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: 1311

1991 Federal Competition For Advanced Students, P2, 2
Tags: function , modular arithmetic , algebra proposed , algebra
Find all functions $ f: \mathbb{Z} \minus{} \{ 0 \} \rightarrow \mathbb{Q}$ satisfying: $ f \left( \frac{x\plus{}y}{3} \right)\equal{}\frac {f(x)\plus{}f(y)}{2},$ whenever $ x,y,\frac{x\plus{}y}{3} \in \mathbb{Z} \minus{} \{ 0 \}.$

2008 District Olympiad, 3
Tags: function , parameterization , Euler , algebra proposed , algebra
For any real $ a$ define $ f_a : \mathbb{R} \rightarrow \mathbb{R}^2$ by the law $ f_a(t) \equal{} \left( \sin(t), \cos(at) \right)$. a) Prove that $ f_{\pi}$ is not periodic. b) Determine the values of the parameter $ a$ for which $ f_a$ is periodic. [b]Remark[/b]. L. Euler proved in $ 1737$ that $ \pi$ is irrational.

2008 China Team Selection Test, 3
Tags: algebra , polynomial , algebra proposed
Let $ n>m>1$ be odd integers, let $ f(x)\equal{}x^n\plus{}x^m\plus{}x\plus{}1$. Prove that $ f(x)$ can't be expressed as the product of two polynomials having integer coefficients and positive degrees.

2010 Baltic Way, 1
Tags: algebra , system of equations , algebra proposed
Find all quadruples of real numbers $(a,b,c,d)$ satisfying the system of equations \[\begin{cases}(b+c+d)^{2010}=3a\\ (a+c+d)^{2010}=3b\\ (a+b+d)^{2010}=3c\\ (a+b+c)^{2010}=3d\end{cases}\]

2013 Ukraine Team Selection Test, 9
Tags: function , algebra proposed , algebra
Determine all functions $f:\Bbb{R}\to\Bbb{R}$ such that \[ f^2(x+y)=f^2(x)+2f(xy)+f^2(y), \] for all $x,y\in \Bbb{R}.$

2011 Iran MO (3rd Round), 2
Tags: algebra , polynomial , algebra proposed
[b]a)[/b] Prove that for every natural numbers $n$ and $k$, we have monic polynomials of degree $n$, with integer coefficients like $A=\{P_1(x),.....,P_k(x)\}$ such that no two of them have a common factor and for every subset of $A$, the sum of elements of $A$ has all its roots real. [b]b)[/b] Are there infinitely many monic polynomial of degree $n$ with integer coefficients like $P_1(x),P_2(x),....$ such that no two of them have a common factor and the sum of a finite number of them has all it's roots real? [i]proposed by Mohammad Mansouri[/i]

2014 International Zhautykov Olympiad, 2
Tags: function , algebra , functional equation , algebra proposed
Does there exist a function $f: \mathbb R \to \mathbb R $ satisfying the following conditions: (i) for each real $y$ there is a real $x$ such that $f(x)=y$ , and (ii) $f(f(x)) = (x - 1)f(x) + 2$ for all real $x$ ? [i]Proposed by Igor I. Voronovich, Belarus[/i]

2007 Nicolae Coculescu, 1
Tags: function , algebra proposed , algebra
Let $w\in \mathbb{C}\setminus \mathbb{R}$, $|w|\neq 1$. Prove that $f\colon \mathbb{C} \to \mathbb{C}$, given by $f(z)= z+w\overline{z}$, is a bijection, and find its inverse.

1999 Hungary-Israel Binational, 1
Tags: algebra , polynomial , algebra proposed
$ f(x)$ is a given polynomial whose degree at least 2. Define the following polynomial-sequence: $ g_1(x)\equal{}f(x), g_{n\plus{}1}(x)\equal{}f(g_n(x))$, for all $ n \in N$. Let $ r_n$ be the average of $ g_n(x)$'s roots. If $ r_{19}\equal{}99$, find $ r_{99}$.

2004 Regional Olympiad - Republic of Srpska, 1
Tags: algebra proposed , algebra
Find all real solutions of the equation \[\sqrt[3]{x-1}+\sqrt[3]{3x-1}=\sqrt[3]{x+1}.\]

2006 Germany Team Selection Test, 1
Tags: trigonometry , percent , function , algebra proposed , algebra
Find all real solutions $x$ of the equation $\cos\cos\cos\cos x=\sin\sin\sin\sin x$. (Angles are measured in radians.)

2012 China Team Selection Test, 1
Tags: inequalities , complex numbers , triangle inequality , algebra proposed , algebra
Complex numbers ${x_i},{y_i}$ satisfy $\left| {{x_i}} \right| = \left| {{y_i}} \right| = 1$ for $i=1,2,\ldots ,n$. Let $x=\frac{1}{n}\sum\limits_{i=1}^n{{x_i}}$, $y=\frac{1}{n}\sum\limits_{i=1}^n{{y_i}}$ and $z_i=x{y_i}+y{x_i}-{x_i}{y_i}$. Prove that $\sum\limits_{i=1}^n{\left| {{z_i}}\right|}\leqslant n$.

2005 Iran MO (2nd round), 3
Tags: function , algebra proposed , algebra , functional equation
Find all functions $f:\mathbb{R}^{+}\to \mathbb{R}^{+}$ such that for all positive real numbers $x$ and $y$, the following equation holds: \[(x+y)f(f(x)y)=x^2f(f(x)+f(y)).\]

1986 India National Olympiad, 1
Tags: algebra proposed , algebra
A person who left home between 4 p.m. and 5 p.m. returned between 5 p.m. and 6 p.m. and found that the hands of his watch had exactly exchanged place, when did he go out ?

2016 Middle European Mathematical Olympiad, 2
Tags: algebra , functional equation , algebra proposed
Let $\mathbb{R}$ denote the set of the reals. Find all $f : \mathbb{R} \to \mathbb{R}$ such that $$ f(x)f(y) = xf(f(y-x)) + xf(2x) + f(x^2) $$ for all real $x, y$.

2014 Contests, 2
Tags: calculus , integration , arithmetic sequence , algebra proposed , algebra
The roots of the equation \[ x^3-3ax^2+bx+18c=0 \] form a non-constant arithmetic progression and the roots of the equation \[ x^3+bx^2+x-c^3=0 \] form a non-constant geometric progression. Given that $a,b,c$ are real numbers, find all positive integral values $a$ and $b$.

2008 Mongolia Team Selection Test, 1
Tags: function , induction , limit , algebra proposed , algebra
Find all function $ f: R^\plus{} \rightarrow R^\plus{}$ such that for any $ x,y,z \in R^\plus{}$ such that $ x\plus{}y \ge z$ , $ f(x\plus{}y\minus{}z) \plus{}f(2\sqrt{xz})\plus{}f(2\sqrt{yz}) \equal{} f(x\plus{}y\plus{}z)$

2014 Switzerland - Final Round, 3
Tags: function , induction , algebra proposed , algebra
Find all such functions $f :\mathbb{R}\to \mathbb{R}$ such that for all $x,y\in\mathbb{R}$ the following holds : \[ f(x^2)+f(xy)=f(x)f(y)+yf(x)+xf(x+y) \]

2003 Bundeswettbewerb Mathematik, 2
Tags: function , algebra , system of equations , algebra proposed
Find all triples $\left(x,\ y,\ z\right)$ of integers satisfying the following system of equations: $x^3-4x^2-16x+60=y$; $y^3-4y^2-16y+60=z$; $z^3-4z^2-16z+60=x$.

2000 Baltic Way, 19
Tags: inequalities , algebra proposed , algebra
Let $t\ge\frac{1}{2}$ be a real number and $n$ a positive integer. Prove that \[t^{2n}\ge (t-1)^{2n}+(2t-1)^n\]

2001 Romania National Olympiad, 1
Tags: algebra , polynomial , algebra proposed
Let $a$ and $b$ be complex non-zero numbers and $z_1,z_2$ the roots of the polynomials $X^2+aX+b$. Show that $|z_1+z_2|=|z_1|+|z_2|$ if and only if there exists a real number $\lambda\ge 4$ such that $a^2=\lambda b$.

2015 Iran Team Selection Test, 1
Tags: algebra , polynomial , algebra proposed
Find all polynomials $P,Q\in \Bbb{Q}\left [ x \right ]$ such that $$P(x)^3+Q(x)^3=x^{12}+1.$$

2009 Indonesia TST, 4
Tags: function , algebra proposed , algebra
Let $ S$ be the set of nonnegative real numbers. Find all functions $ f: S\rightarrow S$ which satisfy $ f(x\plus{}y\minus{}z)\plus{}f(2\sqrt{xz})\plus{}f(2\sqrt{yz})\equal{}f(x\plus{}y\plus{}z)$ for all nonnegative $ x,y,z$ with $ x\plus{}y\ge z$.

1974 IMO Longlists, 20
Tags: algebra proposed , algebra
For which natural numbers $n$ do there exist $n$ natural numbers $a_i\ (1\le i\le n)$ such that $\sum_{i=1}^n a_i^{-2}=1$?

2011 IMAC Arhimede, 1
Tags: function , algebra proposed , algebra
Find all functions $f: \mathbb{N} \rightarrow [0, +\infty)$ such that $f(1000)=10$ and $f(n+1)= \sum_{k=1}^n \frac{1}{f^2(k) + f(k)f(k+1) + f^2(k+1)}$ for all $n \in \mathbb{N}$. (Here, $f^2(i)$ means $(f(i))^2$.)