Found problems: 3597
1983 AIME Problems, 5
Suppose that the sum of the squares of two complex numbers $x$ and $y$ is 7 and the sum of the cubes is 10. What is the largest real value that $x + y$ can have?
2016 Saudi Arabia BMO TST, 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 $Q(x) = x^2 + 1$.
2017 CIIM, Problem 1
Determine all the complex numbers $w = a + bi$ with $a, b \in \mathbb{R}$, such that there exists a polinomial $p(z)$ whose coefficients are real and positive such that $p(w) = 0.$
1995 Poland - Second Round, 1
For a polynomial $P$ with integer coefficients, $P(5)$ is divisible by $2$ and $P(2)$ is divisible by $5$. Prove that $P(7)$ is divisible by $10$.
2006 Romania Team Selection Test, 3
Let $n>1$ be an integer. A set $S \subset \{ 0,1,2, \ldots, 4n-1\}$ is called [i]rare[/i] if, for any $k\in\{0,1,\ldots,n-1\}$, the following two conditions take place at the same time
(1) the set $S\cap \{4k-2,4k-1,4k, 4k+1, 4k+2 \}$ has at most two elements;
(2) the set $S\cap \{4k+1,4k+2,4k+3\}$ has at most one element.
Prove that the set $\{0,1,2,\ldots,4n-1\}$ has exactly $8 \cdot 7^{n-1}$ rare subsets.
2019 Serbia National MO, 2
For the sequence of real numbers $a_1,a_2,\dots ,a_k$ we say it is [i]invested[/i] on the interval $[b,c]$ if there exists numbers $x_0,x_1,\dots ,x_k$ in the interval $[b,c]$ such that $|x_i-x_{i-1}|=a_i$ for $i=1,2,3,\dots k$ .
A sequence is [i]normed[/i] if all its members are not greater than $1$ . For a given natural $n$ , prove :
a)Every [i]normed[/i] sequence of length $2n+1$ is [i]invested[/i] in the interval $\left[ 0, 2-\frac{1}{2^n} \right ]$.
b) there exists [i]normed[/i] sequence of length $4n+3$ wich is not [i]invested[/i] on $\left[ 0, 2-\frac{1}{2^n} \right ]$.
2020 Tuymaada Olympiad, 8
The degrees of polynomials $P$ and $Q$ with real coefficients do not exceed $n$. These polynomials satisfy the identity
\[ P(x) x^{n + 1} + Q(x) (x+1)^{n + 1} = 1. \]
Determine all possible values of $Q \left( - \frac{1}{2} \right)$.
2003 APMO, 1
Let $a,b,c,d,e,f$ be real numbers such that the polynomial
\[ p(x)=x^8-4x^7+7x^6+ax^5+bx^4+cx^3+dx^2+ex+f \]
factorises into eight linear factors $x-x_i$, with $x_i>0$ for $i=1,2,\ldots,8$. Determine all possible values of $f$.
2005 Gheorghe Vranceanu, 2
Let be a twice-differentiable function $ f:(0,\infty )\longrightarrow\mathbb{R} $ that admits a polynomial function of degree $ 1 $ or $ 2, $ namely, $ \alpha :(0,\infty )\longrightarrow\mathbb{R} $ as its asymptote. Prove the following propositions:
[b]a)[/b] $ f''>0\implies f-\alpha >0 $
[b]b)[/b] $ \text{supp} f''=(0,\infty )\wedge f-\alpha >0\implies f''=0 $
1991 Irish Math Olympiad, 2
Problem:
Find all polynomials satisfying the equation
$ f(x^2) = (f(x))^2 $
for all real numbers x.
I'm not exactly sure where to start though it doesn't look too difficult. Thanks!
2007 Princeton University Math Competition, 8
$f(x) = x^3+3x^2 - 1$. Find the number of real solutions of $f(f(x)) = 0$.
2002 Estonia National Olympiad, 1
Find all real parameters $a$ for which the equation $x^8 +ax^4 +1 = 0$ has four real roots forming an arithmetic progression.
1999 Israel Grosman Mathematical Olympiad, 4
Consider a polynomial $f(x) = x^4 +ax^3 +bx^2 +cx+d$ with integer coefficients.
Prove that if $f(x)$ has exactly one real root, then it can be factored into nonconstant polynomials with rational coefficients
2014 Online Math Open Problems, 27
Let $p = 2^{16}+1$ be a prime, and let $S$ be the set of positive integers not divisible by $p$.
Let $f: S \to \{0, 1, 2, ..., p-1\}$ be a function satisfying
\[ f(x)f(y) \equiv f(xy)+f(xy^{p-2}) \pmod{p} \quad\text{and}\quad f(x+p) = f(x) \]
for all $x,y \in S$.
Let $N$ be the product of all possible nonzero values of $f(81)$.
Find the remainder when when $N$ is divided by $p$.
[i]Proposed by Yang Liu and Ryan Alweiss[/i]
1981 AMC 12/AHSME, 8
For all positive numbers $x,y,z$ the product $(x+y+z)^{-1}(x^{-1}+y^{-1}+z^{-1})(xy+yz+xz)^{-1}[(xy)^{-1}+(yz)^{-1}+(xz)^{-1}]$ equals
$\text{(A)}\ x^{-2}y^{-2}z^{-2} \qquad \text{(B)}\ x^{-2}+y^{-2}+z^{-2} \qquad \text{(C)}\ (x+y+z)^{-1}$
$\text{(D)}\ \frac{1}{xyz} \qquad \text{(E)}\ \frac{1}{xy+yz+xz}$
2023 Puerto Rico Team Selection Test, 3
Let $p(x)$ be a polynomial of degree $2022$ such that:
$$p(k) =\frac{1}{k+1}\,\,\, \text{for }\,\,\, k = 0, 1, . . . , 2022$$
Find $p(2023)$.
2005 Poland - Second Round, 1
The polynomial $W(x)=x^2+ax+b$ with integer coefficients has the following property: for every prime number $p$ there is an integer $k$ such that both $W(k)$ and $W(k+1)$ are divisible by $p$. Show that there is an integer $m$ such that $W(m)=W(m+1)=0$.
2001 APMO, 4
A point in the plane with a cartesian coordinate system is called a [i]mixed point[/i] if one of its coordinates is rational and the other one is irrational. Find all polynomials with real coefficients such that their graphs do not contain any mixed point.
1986 Polish MO Finals, 4
Find all $n$ such that there is a real polynomial $f(x)$ of degree $n$ such that $f(x) \ge f'(x)$ for all real $x$.
1978 IMO Longlists, 3
Find all numbers $\alpha$ for which the equation
\[x^2 - 2x[x] + x -\alpha = 0\]
has two nonnegative roots. ($[x]$ denotes the largest integer less than or equal to x.)
1999 National Olympiad First Round, 12
\[ \begin{array}{c} {x^{2} \plus{} y^{2} \plus{} z^{2} \equal{} 21} \\
{x \plus{} y \plus{} z \plus{} xyz \equal{} \minus{} 3} \\
{x^{2} yz \plus{} y^{2} xz \plus{} z^{2} xy \equal{} \minus{} 40} \end{array}
\]
The number of real triples $ \left(x,y,z\right)$ satisfying above system is
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ \text{None}$
2016 Estonia Team Selection Test, 6
A circle is divided into arcs of equal size by $n$ points ($n \ge 1$). For any positive integer $x$, let $P_n(x)$ denote the number of possibilities for colouring all those points, using colours from $x$ given colours, so that any rotation of the colouring by $ i \cdot \frac{360^o}{n}$ , where i is a positive integer less than $n$, gives a colouring that differs from the original in at least one point. Prove that the function $P_n(x)$ is a polynomial with respect to $x$.
2017 Romania National Olympiad, 2
Let be a natural number $ n $ and $ 2n $ real numbers $ b_1,b_2,\ldots ,b_n,a_1<a_2<\cdots <a_n. $ Show that
[b]a)[/b] if $ b_1,b_2,\ldots ,b_n>0, $ then there exists a polynomial $ f\in\mathbb{R}[X] $ irreducible in $ \mathbb{R}[X] $ such that $$ f\left( a_i \right) =b_i,\quad\forall i\in\{ 1,2,\ldots ,n \} . $$
[b]b)[/b] there exists a polynom $ g\in\mathbb{R} [X] $ of degree at least $ 1 $ which has only real roots and such that
$$ g\left( a_i \right) =b_i,\quad\forall i\in\{ 1,2,\ldots ,n \} . $$
2006 MOP Homework, 3
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.
2009 CHKMO, 1
Let $ f(x) \equal{} c_m x^m \plus{} c_{m\minus{}1} x^{m\minus{}1} \plus{}...\plus{} c_1 x \plus{} c_0$, where each $ c_i$ is a non-zero integer. Define a sequence $ \{ a_n \}$ by $ a_1 \equal{} 0$ and $ a_{n\plus{}1} \equal{} f(a_n)$ for all positive integers $ n$.
(a) Let $ i$ and $ j$ be positive integers with $ i<j$. Show that $ a_{j\plus{}1} \minus{} a_j$ is a multiple of $ a_{i\plus{}1} \minus{} a_i$.
(b) Show that $ a_{2008} \neq 0$