Let $a$ and $b$ be positive real numbers, with $a < b$ and let $n$ be a positive integer. Prove that for all real numbers $x_1, x_2, \ldots , x_n \in [a, b]$: $$ |x_1 - x_2| + |x_2 - x_3| + \cdots + |x_{n-1} - x_n| + |x_n - x_1| \leq \frac{2(b - a)}{b + a}(x_1 + x_2 + \cdots + x_n)$$ And determine for what values of $n$ and $x_1, x_2, \ldots , x_n$ the equality holds.

Prove that, for every pair $ n$, $r$ of positive integers, there can be found a polynomial $ f(x)$ of degree $ n$ with integer coefficients, so that every polynomial $ g(x)$ of degree at most $ n$, for which the coefficients of the polynomial $ f(x)\minus{}g(x)$ are integers with absolute value not greater than $ r$, is irreducible over the field of rational numbers.

Let $f : \mathbb{C} \to \mathbb{C} $ be defined by $ f(z) = z^2 + iz + 1 $. How many complex numbers $z $ are there such that $ \text{Im}(z) > 0 $ and both the real and the imaginary parts of $f(z)$ are integers with absolute value at most $ 10 $? ${ \textbf{(A)} \ 399 \qquad \textbf{(B)} \ 401 \qquad \textbf{(C)} \ 413 \qquad \textbf{(D}} \ 431 \qquad \textbf{(E)} \ 441 $

With $400$ members voting the House of Representatives defeated a bill. A re-vote, with the same members voting, resulted in passage of the bill by twice the margin$\dagger$ by which it was originally defeated. The number voting for the bill on the re-vote was $\frac{12}{11}$ of the number voting against it originally. How many more members voted for the bill the second time than voted for it the first time? $\textbf{(A)}\ 75 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 50 \qquad\textbf{(D)}\ 45 \qquad\textbf{(E)}\ 20$ $\dagger$ In this context, margin of defeat (passage) is defined as the number of nays minus the number of ayes (nays-ayes).

Find the number of real numbers which satisfy the equation $x|x-1|-4|x|+3=0$. $ \textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4 $

Let $k$ be an integer. If the equation $(x-1)|x+1|=x+\frac{k}{2020}$ has three distinct real roots, how many different possible values of $k$ are there?

For $a>0$, let $f(a)=\lim_{t\rightarrow +0} \int_{t}^{1} |ax+x\ln x|\ dx.$ Let $a$ vary in the range $0 <a< +\infty$, find the minimum value of $f(a)$.

Done with her new problems, Wendy takes a break from math. Still without any fresh reading material, she feels a bit antsy. She starts to feel annoyed that Michael's loose papers clutter the family van. Several of them are ripped, and bits of paper litter the floor. Tired of trying to get Michael to clean up after himself, Wendy spends a couple of minutes putting Michael's loose papers in the trash. "That seems fair to me," confirms Hannah encouragingly. While collecting Michael's scraps, Wendy comes across a corner of a piece of paper with part of a math problem written on it. There is a monic polynomial of degree $n$, with real coefficients. The first two terms after $x^n$ are $a_{n-1}x^{n-1}$ and $a_{n-2}x^{n-2}$, but the rest of the polynomial is cut off where Michael's page is ripped. Wendy barely makes out a little of Michael's scribbling, showing that $a_{n-1}=-a_{n-2}$. Wendy deciphers the goal of the problem, which is to find the sum of the squares of the roots of the polynomial. Wendy knows neither the value of $n$, nor the value of $a_{n-1}$, but still she finds a [greatest] lower bound for the answer to the problem. Find the absolute value of that lower bound.

Let $ a,b,c\in\mathbb{C}^* $ pairwise distinct, having the same absolute value, and satisfying: $$ a^2+b^2+c^2-ab-bc-ca=0. $$ Prove that $ a,b,c $ represents the affixes of the vertices of a right or equilateral triangle.

Let $x_1, ..., x_n$ ($n \ge 2$) be real numbers from the interval $[1, 2]$. Prove that $|x_1 - x_2| + ... + |x_n - x_1| \le \frac{2}{3}(x_1 + ... + x_n)$, with equality holding if and only if $n$ is even and the $n$-tuple $(x_1, x_2, ..., x_{n - 1}, x_n)$ is equal to $(1, 2, ..., 1, 2)$ or $(2, 1, ..., 2, 1)$.

The set of all real solutions of the inequality \[ |x \minus{} 1| \plus{} |x \plus{} 2| < 3\] is $ \textbf{(A)}\ x \in ( \minus{} 3,2) \qquad \textbf{(B)}\ x \in ( \minus{} 1,2) \qquad \textbf{(C)}\ x \in ( \minus{} 2,1) \qquad$ $ \textbf{(D)}\ x \in \left( \minus{} \frac32,\frac72\right) \qquad \textbf{(E)}\ \O \text{ (empty})$ Note: I updated the notation on this problem.

The symbol $|a|$ means $+a$ if $a$ is greater than or equal to zero, and $-a$ if $a$ is less than or equal to zero; the symbol $<$ means "less than"; the symbol $>$ means "greater than." The set of values $x$ satisfying the inequality $|3-x|<4$ consists of all $x$ such that: $ \textbf{(A)}\ x^2<49 \qquad\textbf{(B)}\ x^2>1 \qquad\textbf{(C)}\ 1<x^2<49\qquad\textbf{(D)}\ -1<x<7\qquad\textbf{(E)}\ -7<x<1 $

Fix two positive integers $a,k\ge2$, and let $f\in\mathbb{Z}[x]$ be a nonconstant polynomial. Suppose that for all sufficiently large positive integers $n$, there exists a rational number $x$ satisfying $f(x)=f(a^n)^k$. Prove that there exists a polynomial $g\in\mathbb{Q}[x]$ such that $f(g(x))=f(x)^k$ for all real $x$. [i]Victor Wang.[/i]

How many integers $n$ are there such that $n^3+8$ has at most $3$ positive divisors? $ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ \text{None of the above} $

Let $n$ and $k$ be integers with $k\leq n-2$. The absolute value of the sum of elements of any $k$-element subset of $\{a_1,a_2,\cdots,a_n\}$ is less than or equal to 1. Show that: If $|a_1|\geq1$, then for any $2\leq i \leq n$, we have: $$|a_1|+|a_i|\leq2$$

Suppose we have a simple polygon (that is it does not intersect itself, but not necessarily convex). Show that this polygon has a diameter which is completely inside the polygon and the two arcs it creates on the polygon perimeter (the two arcs have 2 vertices in common) both have at least one third of the vertices of the polygon.

Let $1,2,3,4,5,6,7,8,9,11,12,\cdots$ be the sequence of all positive integers which do not contain the digit zero. Write $\{a_n\}$ for this sequence. By comparing with a geometric series, show that $\sum_{k=1}^n \frac{1}{a_k} < 90$.

Nine balls numbered $1,2,\cdots,9$ are put on nine poines that divide the circle into nine equal parts. The sum of absolute values of the difference between the number of two adjacent balls is $S$. Find the probablity of $S$ takes its minumum value. Note: If one way of putting balls can be the same as another one by rotating or specular-reflecting, then they are considered the same way.

For a function $f(x)=x^3-x^2+x$, find the limit $\lim_{n\to\infty} \int_{n}^{2n}\frac{1}{f^{-1}(x)^3+|f^{-1}(x)|}\ dx.$

Prove that for any three real numbers $x,y,z$ the following inequality holds: $$|x|+|y|+|z|-|x+y|-|y+z|-|z+x|+|x+y+z|\ge0.$$

Let $n$ be a positive integer and let $(a_1,a_2,\ldots ,a_{2n})$ be a permutation of $1,2,\ldots ,2n$ such that the numbers $|a_{i+1}-a_i|$ are pairwise distinct for $i=1,\ldots ,2n-1$. Prove that $\{a_2,a_4,\ldots ,a_{2n}\}=\{1,2,\ldots ,n\}$ if and only if $a_1-a_{2n}=n$.

Let $n$ be an integer, and let $X$ be a set of $n+2$ integers each of absolute value at most $n$. Show that there exist three distinct numbers $a, b, c \in X$ such that $c=a+b$.

Let $ f(x)\equal{}x^2\plus{}3$ and $ y\equal{}g(x)$ be the equation of the line with the slope $ a$, which pass through the point $ (0,\ f(0))$ . Find the maximum and minimum values of $ I(a)\equal{}3\int_{\minus{}1}^1 |f(x)\minus{}g(x)|\ dx$.

Let $ a,b$ and $ c$ real numbers such that the equation $ ax^2\plus{}bx\plus{}c\equal{}0$ has two distinct real solutions $ p_1,p_2$ and the equation $ cx^2\plus{}bx\plus{}a\equal{}0$ has two distinct real solutions $ q_1,q_2$. We know that the numbers $ p_1,q_1,p_2,q_2$ in that order, form an arithmetic progression. Show that $ a\plus{}c\equal{}0$.

Prove that if $|x| < 1$ and $|y| < 1$, then $\left|\frac{x - y}{1 -xy}\right|< 1$.