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

2019 Ramnicean Hope, 3
Tags: algebra , polynomial , divisibility
Let be two polynoms $ P,Q\in\mathbb{C} [X] $ with degree at least $ 1, $ and such that $ P $ has only simple roots. Prove that the following affirmations are equivalent: $ \text{(i)} P\circ Q $ is divisible by $ P. $ $ \text{(ii)} $ The evaluation of $ Q $ at any root of $ P $ is a root of $ P. $ [i]Marcel Țena[/i]

1988 IMO Shortlist, 16
Tags: algebra , vieta , inequality , root , polynomial
Show that the solution set of the inequality \[ \sum^{70}_{k \equal{} 1} \frac {k}{x \minus{} k} \geq \frac {5}{4} \] is a union of disjoint intervals, the sum of whose length is 1988.

2011 NIMO Problems, 9
Tags: algebra , polynomial , vieta
The roots of the polynomial $P(x) = x^3 + 5x + 4$ are $r$, $s$, and $t$. Evaluate $(r+s)^4 (s+t)^4 (t+r)^4$. [i]Proposed by Eugene Chen [/i]

2020 USA TSTST, 7
Tags: polynomial , algebra , complex number
Find all nonconstant polynomials $P(z)$ with complex coefficients for which all complex roots of the polynomials $P(z)$ and $P(z) - 1$ have absolute value 1. [i]Ankan Bhattacharya[/i]

2018 CMIMC Individual Finals, 3
Tags: algebra , polynomial
Let $a$ be a complex number, and set $\alpha$, $\beta$, and $\gamma$ to be the roots of the polynomial $x^3 - x^2 + ax - 1$. Suppose \[(\alpha^3+1)(\beta^3+1)(\gamma^3+1) = 2018.\] Compute the product of all possible values of $a$.

1971 IMO Longlists, 31
Tags: polynomial , equation , algebra , root
Determine whether there exist distinct real numbers $a, b, c, t$ for which: [i](i)[/i] the equation $ax^2 + btx + c = 0$ has two distinct real roots $x_1, x_2,$ [i](ii)[/i] the equation $bx^2 + ctx + a = 0$ has two distinct real roots $x_2, x_3,$ [i](iii)[/i] the equation $cx^2 + atx + b = 0$ has two distinct real roots $x_3, x_1.$

2020 Greece National Olympiad, 1
Tags: functional equation , functional , algebra , polynomial , polynomial equation
Find all non constant polynomials $P(x),Q(x)$ with real coefficients such that: $P((Q(x))^3)=xP(x)(Q(x))^3$

2017 China Team Selection Test, 4
Tags: polynomial , algebra
Show that there exists a degree $58$ monic polynomial $$P(x) = x^{58} + a_1x^{57} + \cdots + a_{58}$$ such that $P(x)$ has exactly $29$ positive real roots and $29$ negative real roots and that $\log_{2017} |a_i|$ is a positive integer for all $1 \leq i \leq 58$.

1978 IMO Longlists, 25
Tags: polynomial , algebra , vieta
Consider a polynomial $P(x) = ax^2 + bx + c$ with $a > 0$ that has two real roots $x_1, x_2$. Prove that the absolute values of both roots are less than or equal to $1$ if and only if $a + b + c \ge 0, a -b + c \ge 0$, and $a - c \ge 0$.

2001 Manhattan Mathematical Olympiad, 3
Tags: modular arithmetic , polynomial , algebra
Let $x_1$ and $x_2$ be roots of the equation $x^2 - 6x + 1 = 0$. Prove that for any integer $n \ge 1$ the number $x_1^n + x_2^n$ is integer and is not divisible by $5$.

1963 All Russian Mathematical Olympiad, 038
Tags: polynomial , algebra
Find such real $p, q, a, b$, that for all $x$ an equality is held: $$(2x-1)^{20} - (ax+b)^{20} = (x^2+px+q)^{10}$$

2017 Harvard-MIT Mathematics Tournament, 2
Tags: polynomial , algebra
Does there exist a two-variable polynomial $P(x, y)$ with real number coefficients such that $P(x, y)$ is positive exactly when $x$ and $y$ are both positive?

2010 AMC 12/AHSME, 21
Tags: number theory , polynomial , algebra , least common multiple , binomial theorem , floor function
Let $ a>0$, and let $ P(x)$ be a polynomial with integer coefficients such that \[ P(1)\equal{}P(3)\equal{}P(5)\equal{}P(7)\equal{}a\text{, and}\] \[ P(2)\equal{}P(4)\equal{}P(6)\equal{}P(8)\equal{}\minus{}a\text{.}\] What is the smallest possible value of $ a$? $ \textbf{(A)}\ 105 \qquad \textbf{(B)}\ 315 \qquad \textbf{(C)}\ 945 \qquad \textbf{(D)}\ 7! \qquad \textbf{(E)}\ 8!$

2014 Putnam, 4
Tags: algebra , polynomial , probability , inequalities , expected value
Suppose $X$ is a random variable that takes on only nonnegative integer values, with $E[X]=1,$ $E[X^2]=2,$ and $E[X^3]=5.$ (Here $E[Y]$ denotes the expectation of the random variable $Y.$) Determine the smallest possible value of the probability of the event $X=0.$

2013 ELMO Shortlist, 7
Tags: algebra , number theory , polynomial , modular arithmetic
Let $p$ be a prime satisfying $p^2\mid 2^{p-1}-1$, and let $n$ be a positive integer. Define \[ f(x) = \frac{(x-1)^{p^n}-(x^{p^n}-1)}{p(x-1)}. \] Find the largest positive integer $N$ such that there exist polynomials $g(x)$, $h(x)$ with integer coefficients and an integer $r$ satisfying $f(x) = (x-r)^N g(x) + p \cdot h(x)$. [i]Proposed by Victor Wang[/i]

1979 IMO Longlists, 42
Tags: polynomial , algebra , quadratic
Let a quadratic polynomial $g(x) = ax^2 + bx + c$ be given and an integer $n \ge 1$. Prove that there exists at most one polynomial $f(x)$ of $n$th degree such that $f(g(x)) = g(f(x)).$

2007 All-Russian Olympiad Regional Round, 9.1
Tags: geometric transformation , geometry , algebra , polynomial , quadratic
Pete chooses $ 1004$ monic quadratic polynomial $ f_{1},\cdots,f_{1004}$, such that each integer from $ 0$ to $ 2007$ is a root of at least one of them. Vasya considers all equations of the form $ f_{i}\equal{}f_{j}(i\not \equal{}j)$ and computes their roots; for each such root , Pete has to pay to Vasya $ 1$ ruble . Find the least possible value of Vasya's income.

2009 China Team Selection Test, 2
Tags: algebra , combinatorics , number theory , polynomial , modular arithmetic , additive combinatorics
Find all the pairs of integers $ (a,b)$ satisfying $ ab(a \minus{} b)\not \equal{} 0$ such that there exists a subset $ Z_{0}$ of set of integers $ Z,$ for any integer $ n$, exactly one among three integers $ n,n \plus{} a,n \plus{} b$ belongs to $ Z_{0}$.

2015 India Regional MathematicaI Olympiad, 2
Tags: polynomial , quadratic , algebra
Let $P(x) = x^2 + ax + b$ be a quadratic polynomial with real coefficients. Suppose there are real numbers $ s \neq t$ such that $P(s) = t$ and $P(t) = s$. Prove that $b-st$ is a root of $x^2 + ax + b - st$.

STEMS 2023 Math Cat A, 3
Tags: algebra , polynomial
Suppose $f$ is a nonconstant polynomial with integer coefficients with the following property: [list] [*]$f(0)$ and $f(1)$ are both odd. [*]Define a sequence of integers with $a_k = f(1)f(2) \cdots f(k)+1$ [/list] Prove that there are infinitely many prime numbers dividing at least one element of the sequence. [i]Proposed by Sayandeep Shee[/i]

MathLinks Contest 1st, 2
Tags: polynomial , algebra
Let a be a non-zero integer, and $n \ge 3$ another integer. Prove that the following polynomial is irreducible in the ring of integer polynomials (i.e. it cannot be written as a product of two non-constant integer polynomials): $$f(x) = x^n + ax^{n-1} + ax^{n-2} +... + ax -1$$

2008 Mathcenter Contest, 9
Tags: polynomial , algebra
Set $P$ as a polynomial function by $p_n(x)=\sum_{k=0}^{n-1} x^k$. a) Prove that for $m,n\in N$, when dividing $p_n(x)$ by $p_m(x)$, the remainder is $$p_i(x),\forall i=0,1,...,m-1.$$ b) Find all the positive integers $i,j,k$ that make $$p_i(x)+p_j(x^2)+p_k(x^4)=p_{100}(x).$$ [i](square1zoa)[/i]

1982 IMO Shortlist, 4
Tags: algebra , polynomial , diophantine equation , parametric equation , geometric progression
Determine all real values of the parameter $a$ for which the equation \[16x^4 -ax^3 + (2a + 17)x^2 -ax + 16 = 0\] has exactly four distinct real roots that form a geometric progression.

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 $f = g.$

2009 Iran Team Selection Test, 4
Tags: algebra , polynomial , number theory
Find all polynomials $f$ with integer coefficient such that, for every prime $p$ and natural numbers $u$ and $v$ with the condition: \[ p \mid uv - 1 \] we always have $p \mid f(u)f(v) - 1$.