Found problems: 3597
1973 IMO Shortlist, 16
Given $a, \theta \in \mathbb R, m \in \mathbb N$, and $P(x) = x^{2m}- 2|a|^mx^m \cos \theta +a^{2m}$, factorize $P(x)$ as a product of $m$ real quadratic polynomials.
2016 Romanian Master of Mathematics Shortlist, A2
Let $p > 3$ be a prime number, and let $F_p$ denote the (fi nite) set of residue classes modulo $p$.
Let $S_d$ denote the set of $2$-variable polynomials $P(x, y)$ with coefficients in $F_p$, total degree $\le d$, and satisfying $P(x, y) = P(y,- x -y)$. Show that $$|S_d| = p^{\lceil (d+1)(d+2)/6 \rceil}$$.
[i]The total degree of a $2$-variable polynomial $P(x, y)$ is the largest value of $i + j$ among monomials $x^iy^j$
[/i] appearing in $P$.
PEN K Problems, 6
Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f^{(19)}(n)+97f(n)=98n+232.\]
2018 China Team Selection Test, 5
Suppose the real number $\lambda \in \left( 0,1\right),$ and let $n$ be a positive integer. Prove that the modulus of all the roots of the polynomial $$f\left ( x \right )=\sum_{k=0}^{n}\binom{n}{k}\lambda^{k\left ( n-k \right )}x^{k}$$ are $1.$
2002 India National Olympiad, 3
If $x$, $y$ are positive reals such that $x + y = 2$ show that $x^3y^3(x^3+ y^3) \leq 2$.
2010 All-Russian Olympiad, 3
Polynomial $P(x)$ with degree $n \geq 3$ has $n$ real roots $x_1 < x_2 < x_3 <...< x_n$, such that $x_2-x_1<x_3-x_2<....<x_n-x_{n-1}$. Prove that the maximum of the function $y=|P(x)|$ where $x$ is on the interval $[ x_1, x_n ]$, is in the interval $[x_n-1, x_n]$.
1980 AMC 12/AHSME, 2
The degree of $(x^2+1)^4 (x^3+1)^3$ as a polynomial in $x$ is
$\text{(A)} \ 5 \qquad \text{(B)} \ 7 \qquad \text{(C)} \ 12 \qquad \text{(D)} \ 17 \qquad \text{(E)} \ 72$
PEN Q Problems, 4
A prime $p$ has decimal digits $p_{n}p_{n-1} \cdots p_0$ with $p_{n}>1$. Show that the polynomial $p_{n}x^{n} + p_{n-1}x^{n-1}+\cdots+ p_{1}x + p_0$ cannot be represented as a product of two nonconstant polynomials with integer coefficients
2009 Polish MO Finals, 3
Let $P,Q,R$ be polynomials of degree at least $1$ with integer coefficients such that for any real number $x$ holds: $P(Q(x))\equal{}Q(R(x))\equal{}R(P(x))$. Show that the polynomials $P,Q,R$ are equal.
2018 Baltic Way, 5
A polynomial $f(x)$ with real coefficients is called [i]generating[/i], if for each polynomial $\varphi(x)$ with real coefficients there exists a positive integer $k$ and polynomials $g_1(x),\dotsc,g_k(x)$ with real coefficients such that
\[\varphi(x)=f(g_1(x))+\dotsc+f(g_k(x)).\]
Find all generating polynomials.
2013 Romania National Olympiad, 2
Given a ring $\left( A,+,\cdot \right)$ that meets both of the following conditions:
(1) $A$ is not a field, and
(2) For every non-invertible element $x$ of $ A$, there is an integer $m>1$ (depending on $x$) such that $x=x^2+x^3+\ldots+x^{2^m}$.
Show that
(a) $x+x=0$ for every $x \in A$, and
(b) $x^2=x$ for every non-invertible $x\in A$.
1965 AMC 12/AHSME, 22
If $ a_2 \neq 0$ and $ r$ and $ s$ are the roots of $ a_0 \plus{} a_1x \plus{} a_2x^2 \equal{} 0$, then the equality $ a_0 \plus{} a_1x \plus{} a_2x^2 \equal{} a_0\left (1 \minus{} \frac {x}{r} \right ) \left (1 \minus{} \frac {x}{s} \right )$ holds:
$ \textbf{(A)}\ \text{for all values of }x, a_0\neq 0$
$ \textbf{(B)}\ \text{for all values of }x$
$ \textbf{(C)}\ \text{only when }x \equal{} 0$
$ \textbf{(D)}\ \text{only when }x \equal{} r \text{ or }x \equal{} s$
$ \textbf{(E)}\ \text{only when }x \equal{} r \text{ or }x \equal{} s, a_0 \neq 0$
1993 Hungary-Israel Binational, 2
Determine all polynomials $f (x)$ with real coeffcients that satisfy
\[f (x^{2}-2x) = f^{2}(x-2)\]
for all $x.$
2008 Harvard-MIT Mathematics Tournament, 5
Let $ f(x) \equal{} x^3 \plus{} x \plus{} 1$. Suppose $ g$ is a cubic polynomial such that $ g(0) \equal{} \minus{} 1$, and the roots of $ g$ are the squares of the roots of $ f$. Find $ g(9)$.
2017 Purple Comet Problems, 15
For real numbers $a, b$, and $c$ the polynomial $p(x) = 3x^7 - 291x^6 + ax^5 + bx^4 + cx^2 + 134x - 2$ has $7$ real roots whose sum is $97$. Find the sum of the reciprocals of those $7$ roots.
2017 AMC 12/AHSME, 23
The graph of $y=f(x)$, where $f(x)$ is a polynomial of degree $3$, contains points $A(2,4)$, $B(3,9)$, and $C(4,16)$. Lines $AB$, $AC$, and $BC$ intersect the graph again at points $D$, $E$, and $F$, respectively, and the sum of the $x$-coordinates of $D$, $E$, and $F$ is $24$. What is $f(0)$?
$\textbf{(A) } -2 \qquad \textbf{(B) } 0 \qquad \textbf{(C) } 2 \qquad \textbf{(D) } \frac{24}{5} \qquad \textbf{(E) } 8$
1994 Poland - Second Round, 1
Find all real polynomials $P(x)$ of degree $5$ such that $(x-1)^3| P(x)+1$ and $(x+1)^3| P(x)-1$.
2000 National Olympiad First Round, 16
What is the sum of real roots of $(2+(2+(2+x)^2)^2)^2=2000$ ?
$ \textbf{(A)}\ -4
\qquad\textbf{(B)}\ -2
\qquad\textbf{(C)}\ 0
\qquad\textbf{(D)}\ 2
\qquad\textbf{(E)}\ 4
$
2002 China Western Mathematical Olympiad, 3
In the complex plane, consider squares having the following property: the complex numbers its vertex correspond to are exactly the roots of integer coefficients equation $ x^4 \plus{} px^3 \plus{} qx^2 \plus{} rx \plus{} s \equal{} 0$. Find the minimum of square areas.
2012 Iran Team Selection Test, 2
Do there exist $2000$ real numbers (not necessarily distinct) such that all of them are not zero and if we put any group containing $1000$ of them as the roots of a monic polynomial of degree $1000$, the coefficients of the resulting polynomial (except the coefficient of $x^{1000}$) be a permutation of the $1000$ remaining numbers?
[i]Proposed by Morteza Saghafian[/i]
2011 NIMO Problems, 7
Let $P(x) = x^2 - 20x - 11$. If $a$ and $b$ are natural numbers such that $a$ is composite, $\gcd(a, b) = 1$, and $P(a) = P(b)$, compute $ab$.
Note: $\gcd(m, n)$ denotes the greatest common divisor of $m$ and $n$.
[i]Proposed by Aaron Lin
[/i]
1994 All-Russian Olympiad, 1
Let be given three quadratic polynomials:
$P_1(x) = x^2 + p_1x+q_1, P_2(x) = x^2+ p_2x+q_2, P_3(x) = x^2 + p_3x+q_3$.
Prove that the equation $|P_1(x)|+|P_2(x)| = |P_3(x)|$ has at most eight real roots.
2010 ELMO Shortlist, 4
Let $r$ and $s$ be positive integers. Define $a_0 = 0$, $a_1 = 1$, and $a_n = ra_{n-1} + sa_{n-2}$ for $n \geq 2$. Let $f_n = a_1a_2\cdots a_n$. Prove that $\displaystyle\frac{f_n}{f_kf_{n-k}}$ is an integer for all integers $n$ and $k$ such that $0 < k < n$.
[i]Evan O' Dorney.[/i]
PEN S Problems, 5
Suppose that both $x^{3}-x$ and $x^{4}-x$ are integers for some real number $x$. Show that $x$ is an integer.
1970 Swedish Mathematical Competition, 4
Let $p(x) = (x- x_1)(x- x_2)(x- x_3)$, where $x_1, x_2$ and $x_3$ are real. Show that $p(x) p''(x) \le p'(x)^2$ for all $x$.