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: 1269

2004 Czech-Polish-Slovak Match, 1
Tags: algebra , polynomial , quadratics , algebra unsolved
Show that real numbers, $p, q, r$ satisfy the condition $p^4(q-r)^2 + 2p^2(q+r) + 1 = p^4$ if and only if the quadratic equations $x^2 + px + q = 0$ and $y^2 - py + r = 0$ have real roots (not necessarily distinct) which can be labeled by $x_1,x_2$ and $y_1,y_2$, respectively, in such a way that $x_1y_1 - x_2y_2 = 1$.

2002 Romania National Olympiad, 4
Tags: function , inequalities , algebra unsolved , algebra
Find all functions $f: \mathbb{N}\to\mathbb{N}$ which satisfy the inequality: \[f(3x+2y)=f(x)f(y)\] for all non-negative integers $x,y$.

2006 MOP Homework, 3
Tags: algebra unsolved , algebra
Let $a_{1},a_{2},...,a_{n}$ be positive real numbers with $a_{1}\leq a_{2}\leq ... a_{n}$ such that the arithmetic mean of $a_{1}^{2},...,a_{n}^{2}$ is 1. If the arithmetic mean of $a_{1}, a_{2},...,a_{n}$ is $m$. Prove that if $a_{i}\leq$ m for some $1 \leq i \leq n$, then $n(m-a_{i})^2\leq n-i$

1995 Irish Math Olympiad, 2
Tags: abstract algebra , trigonometry , algebra unsolved , algebra
Let $ a,b,c$ be complex numbers. Prove that if all the roots of the equation $ x^3\plus{}ax^2\plus{}bx\plus{}c\equal{}0$ are of module $ 1$, then so are the roots of the equation $ x^3\plus{}|a|x^2\plus{}|b|x\plus{}|c|\equal{}0$.

1991 IMTS, 2
Tags: algebra unsolved , algebra
Find the smallest positive integer, $n$, which can be expressed as the sum of distinct positive integers $a,b,c$ such that $a+b,a+c,b+c$ are perfect squares.

2005 China Girls Math Olympiad, 2
Tags: trigonometry , geometry , trig identities , Law of Sines , algebra unsolved , algebra
Find all ordered triples $ (x, y, z)$ of real numbers such that \[ 5 \left(x \plus{} \frac{1}{x} \right) \equal{} 12 \left(y \plus{} \frac{1}{y} \right) \equal{} 13 \left(z \plus{} \frac{1}{z} \right),\] and \[ xy \plus{} yz \plus{} zy \equal{} 1.\]

2006 Hong kong National Olympiad, 4
Tags: induction , algebra unsolved , algebra
Let $(a_n)_{n\ge 1}$ be a sequence of positive numbers. If there is a constant $M > 0$ such that $a_2^2 + a_2^2 +\ldots + a_n^2 < Ma_{n+1}^2$ for all $n$, then prove that there is a constant $M ' > 0$ such that $a_1 + a_2 +\ldots + a_n < M ' a_{n+1}$ .

2005 India IMO Training Camp, 3
Tags: function , trigonometry , inequalities , algebra unsolved , algebra
For real numbers $a,b,c,d$ not all equal to $0$ , define a real function $f(x) = a +b\cos{2x} + c\sin{5x} +d \cos{8x}$. Suppose $f(t) = 4a$ for some real $t$. prove that there exist a real number $s$ s.t. $f(s)<0$

2004 Austrian-Polish Competition, 5
Tags: induction , analytic geometry , algebra unsolved , algebra
Determine all $n$ for which the system with of equations can be solved in $\mathbb{R}$: \[\sum^{n}_{k=1} x_k = 27\] and \[\prod^{n}_{k=1} x_k = \left( \frac{3}{2} \right)^{24}.\]

2001 Cono Sur Olympiad, 3
Tags: function , induction , algebra unsolved , algebra
A function $g$ defined for all positive integers $n$ satisfies [list][*]$g(1) = 1$; [*]for all $n\ge 1$, either $g(n+1)=g(n)+1$ or $g(n+1)=g(n)-1$; [*]for all $n\ge 1$, $g(3n) = g(n)$; and [*]$g(k)=2001$ for some positive integer $k$.[/list] Find, with proof, the smallest possible value of $k$.

2004 All-Russian Olympiad, 1
Tags: algebra unsolved , algebra
A sequence of non-negative rational numbers $ a(1), a(2), a(3), \ldots$ satisfies $ a(m) \plus{} a(n) \equal{} a(mn)$ for arbitrary natural $ m$ and $ n$. Show that not all elements of the sequence can be distinct.

2010 Contests, 3
Tags: algebra , polynomial , algebra unsolved
Let $P(x)$ be a polynomial with integer coefficients and roots $1997$ and $2010$. Suppose further that $|P(2005)|<10$. Determine what integer values $P(2005)$ can get.

2010 Malaysia National Olympiad, 6
Tags: Diophantine equation , algebra unsolved , algebra
Find the smallest integer $k\ge3$ with the property that it is possible to choose two of the number $1,2,...,k$ in such a way that their product is equal to the sum of the remaining $k-2$ numbers.

1990 Vietnam Team Selection Test, 3
Tags: function , algebra unsolved , algebra
Prove that there is no real function $ f(x)$ satisfying $ f\left(f(x)\right) \equal{} x^2 \minus{} 2$ for all real number $ x$.

2002 Austrian-Polish Competition, 5
Tags: algebra , polynomial , modular arithmetic , algebra unsolved
Let $A$ be the set $\{2,7,11,13\}$. A polynomial $f$ with integer coefficients possesses the following property: for each integer $n$ there exists $p \in A$ such that $p|f(n)$. Prove that there exists $p \in A$ such that $p|f(n)$ for all integers $n$.

2012 Indonesia TST, 1
Tags: function , induction , floor function , algebra unsolved , algebra
Suppose a function $f : \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ satisfies $f(f(n)) + f(n+1) = n+2$ for all positive integer $n$. Prove that $f(f(n)+n) = n+1$ for all positive integer $n$.

1993 Turkey Team Selection Test, 6
Tags: function , algebra unsolved , algebra
Determine all functions $f: \mathbb{Q^+} \rightarrow \mathbb{Q^+}$ that satisfy: \[f\left(x+\frac{y}{x}\right) = f(x)+f\left(\frac{y}{x}\right)+2y \:\text{for all}\: x, y \in \mathbb{Q^+}\]

2009 Tuymaada Olympiad, 1
Tags: modular arithmetic , algebra unsolved , algebra
A magician asked a spectator to think of a three-digit number $ \overline{abc}$ and then to tell him the sum of numbers $ \overline{acb}$, $ \overline{bac}$, $ \overline{bca}$, $ \overline{cab}$, and $ \overline{cba}$. He claims that when he knows this sum he can determine the original number. Is that so?

1993 APMO, 3
Tags: algebra , polynomial , inequalities , ratio , algebra unsolved
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 $\frac{a}{c} \leq n+1$.

2000 IberoAmerican, 1
Tags: ratio , arithmetic sequence , geometric sequence , algebra unsolved , algebra
From an infinite arithmetic progression $ 1,a_1,a_2,\dots$ of real numbers some terms are deleted, obtaining an infinite geometric progression $ 1,b_1,b_2,\dots$ whose ratio is $ q$. Find all the possible values of $ q$.

1994 Polish MO Finals, 1
Tags: algebra , polynomial , Rational Root Theorem , algebra unsolved
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.

2000 Polish MO Finals, 1
Tags: trigonometry , function , algebra unsolved , algebra
Find number of solutions in non-negative reals to the following equations: \begin{eqnarray*}x_1 + x_n ^2 = 4x_n \\ x_2 + x_1 ^2 = 4x_1 \\ ... \\ x_n + x_{n-1}^2 = 4x_{n-1} \end{eqnarray*}

2005 All-Russian Olympiad, 3
Tags: inequalities , algebra unsolved , algebra
Given three reals $a_1,\,a_2,\,a_3>1,\,S=a_1+a_2+a_3$. Provided ${a_i^2\over a_i-1}>S$ for every $i=1,\,2,\,3$ prove that \[\frac{1}{a_1+a_2}+\frac{1}{a_2+a_3}+\frac{1}{a_3+a_1}>1.\]

1998 Vietnam Team Selection Test, 1
Tags: function , algebra , polynomial , limit , algebra unsolved
Let $f(x)$ be a real function such that for each positive real $c$ there exist a polynomial $P(x)$ (maybe dependent on $c$) such that $| f(x) - P(x)| \leq c \cdot x^{1998}$ for all real $x$. Prove that $f$ is a real polynomial.

2007 Hong Kong TST, 1
Tags: inequalities , conics , hyperbola , algebra unsolved , algebra
[url=http://www.mathlinks.ro/Forum/viewtopic.php?t=107262]IMO 2007 HKTST 1[/url] Problem 1 Let $p,q,r$ and $s$ be real numbers such that $p^{2}+q^{2}+r^{2}-s^{2}+4=0$. Find the maximum value of $3p+2q+r-4|s|$.