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

2013 Hanoi Open Mathematics Competitions, 3
Tags: floor function , polynomial , algebra
What is the largest integer not exceeding $8x^3 +6x - 1$, where $x =\frac12 \left(\sqrt[3]{2+\sqrt5} + \sqrt[3]{2-\sqrt5}\right)$ ? (A): $1$, (B): $2$, (C): $3$, (D): $4$, (E) None of the above.

2014 BAMO, 3
Tags: algebra , polynomial , square root
Suppose that for two real numbers $x$ and $y$ the following equality is true: $$(x+ \sqrt{1+ x^2})(y+\sqrt{1+y^2})=1$$ Find (with proof) the value of $x+y$.

2018 Estonia Team Selection Test, 12
Tags: polynomial , divisible
We call the polynomial $P (x)$ simple if the coefficient of each of its members belongs to the set $\{-1, 0, 1\}$. Let $n$ be a positive integer, $n> 1$. Find the smallest possible number of terms with a non-zero coefficient in a simple $n$-th degree polynomial with all values at integer places are divisible by $n$.

2014 NIMO Problems, 5
Tags: algebra , polynomial , vieta , search
Let $r$, $s$, $t$ be the roots of the polynomial $x^3+2x^2+x-7$. Then \[ \left(1+\frac{1}{(r+2)^2}\right)\left(1+\frac{1}{(s+2)^2}\right)\left(1+\frac{1}{(t+2)^2}\right)=\frac{m}{n} \] for relatively prime positive integers $m$ and $n$. Compute $100m+n$. [i]Proposed by Justin Stevens[/i]

2007 Iran Team Selection Test, 1
Tags: polynomial , inequalities , algebra
Find all polynomials of degree 3, such that for each $x,y\geq 0$: \[p(x+y)\geq p(x)+p(y)\]

2002 Romania Team Selection Test, 2
Tags: algebra , polynomial , induction , diophantine equation , number theory
The sequence $ (a_n)$ is defined by: $ a_0\equal{}a_1\equal{}1$ and $ a_{n\plus{}1}\equal{}14a_n\minus{}a_{n\minus{}1}$ for all $ n\ge 1$. Prove that $ 2a_n\minus{}1$ is a perfect square for any $ n\ge 0$.

2012 ELMO Shortlist, 6
Tags: floor function , algebra , polynomial , vieta , quadratic , inequalities , number theory
Prove that if $a$ and $b$ are positive integers and $ab>1$, then \[\left\lfloor\frac{(a-b)^2-1}{ab}\right\rfloor=\left\lfloor\frac{(a-b)^2-1}{ab-1}\right\rfloor.\]Here $\lfloor x\rfloor$ denotes the greatest integer not exceeding $x$. [i]Calvin Deng.[/i]

1971 IMO Longlists, 55
Tags: polynomial , algebra
Prove that the polynomial $x^4+\lambda x^3+\mu x^2+\nu x+1$ has no real roots if $\lambda, \mu , \nu $ are real numbers satisfying \[|\lambda |+|\mu |+|\nu |\le \sqrt{2} \]

2004 Iran MO (3rd Round), 20
Tags: algebra , polynomial , number theory
$ p(x)$ is a polynomial in $ \mathbb{Z}[x]$ such that for each $ m,n\in \mathbb{N}$ there is an integer $ a$ such that $ n\mid p(a^m)$. Prove that $0$ or $1$ is a root of $ p(x)$.

2006 Austrian-Polish Competition, 2
Tags: polynomial , algebra
Find all polynomials $P(x)$ with real coefficients satisfying the equation \[(x+1)^{3}P(x-1)-(x-1)^{3}P(x+1)=4(x^{2}-1) P(x)\] for all real numbers $x$.

1997 All-Russian Olympiad, 1
Tags: quadratic , modular arithmetic , polynomial , algebra
Do there exist two quadratic trinomials $ax^2 +bx+c$ and $(a+1)x^2 +(b + 1)x + (c + 1)$ with integer coeficients, both of which have two integer roots? [i]N. Agakhanov[/i]

1980 IMO Longlists, 4
Tags: geometry , circumcircle , algebra , polynomial , convex polygon
Determine all positive integers $n$ such that the following statement holds: If a convex polygon with with $2n$ sides $A_1 A_2 \ldots A_{2n}$ is inscribed in a circle and $n-1$ of its $n$ pairs of opposite sides are parallel, which means if the pairs of opposite sides \[(A_1 A_2, A_{n+1} A_{n+2}), (A_2 A_3, A_{n+2} A_{n+3}), \ldots , (A_{n-1} A_n, A_{2n-1} A_{2n})\] are parallel, then the sides \[ A_n A_{n+1}, A_{2n} A_1\] are parallel as well.

1988 Flanders Math Olympiad, 1
Tags: algebra , polynomial
show that the polynomial $x^4+3x^3+6x^2+9x+12$ cannot be written as the product of 2 polynomials of degree 2 with integer coefficients.

2005 Austrian-Polish Competition, 2
Tags: algebra , polynomial , ratio
Determine all polynomials $P$ with integer coefficients satisfying \[P(P(P(P(P(x)))))=x^{28}\cdot P(P(x))\qquad \forall x\in\mathbb{R}\]

2007 Finnish National High School Mathematics Competition, 5
Tags: polynomial , modular arithmetic , algebra
Show that there exists a polynomial $P(x)$ with integer coefficients, such that the equation $P(x) = 0$ has no integer solutions, but for each positive integer $n$ there is an $x \in \Bbb{Z}$ such that $n \mid P(x).$

2002 National Olympiad First Round, 28
Tags: polynomial , algebra
How many positive roots does polynomial $x^{2002} + a_{2001}x^{2001} + a_{2000}x^{2000} + \cdots + a_1x + a_0$ have such that $a_{2001} = 2002$ and $a_k = -k - 1$ for $0\leq k \leq 2000$? $ \textbf{a)}\ 0 \qquad\textbf{b)}\ 1 \qquad\textbf{c)}\ 2 \qquad\textbf{d)}\ 1001 \qquad\textbf{e)}\ 2002 $

2006 IMC, 6
Tags: function , algebra , polynomial , trigonometry , real analysis
Find all sequences $a_{0}, a_{1},\ldots, a_{n}$ of real numbers such that $a_{n}\neq 0$, for which the following statement is true: If $f: \mathbb{R}\to\mathbb{R}$ is an $n$ times differentiable function and $x_{0}<x_{1}<\ldots <x_{n}$ are real numbers such that $f(x_{0})=f(x_{1})=\ldots =f(x_{n})=0$ then there is $h\in (x_{0}, x_{n})$ for which \[a_{0}f(h)+a_{1}f'(h)+\ldots+a_{n}f^{(n)}(h)=0.\]

2000 IMC, 6
Tags: linear algebra , matrix , algebra , polynomial
Let $A$ be a real $n\times n$ Matrix and define $e^{A}=\sum_{k=0}^{\infty} \frac{A^{k}}{k!}$ Prove or disprove that for any real polynomial $P(x)$ and any real matrices $A,B$, $P(e^{AB})$ is nilpotent if and only if $P(e^{BA})$ is nilpotent.

2020 USA EGMO Team Selection Test, 6
Tags: algebra , polynomial , integer polynomial , periodic sequence , modular arithmetic
Find the largest integer $N \in \{1, 2, \ldots , 2019 \}$ such that there exists a polynomial $P(x)$ with integer coefficients satisfying the following property: for each positive integer $k$, $P^k(0)$ is divisible by $2020$ if and only if $k$ is divisible by $N$. Here $P^k$ means $P$ applied $k$ times, so $P^1(0)=P(0), P^2(0)=P(P(0)),$ etc.

2013 IMO Shortlist, N3
Tags: number theory , prime divisor , polynomial
Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.

2003 Belarusian National Olympiad, 2
Tags: polynomial , perfect square , algebra
Let $P(x) =(x+1)^p (x-3)^q=x^n+a_1x^{n-1}+a_2x^{n-2}+...+a_{n-1}x+a_n$ where $p$ and $q$ are positive integers a) Given $a_1=a_2$, prove that $3n$ is a perfect square. b) Prove that there exist infinitely many pairs $(p, q)$ of positive integers p and q such that the equality $a_1=a_2$ is valid for the polynomial $P(x)$. (D. Bazylev)

2009 IMC, 4
Tags: algebra , polynomial
Let $p(z)=a_0+a_1z+a_2z^2+\cdots+a_nz^n$ be a complex polynomial. Suppose that $1=c_0\ge c_1\ge \cdots \ge c_n\ge 0$ is a sequence of real numbers which form a convex sequence. (That is $2c_k\le c_{k-1}+c_{k+1}$ for every $k=1,2,\cdots ,n-1$ ) and consider the polynomial \[ q(z)=c_0a_0+c_1a_1z+c_2a_2z^2+\cdots +c_na_nz^n \] Prove that : \[ \max_{|z|\le 1}q(z)\le \max_{|z|\le 1}p(z) \]

2006 Princeton University Math Competition, 3
Tags: algebra , polynomial , vieta
Let $r_1, \dots , r_5$ be the roots of the polynomial $x^5+5x^4-79x^3+64x^2+60x+144$. What is $r^2_1+\dots+r^2_5$?

2013 JBMO TST - Turkey, 4
Tags: quadratic , algebra , polynomial , inequalities
For all positive real numbers $a, b, c$ satisfying $a+b+c=1$, prove that \[ \frac{a^4+5b^4}{a(a+2b)} + \frac{b^4+5c^4}{b(b+2c)} + \frac{c^4+5a^4}{c(c+2a)} \geq 1- ab-bc-ca \]

MathLinks Contest 6th, 7.1
Tags: algebra , polynomial
Write the following polynomial as a product of irreducible polynomials in $\mathbb{Z}[X]$ \[ f(X) = X^{2005} - 2005 X + 2004 . \]Justify your answer.