Inequalities: all problems

Define

How many integers $n$ are there such that $P(n) \le 0$?

What is the least possible value of

where $x$ is a real number?

What is the least possible value of $(xy - 1)^2 + (x + y)^2$ for real numbers $x$ and $y$?

How many ordered triples of integers $(x, y, z)$ satisfy the following system of inequalities?

Consider functions $f$ that satisfy $|f(x) - f(y)| \le \frac12 |x - y|$ for all real numbers $x$ and $y$. Of all such functions that also satisfy the equation $f(300) = f(900)$, what is the greatest possible value of

How many ordered pairs of positive real numbers $(a, b)$ satisfy the equation

A real-valued function $f$ has the property that for all real numbers $a$ and $b$,

Which one of the following cannot be the value of $f(1)$?

Let $\mathcal{B}$ be the set of rectangular prisms that have volume $23$ and surface area $54$. Suppose $r$ is the least possible radius of a sphere that can fit any element of $\mathcal{B}$ inside it. Then $r^2$ can be expressed as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$.

What is the least possible value of $(xy - 1)^2 + (x + y)^2$ for real numbers $x$ and $y$?

Suppose that $|x_i| < 1$ for $i = 1, 2, \ldots, n$. Suppose further that $|x_1| + |x_2| + \cdots + |x_n| = 19 + |x_1 + x_2 + \cdots + x_n|$. What is the smallest possible value of $n$?

Let $x_1 \le x_2 \le \cdots \le x_{100}$ be real numbers such that $|x_1| + |x_2| + \cdots + |x_{100}| = 1$ and $x_1 + x_2 + \cdots + x_{100} = 0$. Among all such 100-tuples of numbers, the greatest value that $x_{76} - x_{16}$ can achieve is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

There is a smallest positive real number $a$ such that there exists a positive real number $b$ such that all the roots of the polynomial $x^3 - ax^2 + bx - a$ are real. In fact, for this value of $a$ the value of $b$ is unique. What is this value of $b$?

Let $x_1, x_2, \ldots, x_6$ be non-negative real numbers such that $x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 1$, and $x_1x_3x_5 + x_2x_4x_6 \ge \frac{1}{540}$. Let $p$ and $q$ be positive relatively prime integers such that $\frac{p}{q}$ is the maximum possible value of $x_1x_2x_3 + x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_6 + x_5x_6x_1 + x_6x_1x_2$. Find $p + q$.

Alpha and Beta both took part in a two-day problem-solving competition. At the end of the second day, each had attempted questions worth a total of 500 points. Alpha scored 160 points out of 300 points attempted on the first day, and scored 140 points out of 200 points attempted on the second day. Beta, who did not attempt 300 points on the first day, had a positive integer score on each of the two days, and Beta's daily success rate (points scored divided by points attempted) on each day was less than Alpha's on that day. Alpha's two-day success ratio was $\frac{300}{500} = \frac35$. The largest possible two-day success ratio that Beta could achieve is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m + n$?