Let $ P(x,y)$ be a polynomial in two variables $ x,y$ such that $ P(x,y)\equal{}P(y,x)$ for every $ x,y$ (for example, the polynomial $ x^2\minus{}2xy\plus{}y^2$ satisfies this condition). Given that $ (x\minus{}y)$ is a factor of $ P(x,y)$, show that $ (x\minus{}y)^2$ is a factor of $ P(x,y)$.

Suppose that $ R(z)= \sum_{n=-\infty}^{\infty} a_nz^n$ converges in a neighborhood of the unit circle $ \{ z : \;|z|=1\ \}$ in the complex plane, and $ R(z)=P(z) / Q(z)$ is a rational function in this neighborhood, where $ P$ and $ Q$ are polynomials of degree at most $ k$. Prove that there is a constant $ c$ independent of $ k$ such that \[ \sum_{n=-\infty} ^{\infty} |a_n| \leq ck^2 \max_{|z|=1} |R(z)|.\] [i]H. S. Shapiro, G. Somorjai[/i]

Let $P(x) = x^3 + 8x^2 - x + 3$ and let the roots of $P$ be $a, b,$ and $c.$ The roots of a monic polynomial $Q(x)$ are $ab - c^2, ac - b^2, bc - a^2.$ Find $Q(-1).$

$f(x)$ is a nonconstant polynomial. Given that $f(f(x)) + f(x) = f(x)^2$, compute $f(3)$.

Let $ P(X) \equal{} a_{n}X^{n} \plus{} a_{n \minus{} 1}X^{n \minus{} 1} \plus{} \ldots \plus{} a_{1}X \plus{} a_{0}$ be a real polynomial of degree $ n$. Suppose $ n$ is an even number and: a) $ a_{0} > 0$, $ a_{n} > 0$; b) $ a_{1}^{2} \plus{} a_{2}^{2} \plus{} \ldots \plus{} a_{n \minus{} 1}^{2}\leq\frac {4\min(a_{0}^{2} , a_{n}^{2})}{n \minus{} 1}$. Prove that $ P(x)\geq 0$ for all real values $ x$. [i]Laurentiu Panaitopol[/i]

Find all positive integers $p$ and $q$ such that all the roots of the polynomial $(x^2 - px+q)(x^2 -qx+ p)$ are positive integers.

Suppose the polynomial $f(x) = x^{2014}$ is equal to $f(x) =\sum^{2014}_{k=0} a_k {x \choose k}$ for some real numbers $a_0,... , a_{2014}$. Find the largest integer $m$ such that $2^m$ divides $a_{2013}$.

Find all functions $f$, of the form $f(x) = x^3 +px^2 +qx+r$ with $p$, $q$ and $r$ integers, such that $f(s) = 506$ for some integer $s$ and $f(\sqrt3) = 0$.

Let $\mathbb{Z}[x]$ denote the set of single-variable polynomials in $x$ with integer coefficients. Find all functions $\theta : \mathbb{Z}[x] \to \mathbb{Z}[x]$ (i.e. functions taking polynomials to polynomials) such that [list] [*] for any polynomials $p, q \in \mathbb{Z}[x]$, $\theta(p + q) = \theta(p) + \theta(q)$; [*] for any polynomial $p \in \mathbb{Z}[x]$, $p$ has an integer root if and only if $\theta(p)$ does. [/list] [i]Carl Schildkraut[/i]

Let $f(x)=\cos(\cos(\cos(\cos(\cos(\cos(\cos(\cos(x))))))))$, and suppose that the number $a$ satisfies the equation $a=\cos a$. Express $f'(a)$ as a polynomial in $a$.

If the polynomial $P(x)=ax^2+bx+c$ takes value $0$ for three different values of $x$, then prove the polynomial $P(x)$ takes value $0$ for all $x$.

[b]p1.[/b] Steven has just learned about polynomials and he is struggling with the following problem: expand $(1-2x)^7$ as $a_0 +a_1x+...+a_7x^7$ . Help Steven solve this problem by telling him what $a_1 +a_2 +...+a_7$ is. [b]p2.[/b] Each element of the set ${2, 3, 4, ..., 100}$ is colored. A number has the same color as any divisor of it. What is the maximum number of colors? [b]p3.[/b] Fuchsia is selecting $24$ balls out of $3$ boxes. One box contains blue balls, one red balls and one yellow balls. They each have a hundred balls. It is required that she takes at least one ball from each box and that the numbers of balls selected from each box are distinct. In how many ways can she select the $24$ balls? [b]p4.[/b] Find the perfect square that can be written in the form $\overline{abcd} - \overline{dcba}$ where $a, b, c, d$ are non zero digits and $b < c$. $\overline{abcd}$ is the number in base $10$ with digits $a, b, c, d$ written in this order. [b]p5.[/b] Steven has $100$ boxes labeled from $ 1$ to $100$. Every box contains at most $10$ balls. The number of balls in boxes labeled with consecutive numbers differ by $ 1$. The boxes labeled $1,4,7,10,...,100$ have a total of $301$ balls. What is the maximum number of balls Steven can have? [b]p6.[/b] In acute $\vartriangle ABC$, $AB=4$. Let $D$ be the point on $BC$ such that $\angle BAD = \angle CAD$. Let $AD$ intersect the circumcircle of $\vartriangle ABC$ at $X$. Let $\Gamma$ be the circle through $D$ and $X$ that is tangent to $AB$ at $P$. If $AP = 6$, compute $AC$. [b]p7.[/b] Consider a $15\times 15$ square decomposed into unit squares. Consider a coloring of the vertices of the unit squares into two colors, red and blue such that there are $133$ red vertices. Out of these $133$, two vertices are vertices of the big square and $32$ of them are located on the sides of the big square. The sides of the unit squares are colored into three colors. If both endpoints of a side are colored red then the side is colored red. If both endpoints of a side are colored blue then the side is colored blue. Otherwise the side is colored green. If we have $196$ green sides, how many blue sides do we have? [b]p8.[/b] Carl has $10$ piles of rocks, each pile with a different number of rocks. He notices that he can redistribute the rocks in any pile to the other $9$ piles to make the other $9$ piles have the same number of rocks. What is the minimum number of rocks in the biggest pile? [b]p9.[/b] Suppose that Tony picks a random integer between $1$ and $6$ inclusive such that the probability that he picks a number is directly proportional to the the number itself. Danny picks a number between $1$ and $7$ inclusive using the same rule as Tony. What is the probability that Tony’s number is greater than Danny’s number? [b]p10.[/b] Mike wrote on the board the numbers $1, 2, ..., n$. At every step, he chooses two of these numbers, deletes them and replaces them with the least prime factor of their sum. He does this until he is left with the number $101$ on the board. What is the minimum value of $n$ for which this is possible? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a$ and $b$ be integers, where $b$ is not a perfect square. Prove that $x^2 + ax + b$ may be the square of an integer only for finite number of integer values of $x$. (Martin Panák)

Let $ P_1(x), P_2(x), \ldots, P_n(x)$ be real polynomials, i.e. they have real coefficients. Show that there exist real polynomials $ A_r(x),B_r(x) \quad (r \equal{} 1, 2, 3)$ such that \[ \sum^n_{s\equal{}1} \left\{ P_s(x) \right \}^2 \equiv \left( A_1(x) \right)^2 \plus{} \left( B_1(x) \right)^2\] \[ \sum^n_{s\equal{}1} \left\{ P_s(x) \right \}^2 \equiv \left( A_2(x) \right)^2 \plus{} x \left( B_2(x) \right)^2\] \[ \sum^n_{s\equal{}1} \left\{ P_s(x) \right \}^2 \equiv \left( A_3(x) \right)^2 \minus{} x \left( B_3(x) \right)^2\]

Let $P$ be a polynomial such that $(x-4)P(2x) = 4(x-1)P(x)$, for every real $x$. If $P(0) \neq 0$, what is the degree of $P$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{None of the preceding} $