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

Tags were heavily modified to better represent problems.

AND:
OR:
NO:

Found problems: 3597

1991 Arnold's Trivium, 53
Tags: polynomial , algebra
Investigate the singular points of the differential form $dt = dx/y$ on the compact Riemann surface $y^2/2 + U(x) = E$, where $U$ is a polynomial and $E$ is not a critical value.

2022 Korea Winter Program Practice Test, 2
Tags: inequality , inequalities , polynomial , algebra
Let $n\ge 2$ be a positive integer. There are $n$ real coefficient polynomials $P_1(x),P_2(x),\cdots ,P_n(x)$ which is not all the same, and their leading coefficients are positive. Prove that $$\deg(P_1^n+P_2^n+\cdots +P_n^n-nP_1P_2\cdots P_n)\ge (n-2)\max_{1\le i\le n}(\deg P_i)$$ and find when the equality holds.

IMSC 2024, 5
Tags: imsc , polynomial , algebra
Let $\mathbb{R}_{>0}$ be the set of all positive real numbers. Find all strictly monotone (increasing or decreasing) functions $f:\mathbb{R}_{>0} \to \mathbb{R}$ such that there exists a two-variable polynomial $P(x, y)$ with real coefficients satisfying $$ f(xy)=P(f(x), f(y)) $$ for all $x, y\in\mathbb{R}_{>0}$.\\ [i]Proposed by Navid Safaei, Iran[/i]

1974 AMC 12/AHSME, 4
Tags: polynomial , algebra
What is the remainder when $x^{51}+51$ is divided by $x+1$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 49 \qquad\textbf{(D)}\ 50 \qquad\textbf{(E)}\ 51 $

1987 Bulgaria National Olympiad, Problem 1
Tags: sequence , polynomial , algebra
Let $f(x)=x^n+a_1x^{n-1}+\ldots+a_n~(n\ge3)$ be a polynomial with real coefficients and $n$ real roots, such that $\frac{a_{n-1}}{a_n}>n+1$. Prove that if $a_{n-2}=0$, then at least one root of $f(x)$ lies in the open interval $\left(-\frac12,\frac1{n+1}\right)$.

2023 All-Russian Olympiad, 7
Tags: algebra , polynomial
We call a polynomial $P(x)$ good if the numbers $P(k)$ and $P'(k)$ are integers for all integers $k$. Let $P(x)$ be a good polynomial of degree $d$, and let $N_d$ be the product of all composite numbers not exceeding $d$. Prove that the leading coefficient of the polynomial $N_d \cdot P(x)$ is integer.

2009 USA Team Selection Test, 7
Tags: polynomial , system of equations , algebra , trigonometry
Find all triples $ (x,y,z)$ of real numbers that satisfy the system of equations \[ \begin{cases}x^3 \equal{} 3x\minus{}12y\plus{}50, \\ y^3 \equal{} 12y\plus{}3z\minus{}2, \\ z^3 \equal{} 27z \plus{} 27x. \end{cases}\] [i]Razvan Gelca.[/i]

2013 Online Math Open Problems, 42
Tags: modular arithmetic , quadratic , algebra , polynomial , number theory
Find the remainder when \[\prod_{i=0}^{100}(1-i^2+i^4)\] is divided by $101$. [i]Victor Wang[/i]

2014 IFYM, Sozopol, 5
Tags: polynomial , coprime , number theory
Let $f(x)$ be a polynomial with integer coefficients, for which there exist $a,b\in \mathbb{Z}$ ($a\neq b$), such that $f(a)$ and $f(b)$ are coprime. Prove that there exist infinitely many values for $x$, such that each $f(x)$ is coprime with any other.

2023 IMAR Test, P4
Tags: polynomial , algebra
Let $n{}$ be a non-negative integer and consider the standard power expansion of the following polynomial \[\sum_{k=0}^n\binom{n}{k}^2(X+1)^{2k}(X-1)^{2(n-k)}=\sum_{k=0}^{2n}a_kX^k.\]The coefficients $a_{2k+1}$ all vanish since the polynomial is invariant under the change $X\mapsto -X.$ Prove that the coefficients $a_{2k}$ are all positive.

2001 All-Russian Olympiad, 1
Tags: quadratic , polynomial , algebra
Two monic quadratic trinomials $f(x)$ and $g(x)$ take negative values on disjoint intervals. Prove that there exist positive numbers $\alpha$ and $\beta$ such that $\alpha f(x) + \beta g(x) > 0$ for all real $x$.

1970 Poland - Second Round, 5
Tags: polynomial , algebra
Given the polynomial $ P(x) = \frac{1}{2} - \frac{1}{3}x + \frac{1}{6}x^2 $. Let $ Q(x) = \sum_{k=0}^{m} b_k x^k $ be a polynomial given by $$ Q(x) = P(x) \cdot P(x^3) \cdot P(x^9) \cdot P(x^{27}) \cdot P(x^{81}). $$ Calculate $ \sum_{k=0}^m |b_k| $.

2017 Pan-African Shortlist, A6
Tags: inequalities , real root , algebra , polynomial
Let $n \geq 1$ be an integer, and $a_0, a_1, \dots, a_{n-1}$ be real numbers such that \[ 1 \geq a_{n-1} \geq a_{n-2} \geq \dots \geq a_1 \geq a_0 \geq 0. \] We assume that $\lambda$ is a real root of the polynomial \[ x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0. \] Prove that $|\lambda| \leq 1$.

1978 Polish MO Finals, 3
Tags: polynomial , algebra
Prove that if $m$ is a natural number and $P,Q,R$ polynomials of degrees less than $m$ satisfying $$x^{2m}P(x,y)+y^{2m}Q(x,y) = (x+y)^{2m}R(x,y),$$ then each of the polynomials is zero.

2013 India National Olympiad, 3
Tags: calculus , algebra , inequalities , polynomial , integration , function
Let $a,b,c,d \in \mathbb{N}$ such that $a \ge b \ge c \ge d $. Show that the equation $x^4 - ax^3 - bx^2 - cx -d = 0$ has no integer solution.

2008 239 Open Mathematical Olympiad, 6
Tags: function , algebra , polynomial
Given a polynomial $P(x,y)$ with real coefficients, suppose that some real function $f:\mathbb R \to \mathbb R$ satisfies $$P(x,y) = f(x+y)-f(x)-f(y)$$for all $x,y\in\mathbb R$. Show that some polynomial $q$ satisfies $$P(x,y) = q(x+y)-q(x)-q(y)$$

2001 Tuymaada Olympiad, 3
Tags: polynomial , function , quadratic , algebra
Do there exist quadratic trinomials $P, \ \ Q, \ \ R$ such that for every integers $x$ and $y$ an integer $z$ exists satisfying $P(x)+Q(y)=R(z)?$ [i]Proposed by A. Golovanov[/i]

2021 Nigerian MO Round 3, Problem 5
Tags: derivative , function , degree , calculus , algebra , polynomial
Let $f(x)=\frac{P(x)}{Q(x)}$, where $P(x), Q(x)$ are two non-constant polynomials with no common zeros and $P(0)=P(1)=0$. Suppose $f(x)f\left(\frac{1}{x}\right)=f(x)+f\left(\frac{1}{x}\right)$ for infinitely many values of $x$. a) Show that $\text{deg}(P)<\text{deg}(Q)$. b) Show that $P'(1)=2Q'(1)-\text{deg}(Q)\cdot Q(1)$. Here, $P'(x)$ denotes the derivative of $P(x)$ as usual.

2007 China Team Selection Test, 3
Tags: function , polynomial , limit , integration , quadratic , induction , algebra
Prove that for any positive integer $ n$, there exists only $ n$ degree polynomial $ f(x),$ satisfying $ f(0) \equal{} 1$ and $ (x \plus{} 1)[f(x)]^2 \minus{} 1$ is an odd function.

2012 Irish Math Olympiad, 3
Tags: algebra , combi , polynomial
Find, with proof, all polynomials $f$ such that $f$ has nonnegative integer coefficients, $f$($1$) = $8$ and $f$($2$) = $2012$.

2014 Contests, 1
Tags: quadratic , polynomial , algebra
Find all the polynomials with real coefficients which satisfy $ (x^2-6x+8)P(x)=(x^2+2x)P(x-2)$ for all $x\in \mathbb{R}$.

2000 Vietnam National Olympiad, 3
Tags: polynomial , number theory , algebra
Consider the polynomial $ P(x) \equal{} x^3 \plus{} 153x^2 \minus{} 111x \plus{} 38$. (a) Prove that there are at least nine integers $ a$ in the interval $ [1, 3^{2000}]$ for which $ P(a)$ is divisible by $ 3^{2000}$. (b) Find the number of integers $ a$ in $ [1, 3^{2000}]$ with the property from (a).

2010 Irish Math Olympiad, 2
Tags: polynomial , algebra
For each odd integer $p\ge 3$ find the number of real roots of the polynomial $$f_p(x)=(x-1)(x-2)\cdots (x-p+1)+1.$$

2002 Romania Team Selection Test, 2
Tags: algebra , polynomial , number theory
Let $P(x)$ and $Q(x)$ be integer polynomials of degree $p$ and $q$ respectively. Assume that $P(x)$ divides $Q(x)$ and all their coefficients are either $1$ or $2002$. Show that $p+1$ is a divisor of $q+1$. [i]Mihai Cipu[/i]

2007 Putnam, 1
Tags: algebra , polynomial , function
Let $ f$ be a polynomial with positive integer coefficients. Prove that if $ n$ is a positive integer, then $ f(n)$ divides $ f(f(n)\plus{}1)$ if and only if $ n\equal{}1.$