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

2006 Brazil National Olympiad, 6
Tags: algebra unsolved , geometric sequence , floor function
Professor Piraldo takes part in soccer matches with a lot of goals and judges a match in his own peculiar way. A match with score of $m$ goals to $n$ goals, $m\geq n$, is [i]tough[/i] when $m\leq f(n)$, where $f(n)$ is defined by $f(0) = 0$ and, for $n \geq 1$, $f(n) = 2n-f(r)+r$, where $r$ is the largest integer such that $r < n$ and $f(r) \leq n$. Let $\phi ={1+\sqrt 5\over 2}$. Prove that a match with score of $m$ goals to $n$, $m\geq n$, is tough if $m\leq \phi n$ and is not tough if $m \geq \phi n+1$.

2009 Kyrgyzstan National Olympiad, 4
Tags: algebra unsolved , algebra
Find all real $(x,y)$ such that $x + {y^2} = {y^3}$ $y + {x^2} = {x^3}$

2010 Contests, 2
Tags: function , induction , algebra , domain , functional equation , algebra unsolved
Find all the continuous functions $f : \mathbb{R} \mapsto\mathbb{R}$ such that $\forall x,y \in \mathbb{R}$, $(1+f(x)f(y))f(x+y)=f(x)+f(y)$.

2011 China Second Round Olympiad, 9
Tags: logarithms , algebra unsolved , algebra
Let $f(x)=|\log(x+1)|$ and let $a,b$ be two real numbers ($a<b$) satisfying the equations $f(a)=f\left(-\frac{b+1}{a+1}\right)$ and $f\left(10a+6b+21\right)=4\log 2$. Find $a,b$.

1997 Romania National Olympiad, 3
Tags: algebra , polynomial , algebra unsolved , function
Suppose that $a,b,c,d\in\mathbb{R}$ and $f(x)=ax^3+bx^2+cx+d$ such that $f(2)+f(5)<7<f(3)+f(4)$. Prove that there exists $u,v\in\mathbb{R}$ such that $u+v=7 , f(u)+f(v)=7$

1988 IMO Longlists, 93
Tags: algebra , polynomial , algebra unsolved
Given a natural number $n,$ find all polynomials $P(x)$ of degree less than $n$ satisfying the following condition \[ \sum^n_{i=0} P(i) \cdot (-1)^i \cdot \binom{n}{i} = 0. \]

2010 Albania Team Selection Test, 2
Tags: function , induction , algebra , domain , functional equation , algebra unsolved
Find all the continuous functions $f : \mathbb{R} \mapsto\mathbb{R}$ such that $\forall x,y \in \mathbb{R}$, $(1+f(x)f(y))f(x+y)=f(x)+f(y)$.

2005 China Northern MO, 2
Tags: function , inequalities , algebra unsolved , algebra
Let $f$ be a function from R to R. Suppose we have: (1) $f(0)=0$ (2) For all $x, y \in (-\infty, -1) \cup (1, \infty)$, we have $f(\frac{1}{x})+f(\frac{1}{y})=f(\frac{x+y}{1+xy})$. (3) If $x \in (-1,0)$, then $f(x) > 0$. Prove: $\sum_{n=1}^{+\infty} f(\frac{1}{n^2+7n+11}) > f(\frac12)$ with $n \in N^+$.

1992 China National Olympiad, 1
Tags: inequalities , triangle inequality , algebra unsolved , algebra
Let equation $x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\dots +a_1x+a_0=0$ with real coefficients satisfy $0<a_0\le a_1\le a_2\le \dots \le a_{n-1}\le 1$. Suppose that $\lambda$ ($|\lambda|>1$) is a complex root of the equation, prove that $\lambda^{n+1}=1$.

2014 Saudi Arabia IMO TST, 2
Tags: algebra unsolved , algebra
Let $S$ be a set of positive real numbers with five elements such that for any distinct $a,~ b,~ c$ in $S$, the number $ab + bc + ca$ is rational. Prove that for any $a$ and $b$ in $S$, $\tfrac{a}{b}$ is a rational number.

1999 Korea - Final Round, 1
Tags: function , algebra unsolved , algebra
If the equation: $f(\frac{x-3}{x+1}) + f(\frac{3+x}{1-x}) = x$ holds true for all real x but $\pm 1$, find $f(x)$.

2011 Puerto Rico Team Selection Test, 7
Tags: modular arithmetic , induction , algebra unsolved , algebra
Show that for any natural number n, n^3 + (n + 1)^3 + (n + 2)^3 is divisible by 9.

2004 Romania National Olympiad, 3
Tags: logarithms , function , inequalities , calculus , algebra unsolved , algebra
Let $n>2,n \in \mathbb{N}$ and $a>0,a \in \mathbb{R}$ such that $2^a + \log_2 a = n^2$. Prove that: \[ 2 \cdot \log_2 n>a>2 \cdot \log_2 n -\frac{1}{n} . \] [i]Radu Gologan[/i]

1996 Iran MO (3rd Round), 3
Tags: algebra unsolved , algebra
Find all sets of real numbers $\{a_1,a_2,\ldots, _{1375}\}$ such that \[2 \left( \sqrt{a_n - (n-1)}\right) \geq a_{n+1} - (n-1), \quad \forall n \in \{1,2,\ldots,1374\},\] and \[2 \left( \sqrt{a_{1375}-1374} \right) \geq a_1 +1.\]

2013 Argentina Cono Sur TST, 2
Tags: Columbia , algebra unsolved , algebra
If $ x\neq1$, $ y\neq1$, $ x\neq y$ and \[ \frac{yz\minus{}x^{2}}{1\minus{}x}\equal{}\frac{xz\minus{}y^{2}}{1\minus{}y}\] show that both fractions are equal to $ x\plus{}y\plus{}z$.

1989 IMO Longlists, 1
Tags: algebra unsolved , algebra
In the set $ S_n \equal{} \{1, 2,\ldots ,n\}$ a new multiplication $ a*b$ is defined with the following properties: [b](i)[/b] $ c \equal{} a * b$ is in $ S_n$ for any $ a \in S_n, b \in S_n.$ [b](ii)[/b] If the ordinary product $ a \cdot b$ is less than or equal to $ n,$ then $ a*b \equal{} a \cdot b.$ [b](iii)[/b] The ordinary rules of multiplication hold for $ *,$ i.e.: [b](1)[/b] $ a * b \equal{} b * a$ (commutativity) [b](2)[/b] $ (a * b) * c \equal{} a * (b * c)$ (associativity) [b](3)[/b] If $ a * b \equal{} a * c$ then $ b \equal{} c$ (cancellation law). Find a suitable multiplication table for the new product for $ n \equal{} 11$ and $ n \equal{} 12.$

2014 Vietnam National Olympiad, 2
Tags: algebra , polynomial , pigeonhole principle , algebra unsolved
Given the polynomial $P(x)=(x^2-7x+6)^{2n}+13$ where $n$ is a positive integer. Prove that $P(x)$ can't be written as a product of $n+1$ non-constant polynomials with integer coefficients.

2012 Indonesia TST, 1
Tags: function , algebra , polynomial , induction , algebra unsolved
Let $P$ be a polynomial with real coefficients. Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that there exists a real number $t$ such that \[f(x+t) - f(x) = P(x)\] for all $x \in \mathbb{R}$.

2001 China Western Mathematical Olympiad, 3
Tags: trigonometry , algebra unsolved , algebra
Find, with proof, all real numbers $ x \in \lbrack 0, \frac {\pi}{2} \rbrack$, such that $ (2 \minus{} \sin 2x)\sin (x \plus{} \frac {\pi}{4}) \equal{} 1$.

2017 Germany, Landesrunde - Grade 11/12, 6
Tags: algebra , algebra unsolved , system of equations
Find all pairs $(x,y)$ of real numbers that satisfy the system \begin{align*} x \cdot \sqrt{1-y^2} &=\frac14 \left(\sqrt3+1 \right), \\ y \cdot \sqrt{1-x^2} &= \frac14 \left( \sqrt3 -1 \right). \end{align*}

2006 Vietnam National Olympiad, 6
Tags: number theory , algebra unsolved , algebra
Let $S$ be a set of 2006 numbers. We call a subset $T$ of $S$ [i]naughty[/i] if for any two arbitrary numbers $u$, $v$ (not neccesary distinct) in $T$, $u+v$ is [i]not[/i] in $T$. Prove that 1) If $S=\{1,2,\ldots,2006\}$ every naughty subset of $S$ has at most 1003 elements; 2) If $S$ is a set of 2006 arbitrary positive integers, there exists a naughty subset of $S$ which has 669 elements.

2011 Kyrgyzstan National Olympiad, 8
Tags: algebra unsolved , algebra
Given a sequence $x_1,x_2,...,x_n$ of real numbers with ${x_{n + 1}}^3 = {x_n}^3 - 3{x_n}^2 + 3{x_n}$, where $(n=1,2,3,...)$. What must be value of $x_1$, so that $x_{100}$ and $x_{1000}$ becomes equal?

2000 Rioplatense Mathematical Olympiad, Level 3, 6
Tags: function , algebra , algebra unsolved
Let $g(x) = ax^2 + bx + c$ a quadratic function with real coefficients such that the equation $g(g(x)) = x$ has four distinct real roots. Prove that there isn't a function $f$: $R--R$ such that $f(f(x)) = g(x)$ for all $x$ real

1996 Canada National Olympiad, 1
Tags: algebra , polynomial , Vieta , algebra unsolved
If $\alpha$, $\beta$, and $\gamma$ are the roots of $x^3 - x - 1 = 0$, compute $\frac{1+\alpha}{1-\alpha} + \frac{1+\beta}{1-\beta} + \frac{1+\gamma}{1-\gamma}$.

1985 IMO Longlists, 17
Tags: limit , algebra unsolved , algebra
Set \[A_n=\sum_{k=1}^n \frac{k^6}{2^k}.\] Find $\lim_{n\to\infty} A_n.$