Found problems: 3597
Susana and Brenda play a game writing polynomials on the board. Susana starts and they play taking turns.
1) On the preparatory turn (turn 0), Susana choose a positive integer $n_0$ and writes the polynomial $P_0(x)=n_0$.
2) On turn 1, Brenda choose a positive integer $n_1$, different from $n_0$, and either writes the polynomial
$$P_1(x)=n_1x+P_0(x) \textup{ or } P_1(x)=n_1x-P_0(x)$$
3) In general, on turn $k$, the respective player chooses an integer $n_k$, different from $n_0, n_1, \ldots, n_{k-1}$, and either writes the polynomial
$$P_k(x)=n_kx^k+P_{k-1}(x) \textup{ or } P_k(x)=n_kx^k-P_{k-1}(x)$$
The first player to write a polynomial with at least one whole whole number root wins. Find and describe a winning strategy.
Find a method by which one can compute the coefficients of $P(x) = x^6 + a_1x^5 + \cdots+ a_6$ from the roots of $P(x) = 0$ by performing not more than $15$ additions and $15$ multiplications.
Prove that if $f(x)$ is a polynomial with integer coefficients and there exists an integer $k$ such that none of $f(1),\ldots,f(k)$ is divisible by $k$, then $f(x)$ has no integral root.
The polynomials $ P(x)$ and $ Q(x)$ are given. It is known that for a certain polynomial $ R(x, y)$ the identity $ P(x) \minus{} P(y) \equal{} R(x, y) (Q(x) \minus{} Q(y))$ applies. Prove that there is a polynomial $ S(x)$ so that $ P(x) \equal{} S(Q(x)) \quad \forall x.$
Prove that for all polynomials $P \in \mathbb{R}[x]$ and positive integers $n$, $P(x)-x$ divides $P^n(x)-x$ as polynomials.
Let $F$ be the set of all polynomials $\Gamma$ such that all the coefficients of $\Gamma (x)$ are integers and $\Gamma (x) = 1$ has integer roots. Given a positive intger $k$, find the smallest integer $m(k) > 1$ such that there exist $\Gamma \in F$ for which $\Gamma (x) = m(k)$ has exactly $k$ distinct integer roots.
For any set $A=\{a_1,a_2,\cdots,a_m\}$, let $P(A)=a_1a_2\cdots a_m$. Let $n={2010\choose99}$, and let $A_1, A_2,\cdots,A_n$ be all $99$-element subsets of $\{1,2,\cdots,2010\}$. Prove that $2010|\sum^{n}_{i=1}P(A_i)$.
Let $ \mathbb{Z}$ be the ring of rational integers. Construct an integral domain $ I$ satisfying the following conditions:
a)$ \mathbb{Z} \varsubsetneqq I$;
b) no element of $ I \minus{} \mathbb{Z}$ (only in $ I$) is algebraic over $ \mathbb{Z}$ (that is, not a root of a polynomial with coefficients in $ \mathbb{Z}$);
c) $ I$ only has trivial endomorphisms.
[i]E. Fried[/i]
Determine all the integers $a$ and $b$, such that $\sqrt{2010 + 2 \sqrt{2009}}$ be a solution of the equation $x^2 + ax + b = 0$. Prove that for such $a$ and $b$ the number$\sqrt{2010 - 2 \sqrt{2009}}$ is not a solution to the given equation.
Let $ f, g$ and $ a$ be polynomials with real coefficients, $ f$ and $ g$ in one variable and $ a$ in two variables. Suppose
\[ f(x) \minus{} f(y) \equal{} a(x, y)(g(x) \minus{} g(y)) \forall x,y \in \mathbb{R}\]
Prove that there exists a polynomial $ h$ with $ f(x) \equal{} h(g(x)) \text{ } \forall x \in \mathbb{R}.$
Compute the value of the expression
\[ 2009^4 \minus{} 4 \times 2007^4 \plus{} 6 \times 2005^4 \minus{} 4 \times 2003^4 \plus{} 2001^4 \, .\]
Let $P(x) = x^3 + ax^2 + bx + 1$ be a polynomial with real coefficients and three real roots $\rho_1$, $\rho_2$, $\rho_3$ such that $|\rho_1| < |\rho_2| < |\rho_3|$. Let $A$ be the point where the graph of $P(x)$ intersects $yy'$ and the point $B(\rho_1, 0)$, $C(\rho_2, 0)$, $D(\rho_3, 0)$. If the circumcircle of $\vartriangle ABD$ intersects $yy'$ for a second time at $E$, find the minimum value of the length of the segment $EC$ and the polynomials for which this is attained.
[i]Brazitikos Silouanos, Greece[/i]
Let $\, P(z) \,$ be a polynomial with complex coefficients which is of degree $\, 1992 \,$ and has distinct zeros. Prove that there exist complex numbers $\, a_1, a_2, \ldots, a_{1992} \,$ such that $\, P(z) \,$ divides the polynomial \[ \left( \cdots \left( (z-a_1)^2 - a_2 \right)^2 \cdots - a_{1991} \right)^2 - a_{1992}. \]
Find all real numbers $ a,b,c,d$ such that \[ \left\{\begin{array}{cc}a \plus{} b \plus{} c \plus{} d \equal{} 20, \\
ab \plus{} ac \plus{} ad \plus{} bc \plus{} bd \plus{} cd \equal{} 150. \end{array} \right.\]
Let $ x_1,x_2,x_3$ be the roots of: $ x^3\minus{}6x^2\plus{}ax\plus{}a\equal{}0$. Find all real numbers $ a$ for which $ (x_1\minus{}1)^3\plus{}(x_2\minus{}1)^3\plus{}(x_3\minus{}1)^3\equal{}0$. Also, for each such $ a$, determine the corresponding values of $ x_1,x_2,$ and $ x_3$.
Let $m_{1},m_{2},...,m_{n}$ be mutually distinct integers. Prove that there exists a $f(x)\in\mathbb{Z}[x]$ of degree $n$ satisfying the following two conditions:
a)$f(m_{i})=-1\forall i=1,2,...,n$.
b)$f(x)$ is irreducible.
Let $n$ be a positive integer and let $k_0,k_1, \dots,k_{2n}$ be nonzero integers such that $k_0+k_1 +\dots+k_{2n}\neq 0$. Is it always possible to a permutation $(a_0,a_1,\dots,a_{2n})$ of $(k_0,k_1,\dots,k_{2n})$ so that the equation
\begin{align*}
a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\dots+a_0=0
\end{align*}
has not integer roots?
Let $ y_0$ be chosen randomly from $ \{0, 50\}$, let $ y_1$ be chosen randomly from $ \{40, 60, 80\}$, let $ y_2$ be chosen randomly from $ \{10, 40, 70, 80\}$, and let $ y_3$ be chosen randomly from $ \{10, 30, 40, 70, 90\}$. (In each choice, the possible outcomes are equally likely to occur.) Let $ P$ be the unique polynomial of degree less than or equal to $ 3$ such that $ P(0) \equal{} y_0$, $ P(1) \equal{} y_1$, $ P(2) \equal{} y_2$, and $ P(3) \equal{} y_3$. What is the expected value of $ P(4)$?
Let $ f(x)\equal{}a_{2n}x^{2n}\plus{}a_{2n\minus{}1}x^{2n\minus{}1}\plus{}\cdots\plus{}a_1x\plus{}a_0$, with $ a_i\equal{}a_{2n\minus{}1}$ for all $ i\equal{}1,2,\ldots,n$ and $ a_{2n}\ne0$. Prove that there exists a polynomial $ g(x)$ of degree $ n$ such that $ g\left(x\plus{}\frac1x\right)x^n\equal{}f(x)$.
The sequences $a_n$, $b_n$ and $c_n$ are defined recursively in the following way:
$a_0 = 1/6$, $b_0 = 1/2$, $c_0 = 1/3,$
$$a_{n+1}= \frac{(a_n + b_n)(a_n + c_n)}{(a_n - b_n)(a_n - c_n)},\,\,
b_{n+1}= \frac{(b_n + a_n)(b_n + c_n)}{(b_n - a_n)(b_n - c_n)},\,\,
c_{n+1}= \frac{(c_n + a_n)(c_n + b_n)}{(c_n - a_n)(c_n - b_n)}$$
For each natural number $N$, the following polynomials are defined:
$A_n(x) =a_o+a_1 x+ ...+ a_{2N}x^{2N}$
$B_n(x) =b_o+a_1 x+ ...+ a_{2N}x^{2N}$
$C_n(x) =a_o+a_1 x+ ...+ a_{2N}x^{2N}$
Assume the sequences are well defined.
Show that there is no real $c$ such that $A_N(c) = B_N(c) = C_N(c) = 0$.
Let $ n>m>1$ be odd integers, let $ f(x)\equal{}x^n\plus{}x^m\plus{}x\plus{}1$. Prove that $ f(x)$ can't be expressed as the product of two polynomials having integer coefficients and positive degrees.
Find all reals $\lambda$ for which there is a nonzero polynomial $P$ with real coefficients such that
$$\frac{P(1)+P(3)+P(5)+\ldots+P(2n-1)}n=\lambda P(n)$$for all positive integers $n$, and find all such polynomials for $\lambda=2$.
Find all polynomials $P(x)$ with real coefficients such that for all pairwise distinct positive integers $x, y, z, t$ with $x^2 + y^2 + z^2 = 2t^2$ and $\gcd(x, y, z, t ) = 1,$ the following equality holds
\[2P^2(t ) + 2P(xy + yz + zx) = P^2(x + y + z) .\]
[b]Note.[/b] $P^2(k)=\left( P(k) \right)^2.$
Find all two-variable polynomials $p(x,y)$ such that for each $a,b,c\in\mathbb R$:
\[p(ab,c^2+1)+p(bc,a^2+1)+p(ca,b^2+1)=0\]
Let $f_n$ be a polynomial with real coefficients for all $n \in \mathbb{Z}$. Suppose that
\[f_n(k) = f_{n+k}(k) \quad n, k \in \mathbb{Z}.\]
(a) Does $f_n = f_m$ necessarily hold for all $m,n \in \mathbb{Z}$?
(b) If furthermore $f_n$ is a polynomial with integer coefficients for all $n \in\mathbb{Z}$, does $f_n = f_m$ necessarily hold for all $m, n \in\mathbb{Z}$?
[i]Proposed by usjl[/i]