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: 1311

1990 IberoAmerican, 6
Tags: algebra , polynomial , algebra proposed
Let $f(x)$ be a cubic polynomial with rational coefficients. If the graph of $f(x)$ is tangent to the $x$ axis, prove that the roots of $f(x)$ are all rational.

2010 CHKMO, 1
Tags: algebra proposed , algebra
Given that $ \{a_n\}$ is a sequence in which all the terms are integers, and $ a_2$ is odd. For any natural number $ n$, $ n(a_{n \plus{} 1} \minus{} a_n \plus{} 3) \equal{} a_{n \plus{} 1} \plus{} a_n \plus{} 3$. Furthermore, $ a_{2009}$ is divisible by $ 2010$. Find the smallest integer $ n > 1$ such that $ a_n$ is divisible by $ 2010$. P.S.: I saw EVEN instead of ODD. Got only half of the points.

2013 Romania National Olympiad, 3
Tags: function , algebra proposed , algebra
Find all injective functions$f:\mathbb{Z}\to \mathbb{Z}$ that satisfy: $\left| f\left( x \right)-f\left( y \right) \right|\le \left| x-y \right|$ ,for any $x,y\in \mathbb{Z}$.

2005 Kazakhstan National Olympiad, 4
Tags: algebra , polynomial , algebra proposed , Kazakhstan
Find all polynomials $ P(x)$ with real coefficients such that for every positive integer $ n$ there exists a rational $ r$ with $ P(r)=n$.

1949 Miklós Schweitzer, 5
Tags: algebra , polynomial , algebra proposed
Let $ f(x)$ be a polynomial of second degree the roots of which are contained in the interval $ [\minus{}1,\plus{}1]$ and let there be a point $ x_0\in [\minus{}1.\plus{}1]$ such that $ |f(x_0)|\equal{}1$. Prove that for every $ \alpha \in [0,1]$, there exists a $ \zeta \in [\minus{}1,\plus{}1]$ such that $ |f'(\zeta)|\equal{}\alpha$ and that this statement is not true if $ \alpha>1$.

2008 All-Russian Olympiad, 1
Tags: algebra , polynomial , algebra proposed
Numbers $ a,b,c$ are such that the equation $ x^3 \plus{} ax^2 \plus{} bx \plus{} c$ has three real roots.Prove that if $ \minus{} 2\leq a \plus{} b \plus{} c\leq 0$,then at least one of these roots belongs to the segment $ [0,2]$

2000 JBMO ShortLists, 12
Tags: algebra proposed , algebra
Consider a sequence of positive integers $x_n$ such that: \[(\text{A})\ x_{2n+1}=4x_n+2n+2 \] \[(\text{B})\ x_{3n+\color[rgb]{0.9529,0.0980,0.0118}2}=3x_{n+1}+6x_n \] for all $n\ge 0$. Prove that \[(\text{C})\ x_{3n-1}=x_{n+2}-2x_{n+1}+10x_n \] for all $n\ge 0$.

2007 India IMO Training Camp, 3
Tags: function , ratio , algebra proposed , algebra
Find all function(s) $f:\mathbb R\to\mathbb R$ satisfying the equation \[f(x+y)+f(x)f(y)=(1+y)f(x)+(1+x)f(y)+f(xy);\] For all $x,y\in\mathbb R.$

2004 Bulgaria Team Selection Test, 1
Tags: function , algebra proposed , algebra
Find all $k>0$ such that there exists a function $f : [0,1]\times[0,1] \to [0,1]$ satisfying the following conditions: $f(f(x,y),z)=f(x,f(y,z))$; $f(x,y) = f(y,x)$; $f(x,1)=x$; $f(zx,zy) = z^{k}f(x,y)$, for any $x,y,z \in [0,1]$

1992 Baltic Way, 9
Tags: algebra , polynomial , algebra proposed
A polynomial $ f(x)\equal{}x^3\plus{}ax^2\plus{}bx\plus{}c$ is such that $ b<0$ and $ ab\equal{}9c$. Prove that the polynomial $ f$ has three different real roots.

2014 District Olympiad, 1
Tags: inequalities , algebra proposed , algebra
Prove that: [list=a][*]$\displaystyle\left( \frac{1}{2}\right) ^{3}+\left( \frac{2}{3}\right)^{3}+\left( \frac{5}{6}\right) ^{3}=1$ [*]$3^{33}+4^{33}+5^{33}<6^{33}$[/list]

2002 Turkey Team Selection Test, 1
Tags: function , symmetry , algebra proposed , algebra
If a function $f$ defined on all real numbers has at least two centers of symmetry, show that this function can be written as sum of a linear function and a periodic function. [For every real number $x$, if there is a real number $a$ such that $f(a-x) + f(a+x) =2f(a)$, the point $(a,f(a))$ is called a center of symmetry of the function $f$.]

2013 Greece National Olympiad, 1
Tags: induction , algebra proposed , algebra
Let the sequence of real numbers $(a_n),n=1,2,3...$ with $a_1=2$ and $a_n=\left(\frac{n+1}{n-1} \right)\left(a_1+a_2+...+a_{n-1} \right),n\geq 2$. Find the term $a_{2013}$.

2010 Olympic Revenge, 4
Tags: algebra proposed , algebra
Let $a_n$ and $b_n$ to be two sequences defined as below: $i)$ $a_1 = 1$ $ii)$ $a_n + b_n = 6n - 1$ $iii)$ $a_{n+1}$ is the least positive integer different of $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n$. Determine $a_{2009}$.

2011 Serbia National Math Olympiad, 1
Tags: inequalities , induction , algebra proposed , algebra
Let $n \ge 2$ be integer. Let $a_0$, $a_1$, ... $a_n$ be sequence of positive reals such that: $(a_{k-1}+a_k)(a_k+a_{k+1})=a_{k-1}-a_{k+1}$, for $k=1, 2, ..., n-1$. Prove $a_n< \frac{1}{n-1}$.

2013 India IMO Training Camp, 2
Tags: algebra , polynomial , algebra proposed
Let $n \ge 2$ be an integer and $f_1(x), f_2(x), \ldots, f_{n}(x)$ a sequence of polynomials with integer coefficients. One is allowed to make moves $M_1, M_2, \ldots $ as follows: in the $k$-th move $M_k$ one chooses an element $f(x)$ of the sequence with degree of $f$ at least $2$ and replaces it with $(f(x) - f(k))/(x-k)$. The process stops when all the elements of the sequence are of degree $1$. If $f_1(x) = f_2(x) = \cdots = f_n(x) = x^n + 1$, determine whether or not it is possible to make appropriate moves such that the process stops with a sequence of $n$ identical polynomials of degree 1.

2011 ELMO Shortlist, 3
Tags: induction , complex numbers , imaginary numbers , algebra proposed , algebra
Let $N$ be a positive integer. Define a sequence $a_0,a_1,\ldots$ by $a_0=0$, $a_1=1$, and $a_{n+1}+a_{n-1}=a_n(2-1/N)$ for $n\ge1$. Prove that $a_n<\sqrt{N+1}$ for all $n$. [i]Evan O'Dorney.[/i]

2008 Moldova Team Selection Test, 4
Tags: algebra , polynomial , factorial , calculus , integration , algebra proposed
A non-zero polynomial $ S\in\mathbb{R}[X,Y]$ is called homogeneous of degree $ d$ if there is a positive integer $ d$ so that $ S(\lambda x,\lambda y)\equal{}\lambda^dS(x,y)$ for any $ \lambda\in\mathbb{R}$. Let $ P,Q\in\mathbb{R}[X,Y]$ so that $ Q$ is homogeneous and $ P$ divides $ Q$ (that is, $ P|Q$). Prove that $ P$ is homogeneous too.

1998 Baltic Way, 7
Tags: function , algebra , functional equation , algebra proposed
Let $\mathbb{R}$ be the set of all real numbers. Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfying for all $x,y\in\mathbb{R}$ the equation $f(x)+f(y)=f(f(x)f(y))$.

2001 Vietnam National Olympiad, 2
Tags: function , integration , algebra proposed , algebra
Find all real-valued continuous functions defined on the interval $(-1, 1)$ such that $(1-x^{2}) f(\frac{2x}{1+x^{2}}) = (1+x^{2})^{2}f(x)$ for all $x$.

2004 India National Olympiad, 3
Tags: algebra proposed , algebra
If $a$ is a real root of $x^5 - x^3 + x - 2 = 0$, show that $[a^6] =3$

2007 Balkan MO Shortlist, N3
Tags: algebra , polynomial , trigonometry , algebra proposed
i thought that this problem was in mathlinks but when i searched i didn't find it.so here it is: Find all positive integers m for which for all $\alpha,\beta \in \mathbb{Z}-\{0\}$ \[ \frac{2^m \alpha^m-(\alpha+\beta)^m-(\alpha-\beta)^m}{3 \alpha^2+\beta^2} \in \mathbb{Z} \]

2007 Iran MO (3rd Round), 5
Tags: algebra , polynomial , geometry , function , algebra proposed
Prove that for two non-zero polynomials $ f(x,y),g(x,y)$ with real coefficients the system: \[ \left\{\begin{array}{c}f(x,y)\equal{}0\\ g(x,y)\equal{}0\end{array}\right.\] has finitely many solutions in $ \mathbb C^{2}$ if and only if $ f(x,y)$ and $ g(x,y)$ are coprime.

2004 China National Olympiad, 2
Tags: floor function , modular arithmetic , induction , algebra proposed , algebra
Let $c$ be a positive integer. Consider the sequence $x_1,x_2,\ldots$ which satisfies $x_1=c$ and, for $n\ge 2$, \[x_n=x_{n-1}+\left\lfloor\frac{2x_{n-1}-(n+2)}{n}\right\rfloor+1\] where $\lfloor x\rfloor$ denotes the largest integer not greater than $x$. Determine an expression for $x_n$ in terms of $n$ and $c$. [i]Huang Yumin[/i]

2012 Iran Team Selection Test, 2
Tags: function , induction , ceiling function , binomial coefficients , algebra proposed , algebra
The function $f:\mathbb R^{\ge 0} \longrightarrow \mathbb R^{\ge 0}$ satisfies the following properties for all $a,b\in \mathbb R^{\ge 0}$: [b]a)[/b] $f(a)=0 \Leftrightarrow a=0$ [b]b)[/b] $f(ab)=f(a)f(b)$ [b]c)[/b] $f(a+b)\le 2 \max \{f(a),f(b)\}$. Prove that for all $a,b\in \mathbb R^{\ge 0}$ we have $f(a+b)\le f(a)+f(b)$. [i]Proposed by Masoud Shafaei[/i]