Found problems: 1311
2013 Bulgaria National Olympiad, 2
Find all $f : \mathbb{R}\to \mathbb{R}$ , bounded in $(0,1)$ and satisfying:
$x^2 f(x) - y^2 f(y) = (x^2-y^2) f(x+y) -xy f(x-y)$
for all $x,y \in \mathbb{R}$
[i]Proposed by Nikolay Nikolov[/i]
2008 Kazakhstan National Olympiad, 3
Let $ f(x,y,z)$ be the polynomial with integer coefficients. Suppose that for all reals $ x,y,z$ the following equation holds:
\[ f(x,y,z) \equal{} \minus{} f(x,z,y) \equal{} \minus{} f(y,x,z) \equal{} \minus{} f(z,y,x)
\]
Prove that if $ a,b,c\in\mathbb{Z}$ then $ f(a,b,c)$ takes an even value
2008 Macedonia National Olympiad, 1
Find all injective functions $ f : \mathbb{N} \to \mathbb{N}$ which satisfy
\[ f(f(n)) \le\frac{n \plus{} f(n)}{2}\]
for each $ n \in \mathbb{N}$.
2011 Benelux, 3
If $k$ is an integer, let $\mathrm{c}(k)$ denote the largest cube that is less than or equal to $k$. Find all positive integers $p$ for which the following sequence is bounded:
$a_0 = p$ and $a_{n+1} = 3a_n-2\mathrm{c}(a_n)$ for $n \geqslant 0$.
1979 IMO Longlists, 21
Let $E$ be the set of all bijective mappings from $\mathbb R$ to $\mathbb R$ satisfying
\[f(t) + f^{-1}(t) = 2t, \qquad \forall t \in \mathbb R,\]
where $f^{-1}$ is the mapping inverse to $f$. Find all elements of $E$ that are monotonic mappings.
1988 Romania Team Selection Test, 13
Let $a$ be a positive integer. The sequence $\{x_n\}_{n\geq 1}$ is defined by $x_1=1$, $x_2=a$ and $x_{n+2} = ax_{n+1} + x_n$ for all $n\geq 1$. Prove that $(y,x)$ is a solution of the equation \[ |y^2 - axy - x^2 | = 1 \] if and only if there exists a rank $k$ such that $(y,x)=(x_{k+1},x_k)$.
[i]Serban Buzeteanu[/i]
1977 IMO Longlists, 21
Given that $x_1+x_2+x_3=y_1+y_2+y_3=x_1y_1+x_2y_2+x_3y_3=0,$ prove that:
\[ \frac{x_1^2}{x_1^2+x_2^2+x_3^2}+\frac{y_1^2}{y_1^2+y_2^2+y_3^2}=\frac{2}{3}\]
2009 IberoAmerican, 5
Consider the sequence $ \{a_n\}_{n\geq1}$ defined as follows: $ a_1 \equal{} 1$, $ a_{2k} \equal{} 1 \plus{} a_k$ and $ a_{2k \plus{} 1} \equal{} \frac {1}{a_{2k}}$ for every $ k\geq 1$. Prove that every positive rational number appears on the sequence $ \{a_n\}$ exactly once.
2006 All-Russian Olympiad, 3
Given a circle and $2006$ points lying on this circle. Albatross colors these $2006$ points in $17$ colors. After that, Frankinfueter joins some of the points by chords such that the endpoints of each chord have the same color and two different chords have no common points (not even a common endpoint). Hereby, Frankinfueter intends to draw as many chords as possible, while Albatross is trying to hinder him as much as he can. What is the maximal number of chords Frankinfueter will always be able to draw?
2002 Bosnia Herzegovina Team Selection Test, 3
Let $p$ and $q$ be different prime numbers. Solve the following system in integers:
\[\frac{z+ p}x+\frac{z-p}y= q,\\ \frac{z+ p}y -\frac{z-p}x= q.\]
2010 ELMO Shortlist, 4
Let $-2 < x_1 < 2$ be a real number and define $x_2, x_3, \ldots$ by $x_{n+1} = x_n^2-2$ for $n \geq 1$. Assume that no $x_n$ is $0$ and define a number $A$, $0 \leq A \leq 1$ in the following way: The $n^{\text{th}}$ digit after the decimal point in the binary representation of $A$ is a $0$ if $x_1x_2\cdots x_n$ is positive and $1$ otherwise. Prove that $A = \frac{1}{\pi}\cos^{-1}\left(\frac{x_1}{2}\right)$.
[i]Evan O' Dorney.[/i]
2024 German National Olympiad, 1
The five real numbers $v,w,x,y,s$ satisfy the system of equations
\begin{align*}
v&=wx+ys,\\
v^2&=w^2x+y^2s,\\
v^3&=w^3x+y^3s.
\end{align*}
Show that at least two of them are equal.
2005 Iran MO (3rd Round), 3
Find all $\alpha>0$ and $\beta>0$ that for each $(x_1,\dots,x_n)$ and $(y_1,\dots,y_n)\in\mathbb {R^+}^n$ that:\[(\sum x_i^\alpha)(\sum y_i^\beta)\geq\sum x_iy_i\]
2011 CentroAmerican, 5
If $x$, $y$, $z$ are positive numbers satisfying
\[x+\frac{y}{z}=y+\frac{z}{x}=z+\frac{x}{y}=2.\]
Find all the possible values of $x+y+z$.
2014 Contests, 2
Find all functions $f:\mathbb{R}\backslash\{0\}\rightarrow\mathbb{R}$ for which $xf(xy) + f(-y) = xf(x)$ for all non-zero real numbers $x, y$.
2011 Iran MO (3rd Round), 8
We call the sequence $d_1,....,d_n$ of natural numbers, not necessarily distinct, [b]covering[/b] if there exists arithmetic progressions like $c_1+kd_1$,....,$c_n+kd_n$ such that every natural number has come in at least one of them. We call this sequence [b]short[/b] if we can not delete any of the $d_1,....,d_n$ such that the resulting sequence be still covering.
[b]a)[/b] Suppose that $d_1,....,d_n$ is a short covering sequence and suppose that we've covered all the natural numbers with arithmetic progressions $a_1+kd_1,.....,a_n+kd_n$, and suppose that $p$ is a prime number that $p$ divides $d_1,....,d_k$ but it does not divide $d_{k+1},....,d_n$. Prove that the remainders of $a_1,....,a_k$ modulo $p$ contains all the numbers $0,1,.....,p-1$.
[b]b)[/b] Write anything you can about covering sequences and short covering sequences in the case that each of $d_1,....,d_n$ has only one prime divisor.
[i]proposed by Ali Khezeli[/i]
2001 Turkey Team Selection Test, 3
Show that there is no continuous function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for every real number $x$
\[f(x-f(x)) = \dfrac x2.\]
2004 Regional Olympiad - Republic of Srpska, 3
Determine all pairs of positive integers $(a,b)$, such that the roots of the equations \[x^2-ax+a+b-3=0,\]
\[x^2-bx+a+b-3=0,\] are also positive integers.
2007 Peru IMO TST, 1
Let $k$ be a positive number and $P$ a Polynomio with integer coeficients.
Prove that exists a $n$ positive integer such that:
$P(1)+P(2)+\dots+P(N)$ is divisible by $k$.
1979 IMO Longlists, 8
The sequence $(a_n)$ of real numbers is defined as follows:
\[a_1=1, \qquad a_2=2, \quad \text{and} \quad a_n=3a_{n-1}-a_{n-2} , \ \ n \geq 3.\]
Prove that for $n \geq 3$, $a_n=\left[ \frac{a_{n-1}^2}{a_{n-2}} \right] +1$, where $[x]$ denotes the integer $p$ such that $p \leq x < p + 1$.
2007 China Team Selection Test, 1
Find all functions $ f: \mathbb{Q}^{\plus{}} \mapsto \mathbb{Q}^{\plus{}}$ such that:
\[ f(x) \plus{} f(y) \plus{} 2xy f(xy) \equal{} \frac {f(xy)}{f(x\plus{}y)}.\]
2006 Kyiv Mathematical Festival, 3
See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url]
Find all positive integers $a, b, c$ such that $3abc+11(a+b+c)=6(ab+bc+ac)+18.$
2010 District Olympiad, 4
Determine all the functions $ f: \mathbb{N}\rightarrow \mathbb{N}$ such that
\[ f(n)\plus{}f(n\plus{}1)\plus{}f(f(n))\equal{}3n\plus{}1, \quad \forall n\in \mathbb{N}.\]
2005 Iran MO (3rd Round), 2
Suppose $\{x_n\}$ is a decreasing sequence that $\displaystyle\lim_{n \rightarrow\infty}x_n=0$. Prove that $\sum(-1)^nx_n$ is convergent
2011 Romania Team Selection Test, 1
Given a positive integer number $k$, define the function $f$ on the set of all positive integer numbers to itself by
\[f(n)=\begin{cases}1, &\text{if }n\le k+1\\ f(f(n-1))+f(n-f(n-1)), &\text{if }n>k+1\end{cases}\]
Show that the preimage of every positive integer number under $f$ is a finite non-empty set of consecutive positive integers.