Found problems: 3597
Let $f(x)= x^2-C$ where $C$ is a rational constant.
Show that exists only finitely many rationals $x$ such that $\{x,f(x),f(f(x)),\ldots\}$ is finite
Prove that any polynomial of the form $1+a_nx^n + a_{n+1}x^{n+1} + \cdots + a_kx^k$ ($k\ge n$) has at least $n-2$ non-real roots (counting multiplicity), where the $a_i$ ($n\le i\le k$) are real and $a_k\ne 0$.
[i]David Yang.[/i]
Let $P(X)=X^n+a_1X^{n-1}+\ldots+a_n$ be a plynomial with real roots $x_1. x_2,\ldots,x_n$. Denote $E_k=x_1^k+x_2^k+\ldots+x_n^k, \forall k\in\mathbb{N}$. There exists an $m\in\mathbb{N}$ such that $E_m=E_{m+1}=E_{m+2}=1$. Find $\max\{P(-2),P(2)\}$.
If $f$ is a monic cubic polynomial with $f(0)=-64$, and all roots of $f$ are non-negative real numbers, what is the largest possible value of $f(-1)$? (A polynomial is monic if it has a leading coefficient of $1$.)
Given that the polynomial $P(x) = x^5 - x^2 + 1$ has $5$ roots $r_1, r_2, r_3, r_4, r_5$. Find the value of the product $Q(r_1)Q(r_2)Q(r_3)Q(r_4)Q(r_5)$, where $Q(x) = x^2 + 1$.
Let $A$ be an $n\times n$ matrix of real numbers for some $n\ge 1.$ For each positive integer $k,$ let $A^{[k]}$ be the matrix obtained by raising each entry to the $k$th power. Show that if $A^k=A^{[k]}$ for $k=1,2,\cdots,n+1,$ then $A^k=A^{[k]}$ for all $k\ge 1.$
A sequence of polynomials $P_m(x, y, z), m = 0, 1, 2, \cdots$, in $x, y$, and $z$ is defined by $P_0(x, y, z) = 1$ and by
\[P_m(x, y, z) = (x + z)(y + z)P_{m-1}(x, y, z + 1) - z^2P_{m-1}(x, y, z)\]
for $m > 0$. Prove that each $P_m(x, y, z)$ is symmetric, in other words, is unaltered by any permutation of $x, y, z.$
Let $f$ be a polynomial of degree $98$, such that $f (k) =\frac{1}{k}$ for $k=1,2,3,\ldots,99$. Determine $f(100)$.
If for the coefficients of equation $x^3+ax^2+bx+c=0$ whose roots are all real, holds, $a^2= 2(b+1)$ then $|a-c|\le 2$. Prove it.
Find the real number coefficient $c$ of polynomial $x^2+x+c$, if his roots $x_1$ and $x_2$ satisfy following: $$\frac{2x_1^3}{2+x_2}+\frac{2x_2^3}{2+x_1}=-1$$
Let $ABC$ be a triangle and $m$ a line which intersects the sides $AB$ and $AC$ at interior points $D$ and $F$, respectively, and intersects the line $BC$ at a point $E$ such that $C$ lies between $B$ and $E$. The parallel lines from the points $A$, $B$, $C$ to the line $m$ intersect the circumcircle of triangle $ABC$ at the points $A_1$, $B_1$ and $C_1$, respectively (apart from $A$, $B$, $C$). Prove that the lines $A_1E$ , $B_1F$ and $C_1D$ pass through the same point.
[i]Greece[/i]
The real numbers $a$, $b$, $c$, $d$ satisfy simultaneously the equations
\[abc -d = 1, \ \ \ bcd - a = 2, \ \ \ cda- b = 3, \ \ \ dab - c = -6.\] Prove that $a + b + c + d \not = 0$.
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]
Suppose $a$ real number so that there is a non-constant polynomial $P (x)$ such that
$\frac{P(x+1)-P(x)}{P(x+\pi)}= \frac{a}{x+\pi}$ for each real number $x$, with $x+\pi \ne 0$ and $P(x+\pi)\ne 0$.
Show that $a$ is a natural number.
Find all polynomials $P(x)$ with real coefficients such that $P(2014) = 1$ and, for some integer $c$:
$xP(x-c) = (x - 2014)P(x)$
A polynomial $p(x)$ of degree $n\ge 2$ has exactly $n$ real roots, counted with multiplicity. We know that the coefficient of $x^n$ is $1$, all the roots are less than or equal to $1$, and $p(2)=3^n$. What values can $p(1)$ take?
Let $p$ be a prime number, and define a sequence by: $x_i=i$ for $i=,0,1,2...,p-1$ and $x_n=x_{n-1}+x_{n-p}$ for $n \geq p$
Find the remainder when $x_{p^3}$ is divided by $p$.
Given non-constant linear functions $p_1(x), p_2(x), \dots p_n(x)$. Prove that at least $n-2$ of polynomials $p_1p_2\dots p_{n-1}+p_n, p_1p_2\dots p_{n-2} p_n + p_{n-1},\dots p_2p_3\dots p_n+p_1$ have a real root.
5. We call $(P_n)_{n\in \mathbb{N}}$ an arithmetic sequence with common difference $Q(x)$ if $\forall n: P_{n+1} = P_n + Q$
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We have an arithmetic sequence with a common difference $Q(x)$ and the first term $P(x)$ such that $P,Q$ are monic polynomials with integer coefficients and don't share an integer root. Each term of the sequence has at least one integer root. Prove that:
$\newline$
a) $P(x)$ is divisible by $Q(x)$
$\newline$
b) $\text{deg}(\frac{P(x)}{Q(x)}) = 1$
Let $p(x)$ and $q(x)$ be two cubic polynomials such that $p(0)=-24$, $q(0)=30$, and \[p(q(x))=q(p(x))\] for all real numbers $x$. Find the ordered pair $(p(3),q(6))$.
Prove for every irrational real number a, there are irrational numbers b and b' such that a+b and ab' are rational while a+b' and ab are irrational.
Suppose that $P(x)$ is a polynomial with real coefficients, such that for some positive real numbers $c$ and $d$, and for all natural numbers $n$, we have $c|n|^3\leq |P(n)|\leq d|n|^3$.
Prove that $P(x)$ has a real zero.
$p$ is a polynomial with integer coefficients and for every natural $n$ we have $p(n)>n$. $x_k $ is a sequence that: $x_1=1, x_{i+1}=p(x_i)$ for every $N$ one of $x_i$ is divisible by $N.$ Prove that $p(x)=x+1$
Given positive integers $ m,n > 1$. Prove that the equation
$ (x \plus{} 1)^n \plus{} (x \plus{} 2)^n \plus{} ... \plus{} (x \plus{} m)^n \equal{} (y \plus{} 1)^{2n} \plus{} (y \plus{} 2)^{2n} \plus{} ... \plus{} (y \plus{} m)^{2n}$ has finitely number of solutions $ x,y \in N$
Let $P(x)$ be a polynomial of degree $n$ whose roots are $i-1, i-2,\cdot\cdot\cdot, i-n$ (where $i^2=-1$), and let $R(x)$ and $S(x)$ be the polynomials with real coefficients such that $P(x)=R(x)+iS(x)$. Show that the polynomial $R$ has $n$ real roots. (R. Stanojevic)