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

2014 AIME Problems, 9
Tags: algebra , polynomial , vieta , quadratic , trigonometry
Let $x_1<x_2<x_3$ be three real roots of equation $\sqrt{2014}x^3-4029x^2+2=0$. Find $x_2(x_1+x_3)$.

Kvant 2025, M2832
Tags: algebra , polynomial
There are $2024$ points of general position marked on the coordinate plane (i.e., points among which there are no three lying on the same straight line). Is there a polynomial of two variables $f(x,y)$ a) of degree $2025$; b) of degree $2024$ such that it equals to zero exactly at these marked points? [i]Proposed by Navid Safaei[/i]

2019 Iran MO (3rd Round), 2
Tags: algebra , polynomial
$P(x)$ is a monoic polynomial with integer coefficients so that there exists monoic integer coefficients polynomials $p_1(x),p_2(x),\dots ,p_n(x)$ so that for any natural number $x$ there exist an index $j$ and a natural number $y$ so that $p_j(y)=P(x)$ and also $deg(p_j) \ge deg(P)$ for all $j$.Show that there exist an index $i$ and an integer $k$ so that $P(x)=p_i(x+k)$.

2008 Moldova MO 11-12, 1
Tags: algebra , polynomial , calculus , inequalities
Consider the equation $ x^4 \minus{} 4x^3 \plus{} 4x^2 \plus{} ax \plus{} b \equal{} 0$, where $ a,b\in\mathbb{R}$. Determine the largest value $ a \plus{} b$ can take, so that the given equation has two distinct positive roots $ x_1,x_2$ so that $ x_1 \plus{} x_2 \equal{} 2x_1x_2$.

1988 Iran MO (2nd round), 1
Tags: polynomial , induction , factorial , algebra
[b](a)[/b] Prove that for all positive integers $m,n$ we have \[\sum_{k=1}^n k(k+1)(k+2)\cdots (k+m-1)=\frac{n(n+1)(n+2) \cdots (n+m)}{m+1}\] [b](b)[/b] Let $P(x)$ be a polynomial with rational coefficients and degree $m.$ If $n$ tends to infinity, then prove that \[\frac{\sum_{k=1}^n P(k)}{n^{m+1}}\] Has a limit.

1997 Federal Competition For Advanced Students, P2, 6
Tags: polynomial , algebra
For every natural number $ n$, find all polynomials $ x^2\plus{}ax\plus{}b$, where $ a^2 \ge 4b$, that divide $ x^{2n}\plus{}ax^n\plus{}b$.

2009 India IMO Training Camp, 9
Tags: polynomial , algebra
Let $ f(x)\equal{}\sum_{k\equal{}1}^n a_k x^k$ and $ g(x)\equal{}\sum_{k\equal{}1}^n \frac{a_k x^k}{2^k \minus{}1}$ be two polynomials with real coefficients. Let g(x) have $ 0,2^{n\plus{}1}$ as two of its roots. Prove That $ f(x)$ has a positive root less than $ 2^n$.

2004 Iran MO (3rd Round), 12
Tags: algebra , polynomial , algorithm , geometry , geometric transformation , induction , number theory
$\mathbb{N}_{10}$ is generalization of $\mathbb{N}$ that every hypernumber in $\mathbb{N}_{10}$ is something like: $\overline{...a_2a_1a_0}$ with $a_i \in {0,1..9}$ (Notice that $\overline {...000} \in \mathbb{N}_{10}$) Also we easily have $+,*$ in $\mathbb{N}_{10}$. first $k$ number of $a*b$= first $k$ nubmer of (first $k$ number of a * first $k$ number of b) first $k$ number of $a+b$= first $k$ nubmer of (first $k$ number of a + first $k$ number of b) Fore example $\overline {...999}+ \overline {...0001}= \overline {...000}$ Prove that every monic polynomial in $\mathbb{N}_{10}[x]$ with degree $d$ has at most $d^2$ roots.

2005 Bulgaria National Olympiad, 6
Tags: algebra , polynomial , symmetry , modular arithmetic , number theory
Let $a,b$ and $c$ be positive integers such that $ab$ divides $c(c^{2}-c+1)$ and $a+b$ is divisible by $c^{2}+1$. Prove that the sets $\{a,b\}$ and $\{c,c^{2}-c+1\}$ coincide.

2019 International Zhautykov OIympiad, 6
Tags: polynomial , algebra
We define two types of operation on polynomial of third degree: a) switch places of the coefficients of polynomial(including zero coefficients), ex: $ x^3+x^2+3x-2 $ => $ -2x^3+3x^2+x+1$ b) replace the polynomial $P(x)$ with $P(x+1)$ If limitless amount of operations is allowed, is it possible from $x^3-2$ to get $x^3-3x^2+3x-3$ ?

1998 Miklós Schweitzer, 2
Tags: polynomial , real analysis
For any polynomial f, denote by $P_f$ the number of integers n for which f(n) is a (positive) prime number. Let $q_d = max P_f$ , where f runs over all polynomials with integer coefficients with degree d and reducible over $\mathbb{Q}$. Prove that $\forall d\geq 2$ , $q_d = d$.

2016 Kyrgyzstan National Olympiad, 5
Tags: algebra , polynomial
Given two monic polynomials $P(x)$ and $Q(x)$ with degrees 2016. $P(x)=Q(x)$ has no real root. [b]Prove that P(x)=Q(x+1) has at least one real root.[/b]

2008 Mongolia Team Selection Test, 3
Tags: polynomial , algebra
Given positive integers $ m,n > 1$. Prove that the equation $ (x \plus{} 1)^n \plus{} (x \plus{} 2)^n \plus{} ... \plus{} (x \plus{} m)^n \equal{} (y \plus{} 1)^{2n} \plus{} (y \plus{} 2)^{2n} \plus{} ... \plus{} (y \plus{} m)^{2n}$ has finitely number of solutions $ x,y \in N$

2006 Iran MO (3rd Round), 2
Tags: vector , geometry , geometric transformation , rotation , algebra , polynomial , group theory
$n$ is a natural number that $\frac{x^{n}+1}{x+1}$ is irreducible over $\mathbb Z_{2}[x]$. Consider a vector in $\mathbb Z_{2}^{n}$ that it has odd number of $1$'s (as entries) and at least one of its entries are $0$. Prove that these vector and its translations are a basis for $\mathbb Z_{2}^{n}$

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 $g(n)$.

VMEO II 2005, 7
Tags: algebra , functional equation , polynomial , function
Find all function $f:[0,\infty )\to\mathbb{R}$ such that $f$ is monotonic and \[ [f(x)+f(y)]^2=f(x^2-y^2)+f(2xy) \] for all $x\geq y\geq 0$

2000 Harvard-MIT Mathematics Tournament, 6
Tags: algebra , polynomial , vieta
If integers $m,n,k$ satisfy $m^2+n^2+1=kmn$, what values can $k$ have?

2011 Poland - Second Round, 3
Tags: polynomial , calculus , integration , algebra
There are two given different polynomials $P(x),Q(x)$ with real coefficients such that $P(Q(x))=Q(P(x))$. Prove that $\forall n\in \mathbb{Z_{+}}$ polynomial: \[\underbrace{P(P(\ldots P(P}_{n}(x))\ldots))- \underbrace{Q(Q(\ldots Q(Q}_{n}(x))\ldots))\] is divisible by $P(x)-Q(x)$.

1993 APMO, 3
Tags: polynomial , ratio , algebra , inequalities
Let \begin{eqnarray*} f(x) & = & a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0 \ \ \mbox{and} \\ g(x) & = & c_{n+1} x^{n+1} + c_n x^n + \cdots + c_0 \end{eqnarray*} be non-zero polynomials with real coefficients such that $g(x) = (x+r)f(x)$ for some real number $r$. If $a = \max(|a_n|, \ldots, |a_0|)$ and $c = \max(|c_{n+1}|, \ldots, |c_0|)$, prove that $\frac{a}{c} \leq n+1$.

1994 Polish MO Finals, 1
Tags: polynomial , rational root theorem , algebra
Find all triples $(x,y,z)$ of positive rationals such that $x + y + z$, $\dfrac{1}{x} + \dfrac{1}{y} + \dfrac{1}{z}$ and $xyz$ are all integers.

STEMS 2023 Math Cat A, 4
Tags: algebra , polynomial
Alice has $n > 1$ one variable quadratic polynomials written on paper she keeps secret from Bob. On each move, Bob announces a real number and Alice tells him the value of one of her polynomials at this number. Prove that there exists a constant $C > 0$ such that after $Cn^5$ questions, Bob can determine one of Alice’s polynomials. [i]Proposed by Rohan Goyal and Anant Mudgal[/i]

2010 Putnam, B4
Tags: algebra , polynomial , function , induction , rational function , recursion
Find all pairs of polynomials $p(x)$ and $q(x)$ with real coefficients for which \[p(x)q(x+1)-p(x+1)q(x)=1.\]

2001 Korea - Final Round, 1
Tags: polynomial , algebra
For given positive integers $n$ and $N$, let $P_n$ be the set of all polynomials $f(x)=a_0+a_1x+\cdots+a_nx^n$ with integer coefficients such that: [list] (a) $|a_j| \le N$ for $j = 0,1, \cdots ,n$; (b) The set $\{ j \mid a_j = N\}$ has at most two elements. [/list] Find the number of elements of the set $\{f(2N) \mid f(x) \in P_n\}$.

2001 India IMO Training Camp, 3
Tags: polynomial , inequalities , induction , algebra
Let $P(x)$ be a polynomial of degree $n$ with real coefficients and let $a\geq 3$. Prove that \[\max_{0\leq j \leq n+1}\left | a^j-P(j) \right |\geq 1\]

2011 Math Prize For Girls Problems, 18
Tags: algebra , polynomial , quadratic , calculus , integration , number theory , relatively prime
The polynomial $P$ is a quadratic with integer coefficients. For every positive integer $n$, the integers $P(n)$ and $P(P(n))$ are relatively prime to $n$. If $P(3) = 89$, what is the value of $P(10)$?