x > 0$. Find the minimum value of $m^2 + n^2$.

A complex number $z$ satisfies $|z| = 1$ and

Find the least positive integer $n$ for which such a $z$ exists.

In the region

there exist four points whose pairwise distances all equal $d$. Find the maximum value of $d$.

Let $f(x) = \left(x^2 + ax - a\right)e^{1-x}$, where $a \in \mathbb{R}$.

1. Find the number of zeros of $f'(x)$. 2. Prove that "$a \geqslant 0$" is a sufficient but not necessary …

Let $a, b, m > 0$. The inequality

holds for all $a, b > 0$. Find the maximum value of $m$.