Found problems: 1311
2011 Kosovo National Mathematical Olympiad, 1
The complex numbers $z_1$ and $z_2$ are given such that $z_1=-1+i$ and $z_2=2+4i$. Find the complex number $z_3$ such that $z_1,z_2,z_3$ are the points of an equilateral triangle. How many solutions do we have ?
1987 Romania Team Selection Test, 11
Let $P(X,Y)=X^2+2aXY+Y^2$ be a real polynomial where $|a|\geq 1$. For a given positive integer $n$, $n\geq 2$ consider the system of equations: \[ P(x_1,x_2) = P(x_2,x_3) = \ldots = P(x_{n-1},x_n) = P(x_n,x_1) = 0 . \] We call two solutions $(x_1,x_2,\ldots,x_n)$ and $(y_1,y_2,\ldots,y_n)$ of the system to be equivalent if there exists a real number $\lambda \neq 0$, $x_1=\lambda y_1$, $\ldots$, $x_n= \lambda y_n$. How many nonequivalent solutions does the system have?
[i]Mircea Becheanu[/i]
2013 Romania National Olympiad, 4
Consider a nonzero integer number $n$ and the function $f:\mathbb{N}\to \mathbb{N}$ by
\[ f(x) = \begin{cases}
\frac{x}{2} & \text{if } x \text{ is even} \\
\frac{x-1}{2} + 2^{n-1} & \text{if } x \text{ is odd}
\end{cases}.
\] Determine the set: \[
A = \{ x\in \mathbb{N} \mid \underbrace{\left( f\circ f\circ ....\circ f \right)}_{n\ f\text{'s}}\left( x \right)=x \}.
\]
1992 Hungary-Israel Binational, 2
We examine the following two sequences: The Fibonacci sequence: $F_{0}= 0, F_{1}= 1, F_{n}= F_{n-1}+F_{n-2 }$ for $n \geq 2$; The Lucas sequence: $L_{0}= 2, L_{1}= 1, L_{n}= L_{n-1}+L_{n-2}$ for $n \geq 2$. It is known that for all $n \geq 0$ \[F_{n}=\frac{\alpha^{n}-\beta^{n}}{\sqrt{5}},L_{n}=\alpha^{n}+\beta^{n},\] where $\alpha=\frac{1+\sqrt{5}}{2},\beta=\frac{1-\sqrt{5}}{2}$. These formulae can be used without proof.
Prove that \[\sum_{k=1}^{n}[\alpha^{k}F_{k}+\frac{1}{2}]=F_{2n+1}\; \forall n>1.\]
2010 Baltic Way, 5
Let $\mathbb{R}$ denote the set of real numbers. Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that
\[f(x^2)+f(xy)=f(x)f(y)+yf(x)+xf(x+y)\]
for all $x,y\in\mathbb{R}$.
1995 All-Russian Olympiad Regional Round, 11.6
The sequence $ a_n$ satisfies $ a_{m\plus{}n}\plus{} a_{m\minus{}n}\equal{}\frac12(a_{2m}\plus{}a_{2n})$ for all $ m\geq n\geq 0$. If $ a_1\equal{}1$, find $ a_{1995}$.
1994 Canada National Olympiad, 2
Prove that $(\sqrt{2}-1)^n$ $\forall n\in \mathbb{Z}^{+}$ can be represented as $\sqrt{m}-\sqrt{m-1}$ for some $m\in \mathbb{Z}^{+}$.
2004 Singapore Team Selection Test, 3
Find all functions $ f: \mathbb{R} \to \mathbb{R}$ satisfying
\[ f\left(\frac {x \plus{} y}{x \minus{} y}\right) \equal{} \frac {f\left(x\right) \plus{} f\left(y\right)}{f\left(x\right) \minus{} f\left(y\right)}
\]
for all $ x \neq y$.
1992 Baltic Way, 11
Let $ Q^\plus{}$ denote the set of positive rational numbers. Show that there exists one and only one function $f: Q^\plus{}\to Q^\plus{}$ satisfying the following conditions:
(i) If $ 0<q<1/2$ then $ f(q)\equal{}1\plus{}f(q/(1\minus{}2q))$,
(ii) If $ 1<q\le2$ then $ f(q)\equal{}1\plus{}f(q\minus{}1)$,
(iii) $ f(q)\cdot f(1/q)\equal{}1$ for all $ q\in Q^\plus{}$.
2016 Vietnam National Olympiad, 1
Solve the system of equations $\begin{cases}6x-y+z^2=3\\ x^2-y^2-2z=-1\quad\quad (x,y,z\in\mathbb{R}.)\\ 6x^2-3y^2-y-2z^2=0\end{cases}$.
1985 Iran MO (2nd round), 2
Let $x, y$ and $z$ be three positive real numbers for which
\[x^2+y^2+z^2=xy+yz+zx.\]
Find the value of $\frac{\sqrt x}{\sqrt x + \sqrt y+ \sqrt z}.$
2000 Irish Math Olympiad, 1
Prove that if $ x,y$ are nonnegative real numbers with $ x\plus{}y\equal{}2$, then: $ x^2 y^2 (x^2\plus{}y^2) \le 2$.
1987 IberoAmerican, 1
Find the function $f(x)$ such that
\[f(x)^2f\left(\frac{1-x}{x+1}\right) =64x \]
for $x\not=0,x\not=1,x\not=-1$.
2008 Stars Of Mathematics, 1
Let $ P(x) \in \mathbb{Z}[x]$ be a polynomial of degree $ \text{deg} P \equal{} n > 1$. Determine the largest number of consecutive integers to be found in $ P(\mathbb{Z})$.
[i]B. Berceanu[/i]
2011 ELMO Problems, 3
Determine whether there exist two reals $x,y$ and a sequence $\{a_n\}_{n=0}^{\infty}$ of nonzero reals such that $a_{n+2}=xa_{n+1}+ya_n$ for all $n\ge0$ and for every positive real number $r$, there exist positive integers $i,j$ such that $|a_i|<r<|a_j|$.
[i]Alex Zhu.[/i]
1995 Taiwan National Olympiad, 1
Let $P(x)=a_{0}+a_{1}x+...+a_{n}x^{n}\in\mathbb{C}[x]$ , where $a_{n}=1$. The roots of $P(x)$ are $b_{1},b_{2},...,b_{n}$, where $|b_{1}|,|b_{2}|,...,|b_{j}|>1$ and $|b_{j+1}|,...,|b_{n}|\leq 1$. Prove that $\prod_{i=1}^{j}|b_{i}|\leq\sqrt{|a_{0}|^{2}+|a_{1}|^{2}+...+|a_{n}|^{2}}$.
2006 Pre-Preparation Course Examination, 4
If $d\in \mathbb{Q}$, is there always an $\omega \in \mathbb{C}$ such that $\omega ^n=1$ for some $n\in \mathbb{N}$ and $\mathbb{Q}(\sqrt{d})\subseteq \mathbb{Q}(\omega)$?
2013 Brazil National Olympiad, 3
Find all injective functions $f\colon \mathbb{R}^* \to \mathbb{R}^* $ from the non-zero reals to the non-zero reals, such that \[f(x+y) \left(f(x) + f(y)\right) = f(xy)\] for all non-zero reals $x, y$ such that $x+y \neq 0$.
2000 Turkey MO (2nd round), 3
Find all continuous functions $f:[0,1]\to [0,1]$ for which there exists a positive integer $n$ such that $f^{n}(x)=x$ for $x \in [0,1]$ where $f^{0} (x)=x$ and $f^{k+1}=f(f^{k}(x))$ for every positive integer $k$.
2002 All-Russian Olympiad Regional Round, 9.2
A monic quadratic polynomial $f$ with integer coefficients attains prime values at three consecutive integer points.show that it attains a prime value at some other integer point as well.
2010 Federal Competition For Advanced Students, Part 1, 2
For a positive integer $n$, we define the function $f_n(x)=\sum_{k=1}^n |x-k|$ for all real numbers $x$. For any two-digit number $n$ (in decimal representation), determine the set of solutions $\mathbb{L}_n$ of the inequality $f_n(x)<41$.
[i](41st Austrian Mathematical Olympiad, National Competition, part 1, Problem 2)[/i]
2008 Pan African, 1
Determine all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying $f(x+y)\le f(x)+f(y)\le x+y$ for all $x,y\in\mathbb{R}$.
2006 All-Russian Olympiad, 7
Given a quadratic trinomial $f\left(x\right)=x^2+ax+b$. Assume that the equation $f\left(f\left(x\right)\right)=0$ has four different real solutions, and that the sum of two of these solutions is $-1$. Prove that $b\leq -\frac14$.
1994 Turkey Team Selection Test, 1
$f$ is a function defined on integers and satisfies $f(x)+f(x+3)=x^2$ for every integer $x$. If $f(19)=94$, then calculate $f(94)$.
2007 China Team Selection Test, 2
Find all positive integers $ n$ such that there exists sequence consisting of $ 1$ and $ - 1: a_{1},a_{2},\cdots,a_{n}$ satisfying $ a_{1}\cdot1^2 + a_{2}\cdot2^2 + \cdots + a_{n}\cdot n^2 = 0.$