Problem Bank

Nonnegative reals $a, b, c$ satisfy $a + b + c = 1$. Let $M$ and $m$ be the maximum and minimum of

Find $M + m$.

A round table has twelve chairs numbered $1$ through $12$ in order. Four ambassadors, each accompanied by one advisor, are to be seated. Each ambassador must sit in an even-numbere…

Three vertices of a regular $n$-gon are chosen at random. The probability that they form an obtuse triangle is $\dfrac{93}{125}$. Find all possible values of $n$.

Let $E$ be a non-circular conic whose axes of symmetry are parallel or perpendicular to the coordinate axes, and let $A,B,C,D$ be four distinct points on $E$ such that the lines $A…

The function

has three distinct zeros. Find the range of possible values of the real number $a$.

Triangle $ABC$ has side lengths $AB=7$, $BC=8$, $CA=9$. A circle $\omega_1$ passes through $B$ and is tangent to the line $AC$ at $A$; a circle $\omega_2$ passes through $C$ and is…

For functions $f(x), g(x)$ with common domain $D$, a line $y = ax + b$ is a separating line if

Let $f(x)…

Let

If the function $g(x) = f\left(f(x) - 1\right)$ has exactly $6$ zeros, …

In the coordinate plane $xOy$, the ellipse $C:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a>b>0$) has eccentricity $\dfrac{\sqrt 3}2$ and left and right foci $F_1$, $F_2$. The circle of…

Compute

Let $a > 0$ and $f(x) = (1 - ax)\left(\mathrm{e}^x - 1\right)$.

1. If $a = 1$, prove that $f(x) < \ln(x+1)$ for $x > 0$. 2. If $h(x) = \ln(x+1) - f(x)$ has a local minimum point $…

An equilateral triangle $ABC$ has incircle of radius $2$ centered at $I$. A point $P$ satisfies $PI=1$. Find the maximum value of the ratio of the areas of $\triangle APB$ and $\tr…