This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 3597

2017 Harvard-MIT Mathematics Tournament, 1
Tags: algebra , polynomial
Let $P(x)$, $Q(x)$ be nonconstant polynomials with real number coefficients. Prove that if \[\lfloor P(y) \rfloor = \lfloor Q(y) \rfloor\] for all real numbers $y$, then $P(x) = Q(x)$ for all real numbers $x$.

2005 Mediterranean Mathematics Olympiad, 4
Tags: algebra , polynomial , function , geometric transformation , modular arithmetic , number theory , geometry
Let $A$ be the set of all polynomials $f(x)$ of order $3$ with integer coefficients and cubic coefficient $1$, so that for every $f(x)$ there exists a prime number $p$ which does not divide $2004$ and a number $q$ which is coprime to $p$ and $2004$, so that $f(p)=2004$ and $f(q)=0$. Prove that there exists a infinite subset $B\subset A$, so that the function graphs of the members of $B$ are identical except of translations

2024 Ukraine National Mathematical Olympiad, Problem 7
Tags: polynomial , number theory
Find all polynomials $P(x)$ with integer coefficients, such that for any positive integer $n$ number $P(n)$ is a positive integer and a divisor of $n!$. [i]Proposed by Mykyta Kharin[/i]

2021 SAFEST Olympiad, 5
Tags: algebra , polynomial , complex number
Find all polynomials $P$ with real coefficients having no repeated roots, such that for any complex number $z$, the equation $zP(z) = 1$ holds if and only if $P(z-1)P(z + 1) = 0$. Remark: Remember that the roots of a polynomial are not necessarily real numbers.

2010 Bundeswettbewerb Mathematik, 4
Tags: polynomial , function , algebra , functional equation , functional
In the following, let $N_0$ denotes the set of non-negative integers. Find all polynomials $P(x)$ that fulfill the following two properties: (1) All coefficients of $P(x)$ are from $N_0$. (2) Exists a function $f : N_0 \to N_0$ such as $f (f (f (n))) = P (n)$ for all $n \in N_0$.

2015 Germany Team Selection Test, 1
Tags: root , algebra , polynomial , real number , integer
Find the least positive integer $n$, such that there is a polynomial \[ P(x) = a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\dots+a_1x+a_0 \] with real coefficients that satisfies both of the following properties: - For $i=0,1,\dots,2n$ it is $2014 \leq a_i \leq 2015$. - There is a real number $\xi$ with $P(\xi)=0$.

VI Soros Olympiad 1999 - 2000 (Russia), 10.5
Tags: algebra , polynomial
Prove that the polynomial $x^{1999}+x^{1998}+...+x^3+x^2+ax+b$ for any real values of the coefficients $a>b>0$ does not have an integer root.

1979 IMO Longlists, 10
Tags: algebra , polynomial , vieta , functional equation
Find all polynomials $f(x)$ with real coefficients for which \[f(x)f(2x^2) = f(2x^3 + x).\]

2012 IFYM, Sozopol, 4
Tags: algebra , polynomial , calculus , integration , gauss , number theory , prime number
Given distinct prime numbers $p$ and $q$ and a natural number $n \geq 3$, find all $a \in \mathbb{Z}$ such that the polynomial $f(x) = x^n + ax^{n-1} + pq$ can be factored into 2 integral polynomials of degree at least 1.

2016 Iran MO (3rd Round), 1
Tags: algebra , polynomial
Let $P(x) \in \mathbb {Z}[X]$ be a polynomial of degree $2016$ with no rational roots. Prove that there exists a polynomial $T(x) \in \mathbb {Z}[X]$ of degree $1395$ such that for all distinct (not necessarily real) roots of $P(x)$ like $(\alpha ,\beta):$ $$T(\alpha)-T(\beta) \not \in \mathbb {Q}$$ Note: $\mathbb {Q}$ is the set of rational numbers.

1985 Bulgaria National Olympiad, Problem 2
Tags: algebra , polynomial
Find all real parameters $a$ for which all the roots of the equation $$x^6+3x^5+(6-a)x^4+(7-2a)x^3+(6-a)x^2+3x+1$$are real.

2007 Germany Team Selection Test, 1
Tags: algebra , polynomial , number theory
Let $ k \in \mathbb{N}$. A polynomial is called [i]$ k$-valid[/i] if all its coefficients are integers between 0 and $ k$ inclusively. (Here we don't consider 0 to be a natural number.) [b]a.)[/b] For $ n \in \mathbb{N}$ let $ a_n$ be the number of 5-valid polynomials $ p$ which satisfy $ p(3) = n.$ Prove that each natural number occurs in the sequence $ (a_n)_n$ at least once but only finitely often. [b]b.)[/b] For $ n \in \mathbb{N}$ let $ a_n$ be the number of 4-valid polynomials $ p$ which satisfy $ p(3) = n.$ Prove that each natural number occurs infinitely often in the sequence $ (a_n)_n$ .

1992 IMO Longlists, 36
Tags: diophantine equation , algebra , system of equations , polynomial
Find all rational solutions of \[a^2 + c^2 + 17(b^2 + d^2) = 21,\]\[ab + cd = 2.\]

KoMaL A Problems 2023/2024, A. 883
Tags: algebra , polynomial , analysis , series
Let $J\subsetneq I\subseteq \mathbb R$ be non-empty open intervals, and let $f_1, f_2,\ldots$ be real polynomials satisfying the following conditions: [list] [*] $f_i(x)\ge 0$ for all $i\ge 1$ and $x\in I$, [*] $\sum\limits_{i=1}^\infty f_i(x)$ is finite for all $x\in I$, [*] $\sum\limits_{i=1}^\infty f_i(x)=1$ for all $x\in J$. [/list] Do these conditions imply that $\sum\limits_{i=1}^\infty f_i(x)=1$ also for all $x\in I$? [i]Proposed by András Imolay, Budapest[/i]

1964 AMC 12/AHSME, 25
Tags: algebra , polynomial , parameterization , conic
The set of values of $m$ for which $x^2+3xy+x+my-m$ has two factors, with integer coefficients, which are linear in $x$ and $y$, is precisely: $ \textbf{(A)}\ 0, 12, -12\qquad\textbf{(B)}\ 0, 12\qquad\textbf{(C)}\ 12, -12\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 0 $

2006 China Team Selection Test, 2
Tags: function , polynomial , algebra
The function $f(n)$ satisfies $f(0)=0$, $f(n)=n-f \left( f(n-1) \right)$, $n=1,2,3 \cdots$. Find all polynomials $g(x)$ with real coefficient such that \[ f(n)= [ g(n) ], \qquad n=0,1,2 \cdots \] Where $[ g(n) ]$ denote the greatest integer that does not exceed $g(n)$.

2016 Saudi Arabia GMO TST, 2
Tags: algebra , polynomial
Let $c$ be a given real number. Find all polynomials $P$ with real coefficients such that: $(x + 1)P(x - 1) - (x - 1)P(x) = c$ for all $x \in R$

2011 Vietnam National Olympiad, 3
Tags: polynomial , algebra
Let $n\in\mathbb N$ and define $P(x,y)=x^n+xy+y^n.$ Show that we cannot obtain two non-constant polynomials $G(x,y)$ and $H(x,y)$ with real coefficients such that $P(x,y)=G(x,y)\cdot H(x,y).$

2005 IMO Shortlist, 1
Tags: coefficient , algebra , polynomial , root , quadratic
Find all pairs of integers $a,b$ for which there exists a polynomial $P(x) \in \mathbb{Z}[X]$ such that product $(x^2+ax+b)\cdot P(x)$ is a polynomial of a form \[ x^n+c_{n-1}x^{n-1}+\cdots+c_1x+c_0 \] where each of $c_0,c_1,\ldots,c_{n-1}$ is equal to $1$ or $-1$.

2015 NIMO Problems, 5
Tags: algebra , polynomial , complex number
Let $a, b, c, d, e,$ and $f$ be real numbers. Define the polynomials \[ P(x) = 2x^4 - 26x^3 + ax^2 + bx + c \quad\text{ and }\quad Q(x) = 5x^4 - 80x^3 + dx^2 + ex + f. \] Let $S$ be the set of all complex numbers which are a root of [i]either[/i] $P$ or $Q$ (or both). Given that $S = \{1,2,3,4,5\}$, compute $P(6) \cdot Q(6).$ [i]Proposed by Michael Tang[/i]

2005 MOP Homework, 7
Tags: polynomial , algebra
Let $n$ be a natural number and $f_1$, $f_2$, ..., $f_n$ be polynomials with integers coeffcients. Show that there exists a polynomial $g(x)$ which can be factored (with at least two terms of degree at least $1$) over the integers such that $f_i(x)+g(x)$ cannot be factored (with at least two terms of degree at least $1$ over the integers for every $i$.

2004 Iran Team Selection Test, 6
Tags: algebra , polynomial , limit , number theory
$p$ 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 $p(x)=x+1$

2000 IMO Shortlist, 4
Tags: algebra , polynomial , number theory , divisibility
Find all triplets of positive integers $ (a,m,n)$ such that $ a^m \plus{} 1 \mid (a \plus{} 1)^n$.

2003 All-Russian Olympiad, 3
Tags: polynomial , algebra
Let $f(x)$ and $g(x)$ be polynomials with non-negative integer coefficients, and let m be the largest coefficient of $f.$ Suppose that there exist natural numbers $a < b$ such that $f(a) = g(a)$ and $f(b) = g(b)$. Show that if $b > m,$ then $f = g.$

2003 India Regional Mathematical Olympiad, 6
Tags: quadratic , algebra , polynomial
Find all real numbers $a$ for which the equation $x^2a- 2x + 1 = 3 |x|$ has exactly three distinct real solutions in $x$.