Given a polynomial $P$ and a positive integer $k$, we denote the $k$-fold composition of $P$ by $P^{\circ k}$. A polynomial $P$ with real coefficients is called [b]perfect[/b] if for each integer $n$ there is a positive integer $k$ so that $P^{\circ k}(n)$ is an integer. Is it true that for each perfect polynomial $P$, there exists a positive $m$ so that for each integer $n$ there is $0<k\leq m$ for which $P^{\circ k}(n)$ is an integer?

If $ y\equal{}x\plus{}\frac{1}{x}$, then $ x^4\plus{}x^3\minus{}4x^2\plus{}x\plus{}1\equal{}0$ becomes: $ \textbf{(A)}\ x^2(y^2\plus{}y\minus{}2)\equal{}0 \qquad\textbf{(B)}\ x^2(y^2\plus{}y\minus{}3)\equal{}0\\ \textbf{(C)}\ x^2(y^2\plus{}y\minus{}4)\equal{}0 \qquad\textbf{(D)}\ x^2(y^2\plus{}y\minus{}6)\equal{}0\\ \textbf{(E)}\ \text{none of these}$

Let $ A$ and $ B$ be the endpoints of a semicircular arc of radius $ 2$. The arc is divided into seven congruent arcs by six equally spaced points $ C_1,C_2,\ldots,C_6$. All chords of the form $ \overline{AC_i}$ or $ \overline{BC_i}$ are drawn. Let $ n$ be the product of the lengths of these twelve chords. Find the remainder when $ n$ is divided by $ 1000$.

Consider the equilateral triangular lattice in the complex plane defined by the Eisenstein integers; let the ordered pair $(x,y)$ denote the complex number $x+y\omega$ for $\omega=e^{2\pi i/3}$. We define an $\omega$-chessboard polygon to be a (non self-intersecting) polygon whose sides are situated along lines of the form $x=a$ or $y=b$, where $a$ and $b$ are integers. These lines divide the interior into unit triangles, which are shaded alternately black and white so that adjacent triangles have different colors. To tile an $\omega$-chessboard polygon by lozenges is to exactly cover the polygon by non-overlapping rhombuses consisting of two bordering triangles. Finally, a [i]tasteful tiling[/i] is one such that for every unit hexagon tiled by three lozenges, each lozenge has a black triangle on its left (defined by clockwise orientation) and a white triangle on its right (so the lozenges are BW, BW, BW in clockwise order). a) Prove that if an $\omega$-chessboard polygon can be tiled by lozenges, then it can be done so tastefully. b) Prove that such a tasteful tiling is unique. [i]Victor Wang.[/i]

Let $ n \ge 1$ and $ k \ge 3$ be integers. A circle is divided into $ n$ sectors $ a_1,a_2,\dots,a_n$. We will color the $ n$ sectors with $ k$ different colors such that $ a_i$ and $ a_{i \plus{} 1}$ have different color for each $ i \equal{} 1,2,\dots,n$ where $ a_{n \plus{} 1}\equal{}a_1$. Find the number of ways to do such coloring.

Polynomial $w(x)$ of second degree with integer coefficients takes for integer arguments values, which are squares of integers. Prove that polynomial $w(x)$ is a square of a polynomial.

Let $P(n)$ be the number of functions $f: \mathbb{R} \to \mathbb{R}$, $f(x)=a x^2 + b x + c$, with $a,b,c \in \{1,2,\ldots,n\}$ and that have the property that $f(x)=0$ has only integer solutions. Prove that $n<P(n)<n^2$, for all $n \geq 4$. [i]Laurentiu Panaitopol[/i]

Consider the polynomials \[f(x) =\sum_{k=1}^{n}a_{k}x^{k}\quad\text{and}\quad g(x) =\sum_{k=1}^{n}\frac{a_{k}}{2^{k}-1}x^{k},\] where $a_{1},a_{2},\ldots,a_{n}$ are real numbers and $n$ is a positive integer. Show that if 1 and $2^{n+1}$ are zeros of $g$ then $f$ has a positive zero less than $2^{n}$.

Suppose $a_0,\ldots, a_{100}$ are positive reals. Consider the following polynomial for each $k$ in $\{0,1,\ldots, 100\}$: $$a_{100+k}x^{100}+100a_{99+k}x^{99}+a_{98+k}x^{98}+a_{97+k}x^{97}+\dots+a_{2+k}x^2+a_{1+k}x+a_k,$$where indices are taken modulo $101$, [i]i.e.[/i], $a_{100+i}=a_{i-1}$ for any $i$ in $\{1,2,\dots, 100\}$. Show that it is impossible that each of these $101$ polynomials has all its roots real. [i]Proposed by Prithwijit De[/i]

Let $ f(x)$ be a $ n \minus{}$degree polynomial all of whose coefficients are equal to $ \pm 1$, and having $ x \equal{} 1$ as its $ m$ multiple root. If $ m\ge 2^k (k\ge 2,k\in N)$, then $ n\ge 2^{k \plus{} 1} \minus{} 1.$

Show that $ \pi$ is irrational.

Let $P \in \mathbb{R}[x]$. Suppose that the multiset of real roots (where roots are counted with multiplicity) of $P(x)-x$ and $P^3(x)-x$ are distinct. Prove that for all $n\in \mathbb{N}$, $P^n(x)-x$ has at least $\sigma(n)-2$ distinct real roots. (Here $P^n(x):=P(P^{n-1}(x))$ with $P^1(x) = P(x)$, and $\sigma(n)$ is the sum of all positive divisors of $n$). [i]Proposed by Malay Mahajan[/i]

Let $n{}$ be a natural number and $f{}$ be polynomial with integer coefficients. It is known that for any integer $m{}$ there is an integer $k{}$ such that $f(k)-m$ is divisible by $n{}$. Prove that there exists a polynomial $g{}$ with integer coefficients such that $f(g(m))-m$ is divisible by $n{}$ for any integer $m{}$. [i]From the folklore[/i]

Let $P(n)$ be a quadratic trinomial with integer coefficients. For each positive integer $n$, the number $P(n)$ has a proper divisor $d_n$, i.e., $1<d_n<P(n)$, such that the sequence $d_1,d_2,d_3,\ldots$ is increasing. Prove that either $P(n)$ is the product of two linear polynomials with integer coefficients or all the values of $P(n)$, for positive integers $n$, are divisible by the same integer $m>1$.

Determine all polynomials $P(x)$ with integer coefficients such that $P(n)$ is divisible by $\sigma(n)$ for all positive integers $n$. (As usual, $\sigma(n)$ denotes the sum of all positive divisors of $n$.)

For a positive integer $n$ we denote by $s(n)$ the sum of the digits of $n$. Let $P(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ be a polynomial, where $n \geqslant 2$ and $a_i$ is a positive integer for all $0 \leqslant i \leqslant n-1$. Could it be the case that, for all positive integers $k$, $s(k)$ and $s(P(k))$ have the same parity?

The senior coefficient $a$ in the square polynomial $$P(x) = ax^2 + bx + c$$ is more than $100$. What is the maximal number of integer values of $x$, such that $|P(x)|<50$.

Let $n$ be a fixed integer with $n \ge 2$. We say that two polynomials $P$ and $Q$ with real coefficients are [i]block-similar[/i] if for each $i \in \{1, 2, \ldots, n\}$ the sequences \begin{eqnarray*} P(2015i), P(2015i - 1), \ldots, P(2015i - 2014) & \text{and}\\ Q(2015i), Q(2015i - 1), \ldots, Q(2015i - 2014) \end{eqnarray*} are permutations of each other. (a) Prove that there exist distinct block-similar polynomials of degree $n + 1$. (b) Prove that there do not exist distinct block-similar polynomials of degree $n$. [i]Proposed by David Arthur, Canada[/i]

Let $k$ be a real number, such that the equation $kx^2 + k = 3x^2 + 2-2kx$ has two real solutions different. Determine all possible values of $k$, such that the sum of the roots of the equation is equal to the product of the roots of the equation increased by $k$.

Solve the following system of equations in rational numbers: \[ (x^2+1)^3=y+1,\\ (y^2+1)^3=z+1,\\ (z^2+1)^3=x+1.\]

Show that no one $n$-th root of a rational (for $n$ a positive integer) can be a root of the polynomial $x^5 - x^4 - 4x^3 + 4x^2 + 2$.

Let $A$ be an $n\times n$ real matrix with minimal polynomial $x^n + x - 1$. Prove that the trace of $(nA^{n-1} + I)^{-1}A^{n-2}$ is zero.

A polynomial $P(x)$ with integer coefficients satisfies the following: if $F(x)$, $G(x)$, and $Q(x)$ are polynomials with integer coefficients satisfying $P\Big(Q(x)\Big)=F(x)\cdot G(x)$, then $F(x)$ or $G(x)$ is a constant polynomial. Prove that $P(x)$ is a constant polynomial.

Suppose $1, 2, 3$ are the roots of the equation $x^4 + ax^2 + bx = c$. Find the value of $c$.

Which of the following polynomials does not divide $x^{60} - 1$? $ \textbf{a)}\ x^2+x+1 \qquad\textbf{b)}\ x^4-1 \qquad\textbf{c)}\ x^5-1 \qquad\textbf{d)}\ x^{15}-1 \qquad\textbf{e)}\ \text{None of above} $