Let $n \ge 2$ be an integer and $f_1(x), f_2(x), \ldots, f_{n}(x)$ a sequence of polynomials with integer coefficients. One is allowed to make moves $M_1, M_2, \ldots $ as follows: in the $k$-th move $M_k$ one chooses an element $f(x)$ of the sequence with degree of $f$ at least $2$ and replaces it with $(f(x) - f(k))/(x-k)$. The process stops when all the elements of the sequence are of degree $1$. If $f_1(x) = f_2(x) = \cdots = f_n(x) = x^n + 1$, determine whether or not it is possible to make appropriate moves such that the process stops with a sequence of $n$ identical polynomials of degree 1.

There are $6$ irrational numbers. Prove that there are always three of them, suppose $a,b,c$ such that $a+b$,$b+c$,$c+a$ are irrational numbers. [i](Erken)[/i]

Let non-constant polynomial $f(x)$ with real coefficients is given with the following property: for any positive integer $n$ and $k$, the value of expression $$\frac{f(n + 1)f(n + 2)... f(n + k)}{ f(1)f(2) ... f(k)} \in Z$$ Prove that $f(x)$ is divisible by $x$

Let $ \mathcal{H}_n$ be the set of all numbers of the form $ 2 \pm\sqrt{2 \pm\sqrt{2 \pm\ldots\pm\sqrt 2}}$ where "root signs" appear $ n$ times. (a) Prove that all the elements of $ \mathcal{H}_n$ are real. (b) Computer the product of the elements of $ \mathcal{H}_n$. (c) The elements of $ \mathcal{H}_{11}$ are arranged in a row, and are sorted by size in an ascending order. Find the position in that row, of the elements of $ \mathcal{H}_{11}$ that corresponds to the following combination of $ \pm$ signs: \[ \plus{}\plus{}\plus{}\plus{}\plus{}\minus{}\plus{}\plus{}\minus{}\plus{}\minus{}\]

Solve the equation $x^3 - [x] = 3$.

The lateral surface of a cylinder of revolution is divided by $n-1$ planes parallel to the base and $m$ parallel generators into $mn$ cases $( n\ge 1,m\ge 3)$. Two cases will be called neighbouring cases if they have a common side. Prove that it is possible to write a real number in each case such that each number is equal to the sum of the numbers of the neighbouring cases and not all the numbers are zero if and only if there exist integers $k,l$ such that $n+1$ does not divide $k$ and \[ \cos \frac{2l\pi}{m}+\cos\frac{k\pi}{n+1}=\frac{1}{2}\] [i]Ciprian Manolescu[/i]

Consider the polynomials \begin{align*}P(x) &= (x + \sqrt{2})(x^2 - 2x + 2)\\Q(x) &= (x - \sqrt{2})(x^2 + 2x + 2)\\R(x) &= (x^2 + 2)(x^8 + 16).\end{align*} Find the coefficient of $x^4$ in $P(x)\cdot Q(x)\cdot R(x)$.

[i](8 pts)[/i] Let $n$ be a positive integer, and let $a, b, c$ be real numbers. (a) [i](2 pts)[/i] Given that $a\cos x+b\cos 2x +c\cos 3x \geq -1$ for all reals $x$, find, with proof, the maximum possible value of $a+b+c$. (b) [i](6 pts)[/i] Let $f$ be a degree $n$ polynomial with real coefficients. Suppose that $|f(z)| \leq \left|f(z)+\frac{2}{z}\right|$ for all complex $z$ lying on the unit circle. Find, with proof, the maximum possible value of $f(1)$.

[b]p1.[/b] Velociraptor $A$ is located at $x = 10$ on the number line and runs at $4$ units per second. Velociraptor $B$ is located at $x = -10$ on the number line and runs at $3$ units per second. If the velociraptors run towards each other, at what point do they meet? [b]p2.[/b] Let $n$ be a positive integer. There are $n$ non-overlapping circles in a plane with radii $1, 2, ... , n$. The total area that they enclose is at least $100$. Find the minimum possible value of $n$. [b]p3.[/b] How many integers between $1$ and $50$, inclusive, are divisible by $4$ but not $6$? [b]p4.[/b] Let $a \star b = 1 + \frac{b}{a}$. Evaluate $((((((1 \star 1) \star 1) \star 1) \star 1) \star 1) \star 1) \star 1$. [b]p5.[/b] In acute triangle $ABC$, $D$ and $E$ are points inside triangle $ABC$ such that $DE \parallel BC$, $B$ is closer to $D$ than it is to $E$, $\angle AED = 80^o$ , $\angle ABD = 10^o$ , and $\angle CBD = 40^o$. Find the measure of $\angle BAE$, in degrees. [b]p6. [/b]Al is at $(0, 0)$. He wants to get to $(4, 4)$, but there is a building in the shape of a square with vertices at $(1, 1)$, $(1, 2)$, $(2, 2)$, and $(2, 1)$. Al cannot walk inside the building. If Al is not restricted to staying on grid lines, what is the shortest distance he can walk to get to his destination? [b]p7. [/b]Point $A = (1, 211)$ and point $B = (b, 2011)$ for some integer $b$. For how many values of $b$ is the slope of $AB$ an integer? [b]p8.[/b] A palindrome is a number that reads the same forwards and backwards. For example, $1$, $11$ and $141$ are all palindromes. How many palindromes between $1$ and 1000 are divisible by $11$? [b]p9.[/b] Suppose $x, y, z$ are real numbers that satisfy: $$x + y - z = 5$$ $$y + z - x = 7$$ $$z + x - y = 9$$ Find $x^2 + y^2 + z^2$. [b]p10.[/b] In triangle $ABC$, $AB = 3$ and $AC = 4$. The bisector of angle $A$ meets $BC$ at $D$. The line through $D$ perpendicular to $AD$ intersects lines $AB$ and $AC$ at $F$ and $E$, respectively. Compute $EC - FB$. (See the following diagram.) [img]https://cdn.artofproblemsolving.com/attachments/2/7/e26fbaeb7d1f39cb8d5611c6a466add881ba0d.png[/img] [b]p11.[/b] Bob has a six-sided die with a number written on each face such that the sums of the numbers written on each pair of opposite faces are equal to each other. Suppose that the numbers $109$, $131$, and $135$ are written on three faces which share a corner. Determine the maximum possible sum of the numbers on the three remaining faces, given that all three are positive primes less than $200$. [b]p12.[/b] Let $d$ be a number chosen at random from the set $\{142, 143, ..., 198\}$. What is the probability that the area of a rectangle with perimeter $400$ and diagonal length $d$ is an integer? [b]p13.[/b] There are $3$ congruent circles such that each circle passes through the centers of the other two. Suppose that $A, B$, and $C$ are points on the circles such that each circle has exactly one of $A, B$, or $C$ on it and triangle $ABC$ is equilateral. Find the ratio of the maximum possible area of $ABC$ to the minimum possible area of $ABC$. (See the following diagram.) [img]https://cdn.artofproblemsolving.com/attachments/4/c/162554fcc6aa21ce3df3ce6a446357f0516f5d.png[/img] [b]p14.[/b] Let $k$ and $m$ be constants such that for all triples $(a, b, c)$ of positive real numbers, $$\sqrt{ \frac{4}{a^2}+\frac{36}{b^2}+\frac{9}{c^2}+\frac{k}{ab} }=\left| \frac{2}{a}+\frac{6}{b}+\frac{3}{c}\right|$$ if and only if $am^2 + bm + c = 0$. Find $k$. [b]p15.[/b] A bored student named Abraham is writing $n$ numbers $a_1, a_2, ..., a_n$. The value of each number is either $1, 2$, or $3$; that is, $a_i$ is $1, 2$ or $3$ for $1 \le i \le n$. Abraham notices that the ordered triples $$(a_1, a_2, a_3), (a_2, a_3, a_4), ..., (a_{n-2}, a_{n-1}, a_n), (a_{n-1}, a_n, a_1), (a_n, a_1, a_2)$$ are distinct from each other. What is the maximum possible value of $n$? Give the answer n, along with an example of such a sequence. Write your answer as an ordered pair. (For example, if the answer were $5$, you might write $(5, 12311)$.) PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $P(x) = 1-x+x^2-x^3+\dots+x^{18}-x^{19}$ and $Q(x)=P(x-1)$. What is the coefficient of $x^2$ in polynomial $Q$? $ \textbf{(A)}\ 840 \qquad\textbf{(B)}\ 816 \qquad\textbf{(C)}\ 969 \qquad\textbf{(D)}\ 1020 \qquad\textbf{(E)}\ 1140 $

Let $P(x) = x^2 + rx + s$ be a polynomial with real coefficients. Suppose $P(x)$ has two distinct real roots, both of which are less than $-1$ and the difference between the two is less than $2$. Prove that $P(P(x)) > 0$ for all real $x$.

Let $ a$, $ b$, $ c$, $ d$ be positive real numbers such that $ abcd \equal{} 1$ and $ a \plus{} b \plus{} c \plus{} d > \dfrac{a}{b} \plus{} \dfrac{b}{c} \plus{} \dfrac{c}{d} \plus{} \dfrac{d}{a}$. Prove that \[ a \plus{} b \plus{} c \plus{} d < \dfrac{b}{a} \plus{} \dfrac{c}{b} \plus{} \dfrac{d}{c} \plus{} \dfrac{a}{d}\] [i]Proposed by Pavel Novotný, Slovakia[/i]

Prove that every positive rational number can be expressed uniquely as a finite sum of the form $$a_1+\frac{a_2}{2!}+\frac{a_3}{3!}+\dots+\frac{a_n}{n!},$$ where $a_n$ are integers such that $0 \leq a_n \leq n-1$ for all $n > 1$.

The first four terms of an arithmetic sequence are $ p,9,3p\minus{}q,$ and $ 3p\plus{}q$. What is the $ 2010^{\text{th}}$ term of the sequence? $ \textbf{(A)}\ 8041\qquad \textbf{(B)}\ 8043\qquad \textbf{(C)}\ 8045\qquad \textbf{(D)}\ 8047\qquad \textbf{(E)}\ 8049$

For each positive integer $ n$, let $ c(n)$ be the largest real number such that \[ c(n) \le \left| \frac {f(a) \minus{} f(b)}{a \minus{} b}\right|\] for all triples $ (f, a, b)$ such that --$ f$ is a polynomial of degree $ n$ taking integers to integers, and --$ a, b$ are integers with $ f(a) \neq f(b)$. Find $ c(n)$. [i]Shaunak Kishore.[/i]

Let $a$ and $b$ be two real numbers. Find these numbers given that the graphs of $f:\mathbb{R} \to \mathbb{R} , f(x)=2x^4-a^2x^2+b-1$ and $g:\mathbb{R} \to \mathbb{R} ,g(x)=2ax^3-1$ have exactly two points of intersection.

Which of the following polynomials with integer coefficients has $\sin\left(10^{\circ}\right)$ as a root? $\text{(A) }4x^3-4x+1\qquad\text{(B) }6x^3-8x^2+1\qquad\text{(C) }4x^3+4x-1\qquad\text{(D) }8x^3+6x-1\qquad\text{(E) }8x^3-6x+1$

Let $\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\mathcal{B}$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as \begin{align*} (x + y + z)P(x, y, z) + (xy + yz + zx)Q(x, y, z) + xyzR(x, y, z) \end{align*} with $P, Q, R \in \mathcal{A}$. Find the smallest non-negative integer $n$ such that $x^i y^j z^k \in \mathcal{B}$ for all non-negative integers $i, j, k$ satisfying $i + j + k \geq n$.

Find all complex polynomial $ P(x)$ such that for any three integers $ a,b,c$ satisfying $ a \plus{} b \plus{} c\not \equal{} 0, \frac{P(a) \plus{} P(b) \plus{} P(c)}{a \plus{} b \plus{} c}$ is an integer.

How many ordered pairs of integers (x; y) satisfy the equation $x^2 + y^2 + xy = 4(x + y)?$.

Find all functions $f : \mathbb N \to \mathbb N$ satisfying, for all $x \in \mathbb N$, \[ f(2f(x)) = x + 1998 . \]

Let $(a_n )_{n \ge o}$ and $(b_n )_{n \ge o}$ be sequences defined by $a_{n+2} = a_{n+1}+ a_n$ , $n = 0 , 1 , . .. $, $a_0 = 1$, $a_1 = 2$, and $b_{n+2} = b_{n+1} + b_n$ , $n = 0 , 1 , . . .$, $b_0 = 2$, $b_1 = 1$. How many integers do the sequences have in common?

Let $a$, $b$, $c$ be the three roots of the equation $x^3-(k+1)x^2+kx+12=0$, where $k$ is a real number. If $(a-2)^3+(b-2)^3+(c-2)^3=-18$, find the value of $k$.

Prove that, for all integer $a>1$, the prime divisors of $5a^4-5a^2+1$ have the form $20k\pm1,k\in\mathbb{Z}$. [i]Proposed by Géza Kós.[/i]

Find all functions $f : Z \to Z$ for which $f(g(n)) - g(f(n))$ is independent on $n$ for any $g : Z \to Z$.