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

2023 Quang Nam Province Math Contest (Grade 11), Problem 3
Tags: algebra , polynomial
Given a polynomial $P(x)$ with real coefficents satisfying:$$P(x).P(x+1)=P(x^2+x+1),\forall x\in \mathbb{R}.$$ Prove that: $\deg(P)$ is an even number and find $P(x).$

2007 Thailand Mathematical Olympiad, 8
Tags: algebra , sum , polynomial
Let $x_1, x_2,... , x_{84}$ be the roots of the equation $x^{84} + 7x - 6 = 0$. Compute $\sum_{k=1}^{84} \frac{x_k}{x_k-1}$.

2014 239 Open Mathematical Olympiad, 2
Tags: polynomial , algebra
The fourth-degree polynomial $P(x)$ is such that the equation $P(x)=x$ has $4$ roots, and any equation of the form $P(x)=c$ has no more two roots. Prove that the equation $P(x)=-x$ too has no more than two roots.

1940 Putnam, A1
Tags: algebra , polynomial , calculus , integration
Prove that if $f(x)$ is a polynomial with integer coefficients and there exists an integer $k$ such that none of $f(1),\ldots,f(k)$ is divisible by $k$, then $f(x)$ has no integral root.

2004 Rioplatense Mathematical Olympiad, Level 3, 1
Tags: polynomial , algebra
Find all polynomials $P(x)$ with real coefficients such that \[xP\bigg(\frac{y}{x}\bigg)+yP\bigg(\frac{x}{y}\bigg)=x+y\] for all nonzero real numbers $x$ and $y$.

2024 Thailand October Camp, 6
Tags: algebra , polynomial
A polynomial $A(x)$ is said to be [i]simple[/i] if $A(x)$ is divisible by $x$ but not divisible by $x^2$. Suppose that a polynomial $P(x)$ has a simple polynomial $Q(x)$ such that $P(Q(x))-Q(2x)$ is divisible by $x^2$. Prove that there exists a simple polynomial $R(x)$ such that $P(R(x))-R(2x)$ is divisible by $x^{2023}$.

2003 AMC 12-AHSME, 21
Tags: algebra , polynomial , vieta
The graph of the polynomial \[P(x) \equal{} x^5 \plus{} ax^4 \plus{} bx^3 \plus{} cx^2 \plus{} dx \plus{} e\] has five distinct $ x$-intercepts, one of which is at $ (0,0)$. Which of the following coefficients cannot be zero? $ \textbf{(A)}\ a \qquad \textbf{(B)}\ b \qquad \textbf{(C)}\ c \qquad \textbf{(D)}\ d \qquad \textbf{(E)}\ e$

2007 AIME Problems, 8
Tags: polynomial , quadratic , number theory , greatest common divisor , algorithm , least common multiple , algebra
The polynomial $P(x)$ is cubic. What is the largest value of $k$ for which the polynomials $Q_{1}(x) = x^{2}+(k-29)x-k$ and $Q_{2}(x) = 2x^{2}+(2k-43)x+k$ are both factors of $P(x)$?

2011 QEDMO 9th, 2
Tags: polynomial , algebra , cubic
Let $a,b,c$ be the three different solutions of $x^3-x-1 = 0$. Compute $a^4+b^5+c^6-c$.

2005 Postal Coaching, 19
Tags: function , algebra , polynomial
Find all functions $f : \mathbb{R} \mapsto \mathbb{R}$ such that $f(xy+f(x)) = xf(y) +f(x)$ for all $x,y \in \mathbb{R}$.

1997 Brazil National Olympiad, 5
Tags: polynomial , algebra
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

2008 Costa Rica - Final Round, 3
Tags: polynomial , algebra
Find all polinomials $ P(x)$ with real coefficients, such that $ P(\sqrt {3}(a \minus{} b)) \plus{} P(\sqrt {3}(b \minus{} c)) \plus{} P(\sqrt {3}(c \minus{} a)) \equal{} P(2a \minus{} b \minus{} c) \plus{} P( \minus{} a \plus{} 2b \minus{} c) \plus{} P( \minus{} a \minus{} b \plus{} 2c)$ for any $ a$,$ b$ and $ c$ real numbers

2017 Saudi Arabia BMO TST, 2
Tags: algebra , integer polynomial , polynomial
Polynomial P(x) with integer coefficient is called [i]cube-presented[/i] if it can be represented as sum of several cube of polynomials with integer coefficients. Examples: $3x + 3x^2$ is cube-represented because $3x + 3x^2 = (x + 1)^3 +(-x)^3 + (-1)^3$. a) Is $3x^2$ a cube-represented polynomial? b). How many quadratic polynomial P(x) with integer coefficients belong to the set $\{1,2, 3, ...,2017\}$ which is cube-represented?

2017 Kürschák Competition, 2
Tags: algebra , polynomial
Do there exist polynomials $p(x)$ and $q(x)$ with real coefficients such that $p^3(x)-q^2(x)$ is linear but not constant?

2019 Tournament Of Towns, 1
Tags: algebra , polynomial , two variables
The polynomial P(x,y) is such that for every integer n >= 0 each of the polynomials P(x,n) and P(n,y) either is a constant zero or has a degree not greater than n. Is it possible that P(x,x) has an odd degree?

1975 Poland - Second Round, 6
Tags: algebra , polynomial
Let $ f(x) $ and $ g(x) $ be polynomials with integer coefficients. Prove that if for every integer value $ n $ the number $ g(n) $ is divisible by the number $ f(n) $, then $ g(x) = f(x)\cdot h(x) $, where $ h(x) $ is a polynomial,. Show with an example that the coefficients of the polynomial $ h(x) $ do not have to be integer.

2014 India National Olympiad, 4
Tags: quadratic , polynomial , calculus , integration , probability , analytic geometry , algebra
Written on a blackboard is the polynomial $x^2+x+2014$. Calvin and Hobbes take turns alternately (starting with Calvin) in the following game. At his turn, Calvin should either increase or decrease the coefficient of $x$ by $1$. And at this turn, Hobbes should either increase or decrease the constant coefficient by $1$. Calvin wins if at any point of time the polynomial on the blackboard at that instant has integer roots. Prove that Calvin has a winning stratergy.

2022 VJIMC, 1
Tags: algebra , polynomial
Assume that a real polynomial $P(x)$ has no real roots. Prove that the polynomial $$Q(x)=P(x)+\frac{P''(x)}{2!}+\frac{P^{(4)}(x)}{4!}+\ldots$$ also has no real roots.

1968 IMO, 3
Tags: algebra , system of equations , polynomial , discriminant
Let $a,b,c$ be real numbers with $a$ non-zero. It is known that the real numbers $x_1,x_2,\ldots,x_n$ satisfy the $n$ equations: \[ ax_1^2+bx_1+c = x_{2} \]\[ ax_2^2+bx_2 +c = x_3\]\[ \ldots \quad \ldots \quad \ldots \quad \ldots\]\[ ax_n^2+bx_n+c = x_1 \] Prove that the system has [b]zero[/b], [u]one[/u] or [i]more than one[/i] real solutions if $(b-1)^2-4ac$ is [b]negative[/b], equal to [u]zero[/u] or [i]positive[/i] respectively.

2011 Iran MO (3rd Round), 2
Tags: induction , algebra , polynomial , modular arithmetic , calculus , integration , function
Let $n$ and $k$ be two natural numbers such that $k$ is even and for each prime $p$ if $p|n$ then $p-1|k$. let $\{a_1,....,a_{\phi(n)}\}$ be all the numbers coprime to $n$. What's the remainder of the number $a_1^k+.....+a_{\phi(n)}^k$ when it's divided by $n$? [i]proposed by Yahya Motevassel[/i]

2008 Federal Competition For Advanced Students, Part 2, 2
Tags: algebra , polynomial , vieta , number theory
(a) Does there exist a polynomial $ P(x)$ with coefficients in integers, such that $ P(d) \equal{} \frac{2008}{d}$ holds for all positive divisors of $ 2008$? (b) For which positive integers $ n$ does a polynomial $ P(x)$ with coefficients in integers exists, such that $ P(d) \equal{} \frac{n}{d}$ holds for all positive divisors of $ n$?

2006 Romania Team Selection Test, 2
Tags: polynomial , algebra
Let $p$ a prime number, $p\geq 5$. Find the number of polynomials of the form \[ x^p + px^k + p x^l + 1, \quad k > l, \quad k, l \in \left\{1,2,\dots,p-1\right\}, \] which are irreducible in $\mathbb{Z}[X]$. [i]Valentin Vornicu[/i]

2018 Hanoi Open Mathematics Competitions, 2
Tags: polynomial , functional equation
Let $f(x)$ be a polynomial such that $2f(x) + f(2 - x) = 5 + x$ for any real number x. Find the value of $f(0) + f(2)$. A. $4$ B. $0$ C.$ 2$ D. $3$ E. $1$

2015 Belarus Team Selection Test, 4
Tags: polynomial , algebra
Find all pairs of polynomials $p(x),q(x)\in R[x]$ satisfying the equality $p(x^2)=p(x)q(1-x)+p(1-x)q(x)$ for all real $x$. I.Voronovich

Today's calculation of integrals, 852
Tags: calculus , integration , trigonometry , polynomial , absolute value , calculus computations , algebra
Let $f(x)$ be a polynomial. Prove that if $\int_0^1 f(x)g_n(x)\ dx=0\ (n=0,\ 1,\ 2,\ \cdots)$, then all coefficients of $f(x)$ are 0 for each case as follows. (1) $g_n(x)=(1+x)^n$ (2) $g_n(x)=\sin n\pi x$ (3) $g_n(x)=e^{nx}$