Let $p(x)$ be a polynomial with integer coefficients, and let $a, b$ and $c$ be three distinct integers. Show that it is not possible to have $p(a) = b$, $p(b) = c$, and $p(c) = a$.

Find minimal value of $A=\frac{\left(x+\frac{1}{x}\right)^6-\left(x^6+\frac{1}{x^6}\right)-2}{\left(x+\frac{1}{x}\right)^3+\left(x^3+\frac{1}{x^3}\right)}$

Let the sequence $(a_n)_{n \in \mathbb{N}}$, where $\mathbb{N}$ denote the set of natural numbers, is given with $a_1=2$ and $a_{n+1}$ $=$ $a_n^2$ $-$ $a_n+1$. Find the minimum real number $L$, such that for every $k$ $\in$ $\mathbb{N}$ \begin{align*} \sum_{i=1}^k \frac{1}{a_i} < L \end{align*}

Let $ A_1,A_2,\dots,A_n$ be idempotent matrices with real entries. Prove that: \[ \mbox{N}(A_1)\plus{}\mbox{N}(A_2)\plus{}\dots\plus{}\mbox{N}(A_n)\geq \mbox{rank}(I\minus{}A_1A_2\dots A_n)\] $ \mbox{N}(A)$ is $ \mbox{dim}(\mbox{ker(A)})$

The operations below can be applied on any expression of the form \(ax^2+bx+c\). $(\text{I})$ If \(c \neq 0\), replace \(a\) by \(4a-\frac{3}{c}\) and \(c\) by \(\frac{c}{4}\). $(\text{II})$ If \(a \neq 0\), replace \(a\) by \(-\frac{a}{2}\) and \(c\) by \(-2c+\frac{3}{a}\). $(\text{III}_t)$ Replace \(x\) by \(x-t\), where \(t\) is an integer. (Different values of \(t\) can be used.) Is it possible to transform \(x^2-x-6\) into each of the following by applying some sequence of the above operations? $(\text{a})$ \(5x^2+5x-1\) $(\text{b})$ \(x^2+6x+2\)

Suppose that $f:P(\mathbb N)\longrightarrow \mathbb N$ and $A$ is a subset of $\mathbb N$. We call $f$ $A$-predicting if the set $\{x\in \mathbb N|x\notin A, f(A\cup x)\neq x \}$ is finite. Prove that there exists a function that for every subset $A$ of natural numbers, it's $A$-predicting. [i]proposed by Sepehr Ghazi-Nezami[/i]

We have $0\le x_i<b$ for $i=0,1,\ldots,n$ and $x_n>0,x_{n-1}>0$. If $a>b$, and $x_nx_{n-1}\ldots x_0$ represents the number $A$ base $a$ and $B$ base $b$, whilst $x_{n-1}x_{n-2}\ldots x_0$ represents the number $A'$ base $a$ and $B'$ base $b$, prove that $A'B<AB'$.

Let $a, b, c \in R,$ $a \ne 0$, such that $a$ and $4a+3b+2c$ have the same sign. Show that the equation $ax^2+bx+c=0$ cannot have both roots in the interval $(1,2)$.

Let $a,b,c,d$ be positive real numbers. Prove that \[(\frac{a}{a+b})^{5}+(\frac{b}{b+c})^{5}+(\frac{c}{c+d})^{5}+(\frac{d}{d+a})^{5}\ge \frac{1}{8}\]

Does there exist a real $3\times 3$ matrix $A$ such that $\text{tr}(A)=0$ and $A^2+A^t=I?$ ($\text{tr}(A)$ denotes the trace of $A,\ A^t$ the transpose of $A,$ and $I$ is the identity matrix.) [i]Proposed by Moubinool Omarjee, Paris[/i]

Let $ f(x) $ and $ g(x) $ be polynomials with integer coefficients. Prove that if for every integer value $ n $ the number $ g(n) $ is divisible by the number $ f(n) $, then $ g(x) = f(x)\cdot h(x) $, where $ h(x) $ is a polynomial,. Show with an example that the coefficients of the polynomial $ h(x) $ do not have to be integer.

[b]Problem Section #1 a) A set contains four numbers. The six pairwise sums of distinct elements of the set, in no particular order, are $189, 320, 287, 264, x$, and y. Find the greatest possible value of: $x + y$. [color=red]NOTE: There is a high chance that this problems was copied.[/color]

A [i]palindrome[/i], such as $ 83438$, is a number that remains the same when its digits are reversed. The numbers $ x$ and $ x \plus{} 32$ are three-digit and four-digit palindromes, respectively. What is the sum of the digits of x? $ \textbf{(A)}\ 20\qquad \textbf{(B)}\ 21\qquad \textbf{(C)}\ 22\qquad \textbf{(D)}\ 23\qquad \textbf{(E)}\ 24$

Given $\lambda^3 - 2\lambda^2- 1 = 0$ for some real $\lambda$ prove that $[\lambda[\lambda[\lambda n]]] - n$ is odd for any positive integer $n$ . (I Voronovich)

P(x) is polynomial such that, polynomial P(P(x)) is strictly monotone in all real number line. Prove that polynomial P(x) is also strictly monotone in all real number line.

For any polynomial $P(x)=a_0+a_1x+\ldots+a_kx^k$ with integer coefficients, the number of odd coefficients is denoted by $o(P)$. For $i-0,1,2,\ldots$ let $Q_i(x)=(1+x)^i$. Prove that if $i_1,i_2,\ldots,i_n$ are integers satisfying $0\le i_1<i_2<\ldots<i_n$, then: \[ o(Q_{i_1}+Q_{i_2}+\ldots+Q_{i_n})\ge o(Q_{i_1}). \]

[u]Algebra Round[/u] [b]p1.[/b] Given that $x$ and $y$ are nonnegative integers, compute the number of pairs $(x, y)$ such that $5x + y = 20$. [b]p2.[/b] $f(x) = x^2 + bx + c$ is a function with the property that the $x$-coordinate of the vertex is equal to the positive difference of the two roots of $f(x)$. Given that $c = 48$, compute $b$. [b]p3.[/b] Suppose we have a function $f(x)$ such that $f(x)^2 = f(x - 5)f(x + 5)$ for all integers $x$. Given that $f(1) = 1$ and $f(16) = 8$, what is $f(2016)$? [b]p4.[/b] Suppose that we have the following set of equations $$\log_2 x + \log_3 x + \log_4 x = 20$$ $$\log_4 y + \log_9 y + \log_{16} y = 16$$ Compute $\log_x y$. [b]p5.[/b] Let $\{a_n\}$ be the arithmetic sequence defined as $a_n = 2(n - 1) + 6$ for all $n \ge 1$. Compute $$\sum^{\infty}_{i=1} \frac{1}{a_ia_{i+2}}.$$ [b]p6.[/b] Let $a, b, c, d, e, f$ be non-negative real numbers. Suppose that $a + b + c + d + e + f = 1$ and $ad + be + cf \ge \frac{1}{18} $. Find the maximum value of $ab + bc + cd + de + ef + fa$. [b]p7.[/b] Let f be a continuous real-valued function defined on the positive real numbers. Determine all $f$ such that for all positive real $x, y$ we have $f(xy) = xf(y) + yf(x)$ and $f(2016) = 1$. [b]p8.[/b] Find the maximum of the following expression: $$21 cos \theta + 18 sin \theta sin \phi + 14 sin \theta cos \phi $$ [b]p9.[/b] $a, b, c, d$ satisfy the following system of equations $$ab + c + d = 13$$ $$bc + d + a = 27$$ $$cd + a + b = 30$$ $$da + b + c = 17.$$ Compute the value of $a + b + c + d$. [b]p10.[/b] Define a sequence of numbers $a_{n+1} = \frac{(2+\sqrt3)a_n+1}{(2+\sqrt3)-a_{n}}$ for $n > 0$, and suppose that $a_1 = 2$. What is $a_{2016}$? [u]Algebra Tiebreakers[/u] [b]Tie 1.[/b] Mark takes a two digit number $x$ and forms another two digit number by reversing the digits of $x$. He then sums the two values, obtaining a value which is divisible by $13$. Compute the smallest possible value of $x$. [b]Tie 2.[/b] Let $p(x) = x^4 - 10x^3 + cx^2 - 10x + 1$, where $c$ is a real number. Given that $p(x)$ has at least one real root, what is the maximum value of $c$? [b]Tie 3.[/b] $x$ satisfies the equation $(1 + i)x^3 + 8ix^2 + (-8 + 8i)x + 36 = 0$. Compute the largest possible value of $|x|$. PS. You should use hide for answers.

Let $n \in \mathbb{N}, n \ge 3.$ $a)$ Prove that there exist $z_1,z_2,…,z_n \in \mathbb{C}$ such that $$\frac{z_1}{z_2}+ \frac{z_2}{z_3}+…+ \frac{z_{n-1}}{z_n}+ \frac{z_n}{z_1}=n \mathrm{i}.$$ $b)$ Which are the values of $n$ for which there exist the complex numbers $z_1,z_2,…,z_n,$ of the same modulus, such that $$\frac{z_1}{z_2}+ \frac{z_2}{z_3}+…+ \frac{z_{n-1}}{z_n}+ \frac{z_n}{z_1}=n \mathrm{i}?$$

[b]a)[/b] Solve in $ \mathbb{R} $ the equation $ 2^x=x+1. $ [b]b)[/b] If a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ has the property that $$ (f\circ f)(x)=2^x-1,\quad\forall x\in\mathbb{R} , $$ then $ f(0)+f(1)=1. $

Let $\{a_n\}$ be the sequance with $a_0=0$, $a_n=\frac{1}{a_{n-1}-2}$ ($n\in N_+$). Select an arbitrary term $a_k$ in the sequence $\{a_n\}$ and construct the sequence $\{b_n\}$: $b_0=a_k$, $b_n=\frac{2b_{n-1}+1} {b_{n-1}}$ ($n\in N_+$) . Determine whether the sequence $\{b_n\}$ is a finite sequence or an infinite sequence and give proof.

Compute the sum $$\sum_{k=0}^{1273}\frac{1}{1 + tan^{2548}\left(\frac{k\pi}{2548}\right)}$$

Let $a_1,a_2,\ldots a_n,k$, and $M$ be positive integers such that $$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=k\quad\text{and}\quad a_1a_2\cdots a_n=M.$$ If $M>1$, prove that the polynomial $$P(x)=M(x+1)^k-(x+a_1)(x+a_2)\cdots (x+a_n)$$ has no positive roots.

Given an integer $n\ge 2$, prove that \[\lfloor \sqrt n \rfloor + \lfloor \sqrt[3]n\rfloor + \cdots +\lfloor \sqrt[n]n\rfloor = \lfloor \log_2n\rfloor + \lfloor \log_3n\rfloor + \cdots +\lfloor \log_nn\rfloor\]. [hide="Edit"] Thanks to shivangjindal for pointing out the mistake (and sorry for the late edit)[/hide]

We consider the sequence $t_1,t_2,t_3,...$ By $P_n$ we mean the product of the first $n$ terms of the sequence. Given that $t_{n+1} = t_n \cdot t_{n+2}$ for each $n$, and that $P_{40} = P_{80} = 8$. Calculate $t_1$ and $t_2$.

Solve the equation $$16x^3 = (11x^2 + x -1)\sqrt{x^2 - x + 1}.$$