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.

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Found problems: 3597

1998 Baltic Way, 6
Tags: polynomial , algebra
Let $P$ be a polynomial of degree $6$ and let $a,b$ be real numbers such that $0<a<b$. Suppose that $P(a)=P(-a),P(b)=P(-b),P'(0)=0$. Prove that $P(x)=P(-x)$ for all real $x$.

2016 Saudi Arabia BMO TST, 1
Tags: polynomial , complex , algebra
Given two non-constant polynomials $P(x),Q(x)$ with real coefficients. For a real number $a$, we define $$P_a= \{z \in C : P(z) = a\}, Q_a =\{z \in C : Q(z) = a\}$$ Denote by $K$ the set of real numbers $a$ such that $P_a = Q_a$. Suppose that the set $K$ contains at least two elements, prove that $P(x) = Q(x)$.

2023 Romanian Master of Mathematics Shortlist, A1
Tags: algebra , rational number , polynomial
Determine all polynomials $P$ with real coefficients satisfying the following condition: whenever $x$ and $y$ are real numbers such that $P(x)$ and $P(y)$ are both rational, so is $P(x + y)$.

1986 AIME Problems, 11
Tags: algebra , polynomial , ratio , binomial theorem
The polynomial $1-x+x^2-x^3+\cdots+x^{16}-x^{17}$ may be written in the form $a_0+a_1y+a_2y^2+\cdots +a_{16}y^{16}+a_{17}y^{17}$, where $y=x+1$ and thet $a_i$'s are constants. Find the value of $a_2$.

1974 Polish MO Finals, 3
Tags: algebra , polynomial , quadratic polynomial
Let $r$ be a natural number. Prove that the quadratic trinomial $x^2 - rx- 1$ does not divide any nonzero polynomial whose coefficients are integers with absolute values less than $r$.

2004 AIME Problems, 13
Tags: algebra , polynomial , number theory , relatively prime , geometric series
The polynomial \[P(x)=(1+x+x^2+\cdots+x^{17})^2-x^{17}\] has 34 complex roots of the form $z_k=r_k[\cos(2\pi a_k)+i\sin(2\pi a_k)], k=1, 2, 3,\ldots, 34$, with $0<a_1\le a_2\le a_3\le\cdots\le a_{34}<1$ and $r_k>0$. Given that $a_1+a_2+a_3+a_4+a_5=m/n$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.

2004 Putnam, A4
Tags: algebra , polynomial
Show that for any positive integer $n$ there is an integer $N$ such that the product $x_1x_2\cdots x_n$ can be expressed identically in the form \[x_1x_2\cdots x_n=\sum_{i=1}^Nc_i(a_{i1}x_1+a_{i2}x_2+\cdots +a_{in}x_n)^n\] where the $c_i$ are rational numbers and each $a_{ij}$ is one of the numbers, $-1,0,1.$

1973 IMO Shortlist, 11
Tags: algebra , polynomial , root , minimum value , coefficient
Determine the minimum value of $a^{2} + b^{2}$ when $(a,b)$ traverses all the pairs of real numbers for which the equation \[ x^{4} + ax^{3} + bx^{2} + ax + 1 = 0 \] has at least one real root.

1994 Taiwan National Olympiad, 6
Tags: polynomial , algebra
For $-1\leq x\leq 1$ and $n\in\mathbb N$ define $T_{n}(x)=\frac{1}{2^{n}}[(x+\sqrt{1-x^{2}})^{n}+(x-\sqrt{1-x^{2}})^{n}]$. a)Prove that $T_{n}$ is a monic polynomial of degree $n$ in $x$ and that the maximum value of $|T_{n}(x)|$ is $\frac{1}{2^{n-1}}$. b)Suppose that $p(x)=x^{n}+a_{n-1}x^{n-1}+...+a_{1}x+a_{0}\in\mathbb{R}[x]$ is a monic polynomial of degree $n$ such that $p(x)>-\frac{1}{2^{n-1}}$ forall $x$, $-1\leq x\leq 1$. Prove that there exists $x_{0}$, $-1\leq x_{0}\leq 1$ such that $p(x_{0})\geq\frac{1}{2^{n-1}}$.

1982 IMO Longlists, 16
Tags: algebra , polynomial , root , cubic
Let $p(x)$ be a cubic polynomial with integer coefficients with leading coefficient $1$ and with one of its roots equal to the product of the other two. Show that $2p(-1)$ is a multiple of $p(1)+p(-1)-2(1+p(0)).$

1983 Federal Competition For Advanced Students, P2, 2
Tags: polynomial , algebra
Let $ x_1,x_2,x_3$ be the roots of: $ x^3\minus{}6x^2\plus{}ax\plus{}a\equal{}0$. Find all real numbers $ a$ for which $ (x_1\minus{}1)^3\plus{}(x_2\minus{}1)^3\plus{}(x_3\minus{}1)^3\equal{}0$. Also, for each such $ a$, determine the corresponding values of $ x_1,x_2,$ and $ x_3$.

1975 IMO Shortlist, 10
Tags: algebra , polynomial , functional equation
Determine the polynomials P of two variables so that: [b]a.)[/b] for any real numbers $t,x,y$ we have $P(tx,ty) = t^n P(x,y)$ where $n$ is a positive integer, the same for all $t,x,y;$ [b]b.)[/b] for any real numbers $a,b,c$ we have $P(a + b,c) + P(b + c,a) + P(c + a,b) = 0;$ [b]c.)[/b] $P(1,0) =1.$

2001 Nordic, 3
Tags: polynomial equation , algebra , polynomial , real root
Determine the number of real roots of the equation ${x^8 -x^7 + 2x^6- 2x^5 + 3x^4 - 3x^3 + 4x^2 - 4x + \frac{5}{2}= 0}$

1982 All Soviet Union Mathematical Olympiad, 347
Tags: polynomial , algebra
Can you find three polynomials $P,Q,R$ of three variables $x,y,z$, providing the condition: a)$P(x-y+z)^3 + Q(y-z-1)^3 +R(z-2x+1)^3 = 1$ b)$P(x-y+z)^3 + Q(y-z-1)^3 +R(z-x+1)^3 = 1$ for all $x,y,z$?

2018 ELMO Shortlist, 4
Tags: algebra , polynomial
Elmo calls a monic polynomial with real coefficients [i]tasty[/i] if all of its coefficients are in the range $[-1,1]$. A monic polynomial $P$ with real coefficients and complex roots $\chi_1,\cdots,\chi_m$ (counted with multiplicity) is given to Elmo, and he discovers that there does not exist a monic polynomial $Q$ with real coefficients such that $PQ$ is tasty. Find all possible values of $\max\left(|\chi_1|,\cdots,|\chi_m|\right)$. [i]Proposed by Carl Schildkraut[/i]

2012 NIMO Problems, 9
Tags: quadratic , algebra , polynomial
A quadratic polynomial $p(x)$ with integer coefficients satisfies $p(41) = 42$. For some integers $a, b > 41$, $p(a) = 13$ and $p(b) = 73$. Compute the value of $p(1)$. [i]Proposed by Aaron Lin[/i]

1990 IMO Shortlist, 26
Tags: algebra , polynomial , number theory , cubic , iteration
Let $ p(x)$ be a cubic polynomial with rational coefficients. $ q_1$, $ q_2$, $ q_3$, ... is a sequence of rationals such that $ q_n \equal{} p(q_{n \plus{} 1})$ for all positive $ n$. Show that for some $ k$, we have $ q_{n \plus{} k} \equal{} q_n$ for all positive $ n$.

1996 All-Russian Olympiad, 4
Tags: polynomial , algebra
Show that if the integers $a_1$; $\dots$ $a_m$ are nonzero and for each $k =0; 1; \dots ;n$ ($n < m - 1$), $a_1 + a_22^k + a_33^k + \dots + a_mm^k = 0$; then the sequence $a_1, \dots, a_m$ contains at least $n+1$ pairs of consecutive terms having opposite signs. [i]O. Musin[/i]

2014 239 Open Mathematical Olympiad, 2
Tags: algebra , polynomial
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.

2009 Thailand Mathematical Olympiad, 6
Tags: polynomial , algebra
Find all polynomials of the form $P(x) = (-1)^nx^n + a_1x^{n-1} + a_2x^{n-2} + ...+ a_{n-1}x + a_n$ with the following two properties: (i) $\{a_1, a_2, . . . , a_n-1, a_n\} =\{0, 1\}$, and (ii) all roots of $P(x)$ are distinct real numbers

2000 Switzerland Team Selection Test, 14
Tags: polynomial , algebra
The polynomial $P$ of degree $n$ satisfies $P(k) = \frac{k}{k +1}$ for $k = 0,1,2,...,n$. Find $P(n+1)$.

2023 Estonia Team Selection Test, 2
Tags: algebra , polynomial , real coefficients
Let $n$ be a positive integer. Find all polynomials $P$ with real coefficients such that $$P(x^2+x-n^2)=P(x)^2+P(x)$$ for all real numbers $x$.

1994 Putnam, 4
Tags: algebra , polynomial , quadratic
Let $A$ and $B$ be $2\times 2$ matrices with integer entries such that $A, A+B, A+2B, A+3B,$ and $A+4B$ are all invertible matrices whose inverses have integer entries. Show that $A+5B$ is invertible and that its inverse has integer entries.

1975 USAMO, 3
Tags: algebra , polynomial
If $ P(x)$ denotes a polynomial of degree $ n$ such that $ P(k)\equal{}\frac{k}{k\plus{}1}$ for $ k\equal{}0,1,2,\ldots,n$, determine $ P(n\plus{}1)$.

2014 ELMO Shortlist, 3
Tags: algebra , polynomial , induction , binomial theorem , number theory
Let $t$ and $n$ be fixed integers each at least $2$. Find the largest positive integer $m$ for which there exists a polynomial $P$, of degree $n$ and with rational coefficients, such that the following property holds: exactly one of \[ \frac{P(k)}{t^k} \text{ and } \frac{P(k)}{t^{k+1}} \] is an integer for each $k = 0,1, ..., m$. [i]Proposed by Michael Kural[/i]