Found problems: 1311
2005 District Olympiad, 2
Find the functions $f:\mathbb{Z}\times \mathbb{Z}\to\mathbb{R}$ such that
a) $f(x,y)\cdot f(y,z) \cdot f(z,x) = 1$ for all integers $x,y,z$;
b) $f(x+1,x)=2$ for all integers $x$.
1987 Iran MO (2nd round), 2
Find all continuous functions $f: \mathbb R \to \mathbb R$ such that
\[f(x^2-y^2)=f(x)^2 + f(y)^2, \quad \forall x,y \in \mathbb R.\]
1986 India National Olympiad, 4
Find the least natural number whose last digit is 7 such that it becomes 5 times larger when this last digit is carried to the beginning of the number.
1997 Pre-Preparation Course Examination, 3
Let $\omega_1,\omega_2, . . . ,\omega_k$ be distinct real numbers with a nonzero sum. Prove that there exist integers $n_1, n_2, . . . , n_k$ such that $\sum_{i=1}^k n_i\omega_i>0$, and for any non-identical permutation $\pi$ of $\{1, 2,\dots, k\}$ we have
\[\sum_{i=1}^k n_i\omega_{\pi(i)}<0.\]
2013 China Team Selection Test, 2
Find the greatest positive integer $m$ with the following property:
For every permutation $a_1, a_2, \cdots, a_n,\cdots$ of the set of positive integers, there exists positive integers $i_1<i_2<\cdots <i_m$ such that $a_{i_1}, a_{i_2}, \cdots, a_{i_m}$ is an arithmetic progression with an odd common difference.
2005 Taiwan TST Round 2, 1
Prove that for any quadratic polynomial $f(x)=x^2+px+q$ with integer coefficients, it is possible to find another polynomial $q(x)=2x^2+rx+s$ with integer coefficients so that \[\{f(x)|x \in \mathbb{Z} \} \cap \{g(x)|x \in \mathbb{Z} \} = \emptyset .\]
2000 JBMO ShortLists, 14
Let $m$ and $n$ be positive integers with $m\le 2000$ and $k=3-\frac{m}{n}$. Find the smallest positive value of $k$.
2013 Iran MO (3rd Round), 5
Prove that there is no polynomial $P \in \mathbb C[x]$ such that set $\left \{ P(z) \; | \; \left | z \right | =1 \right \}$ in complex plane forms a polygon. In other words, a complex polynomial can't map the unit circle to a polygon.
(30 points)
2006 CentroAmerican, 3
For every natural number $n$ we define \[f(n)=\left\lfloor n+\sqrt{n}+\frac{1}{2}\right\rfloor\] Show that for every integer $k \geq 1$ the equation \[f(f(n))-f(n)=k\] has exactly $2k-1$ solutions.
2010 Kazakhstan National Olympiad, 4
Let $x$- minimal root of equation $x^2-4x+2=0$.
Find two first digits of number $ \{x+x^2+....+x^{20} \}$ after $0$, where $\{a\}$- fractional part of $a$.
2005 USAMO, 2
Prove that the system \begin{align*}
x^6+x^3+x^3y+y & = 147^{157} \\
x^3+x^3y+y^2+y+z^9 & = 157^{147}
\end{align*} has no solutions in integers $x$, $y$, and $z$.
2010 IberoAmerican, 1
The arithmetic, geometric and harmonic mean of two distinct positive integers are different numbers. Find the smallest possible value for the arithmetic mean.
1950 Miklós Schweitzer, 1
Let $ \{k_n\}_{n \equal{} 1}^{\infty}$ be a sequence of real numbers having the properties $ k_1 > 1$ and $ k_1 \plus{} k_2 \plus{} \cdots \plus{} k_n < 2k_n$ for $ n \equal{} 1,2,...$. Prove that there exists a number $ q > 1$ such that $ k_n > q^n$ for every positive integer $ n$.
2005 Taiwan National Olympiad, 3
$f(x)=x^3-6x^2+17x$. If $f(a)=16, f(b)=20$, find $a+b$.
2012 ELMO Problems, 3
Let $f,g$ be polynomials with complex coefficients such that $\gcd(\deg f,\deg g)=1$. Suppose that there exist polynomials $P(x,y)$ and $Q(x,y)$ with complex coefficients such that $f(x)+g(y)=P(x,y)Q(x,y)$. Show that one of $P$ and $Q$ must be constant.
[i]Victor Wang.[/i]
2007 District Olympiad, 1
Three positive reals $x,y,z$ are given so that $xy=\frac{z-x+1}{y}=\frac{z+1}2.$ Prove that one of the numbers is the arithmetic mean of the other two.
2014 Iran MO (3rd Round), 5
We say $p(x,y)\in \mathbb{R}\left[x,y\right]$ is [i]good[/i] if for any $y \neq 0$ we have $p(x,y) = p\left(xy,\frac{1}{y}\right)$ . Prove that there are good polynomials $r(x,y) ,s(x,y)\in \mathbb{R}\left[x,y\right]$ such that for any good polynomial $p$ there is a $f(x,y)\in \mathbb{R}\left[x,y\right]$ such that \[f(r(x,y),s(x,y))= p(x,y)\]
[i]Proposed by Mohammad Ahmadi[/i]
1993 All-Russian Olympiad, 3
Quadratic trinomial $f(x)$ is allowed to be replaced by one of the trinomials $x^2f(1+\frac{1}{x})$ or $(x-1)^2f(\frac{1}{x-1})$. With the use of these operations, is it possible to go from $x^2+4x+3$ to $x^2+10x+9$?
2012 Turkmenistan National Math Olympiad, 7
If $a,b,c$ are positive real numbers and satisfy: $\frac{a_1}{b_1}=\frac{a_2}{b_2}=...=\frac{a_n}{b_n}$ then prove that :$ \sum_{i=1}^{n} a^{2}_i \cdot \sum_{i=1}^{n} b^{2}_i =(\sum_{i=1}^{n} a_{i}b_{i})^2$
2010 CentroAmerican, 5
If $p$, $q$ and $r$ are nonzero rational numbers such that $\sqrt[3]{pq^2}+\sqrt[3]{qr^2}+\sqrt[3]{rp^2}$ is a nonzero rational number, prove that
$\frac{1}{\sqrt[3]{pq^2}}+\frac{1}{\sqrt[3]{qr^2}}+\frac{1}{\sqrt[3]{rp^2}}$
is also a rational number.
1995 Baltic Way, 7
Prove that $\sin^318^{\circ}+\sin^218^{\circ}=\frac18$.
2013 Iran MO (3rd Round), 1
Let $a_0,a_1,\dots,a_n \in \mathbb N$. Prove that there exist positive integers $b_0,b_1,\dots,b_n$ such that for $0 \leq i \leq n : a_i \leq b_i \leq 2a_i$ and polynomial \[P(x) = b_0 + b_1 x + \dots + b_n x^n\] is irreducible over $\mathbb Q[x]$.
(10 points)
1974 IMO Longlists, 49
Determine an equation of third degree with integral coefficients having roots $\sin \frac{\pi}{14}, \sin \frac{5 \pi}{14}$ and $\sin \frac{-3 \pi}{14}.$
2000 Spain Mathematical Olympiad, 3
Show that there is no function $f : \mathbb N \to \mathbb N$ satisfying $f(f(n)) = n + 1$ for each positive integer $n.$
2006 Moldova MO 11-12, 6
Sequences $(x_n)_{n\ge1}$, $(y_n)_{n\ge1}$ satisfy the relations $x_n=4x_{n-1}+3y_{n-1}$ and $y_n=2x_{n-1}+3y_{n-1}$ for $n\ge1$. If $x_1=y_1=5$ find $x_n$ and $y_n$.
Calculate $\lim_{n\rightarrow\infty}\frac{x_n}{y_n}$.