Found problems: 1311
2012 Indonesia MO, 2
Let $\mathbb{R}^+$ be the set of all positive real numbers. Show that there is no function $f:\mathbb{R}^+ \to \mathbb{R}^+$ satisfying
\[f(x+y)=f(x)+f(y)+\dfrac{1}{2012}\]
for all positive real numbers $x$ and $y$.
[i]Proposer: Fajar Yuliawan[/i]
2005 Kazakhstan National Olympiad, 1
Does there exist a solution in real numbers of the system of equations
\[\left\{
\begin{array}{rcl}
(x - y)(z - t)(z - x)(z - t)^2 = A, \\
(y - z)(t - x)(t - y)(x - z)^2 = B,\\
(x - z)(y - t)(z - t)(y - z)^2 = C,\\
\end{array}
\right.\]
when
a) $A=2, B=8, C=6;$
b) $A=2, B=6, C=8.$?
2011 Baltic Way, 3
A sequence $a_1,a_2,a_3,\ldots $ of non-negative integers is such that $a_{n+1}$ is the last digit of $a_n^n+a_{n-1}$ for all $n>2$. Is it always true that for some $n_0$ the sequence $a_{n_0},a_{n_0+1},a_{n_0+2},\ldots$ is periodic?
2003 Bulgaria Team Selection Test, 2
Find all $f:R-R$ such that $f(x^2+y+f(y))=2y+f(x)^2$
2010 Contests, 2
Prove that for any real number $ x$ the following inequality is true:
$ \max\{|\sin x|, |\sin(x\plus{}2010)|\}>\dfrac1{\sqrt{17}}$
2006 Greece National Olympiad, 4
Does there exist a function $f : \mathbb{R} \rightarrow \mathbb{R}$, which satisfies both conditions :
[b]a)[/b] $f( x + y + z) \leq 3(xy + yz + zx)$ for all real numbers $x , y , z$
and
[b]b)[/b] there exist function $g$ and natural number $n$, such that
$g(g(x)) = x ^ {2n + 1}$ and $f(g(x)) = (g(x)) ^2$ for every real number $x$ ?
2024 Abelkonkurransen Finale, 2b
Find all functions $f:\mathbb{R} \to \mathbb{R}$ satisfying
\[xf(f(x)+y)=f(xy)+x^2\]
for all $x,y \in \mathbb{R}$.
2012 Pre - Vietnam Mathematical Olympiad, 2
Let $(a_n)$ defined by: $a_0=1, \; a_1=p, \; a_2=p(p-1)$, $a_{n+3}=pa_{n+2}-pa_{n+1}+a_n, \; \forall n \in \mathbb{N}$. Knowing that
(i) $a_n>0, \; \forall n \in \mathbb{N}$.
(ii) $a_ma_n>a_{m+1}a_{n-1}, \; \forall m \ge n \ge 0$.
Prove that $|p-1| \ge 2$.
2010 Iran MO (3rd Round), 3
prove that for each natural number $n$ there exist a polynomial with degree $2n+1$ with coefficients in $\mathbb{Q}[x]$ such that it has exactly $2$ complex zeros and it's irreducible in $\mathbb{Q}[x]$.(20 points)
1996 All-Russian Olympiad, 8
Goodnik writes 10 numbers on the board, then Nogoodnik writes 10 more numbers, all 20 of the numbers being positive and distinct. Can Goodnik choose his 10 numbers so that no matter what Nogoodnik writes, he can form 10 quadratic trinomials of the form $x^2 +px+q$, whose coeficients $p$ and $q$ run through all of the numbers written, such that the real roots of these trinomials comprise exactly 11 values?
[i]I. Rubanov[/i]
1950 Miklós Schweitzer, 3
For any system $ x_1,x_2,...,x_n$ of positive real numbers, let
$ f_1(x_1,x_2,...,x_n) \equal{} x_1$,
and
$ f_{\nu} \equal{} \frac {x_1 \plus{} 2x_2 \plus{} \cdots \plus{} \nu x_{\nu}}{\nu \plus{} (\nu \minus{} 1)x_1 \plus{} (\nu \minus{} 2)x_2 \plus{} \cdots \plus{} 1\cdot x_{\nu \minus{} 1}}$
for $ \nu \equal{} 2,3,...,n$. Show that for any $ \epsilon > 0$, a positive integer $ n_0 < n_0(\epsilon)$ can be found such that for every $ n > n_0$ there exists a system $ x_1',x_2',...,x_n'$ of positive real numbers
with $ x_1' \plus{} x_2' \plus{} \cdots \plus{} x_n' \equal{} 1$ and $ f_{\nu}(x_1',x_2',...,x_n')\le \epsilon$ for $ \nu \equal{} 1,2,...,n$ .
2012 Romania National Olympiad, 1
[color=darkred]Let $M=\{x\in\mathbb{C}\, |\, |z|=1,\ \text{Re}\, z\in\mathbb{Q}\}\, .$ Prove that there exist infinitely many equilateral triangles in the complex plane having all affixes of their vertices in the set $M$ .[/color]
2012 Pre - Vietnam Mathematical Olympiad, 2
Compute $\mathop {\lim }\limits_{n \to \infty } \left\{ {{{\left( {2 + \sqrt 3 } \right)}^n}} \right\}$
2010 Switzerland - Final Round, 6
Find all functions $ f: \mathbb{R}\mapsto\mathbb{R}$ such that for all $ x$, $ y$ $ \in\mathbb{R}$,
\[ f(f(x))\plus{}f(f(y))\equal{}2y\plus{}f(x\minus{}y)\]
holds.
2012 ELMO Problems, 4
Let $a_0,b_0$ be positive integers, and define $a_{i+1}=a_i+\lfloor\sqrt{b_i}\rfloor$ and $b_{i+1}=b_i+\lfloor\sqrt{a_i}\rfloor$ for all $i\ge0$. Show that there exists a positive integer $n$ such that $a_n=b_n$.
[i]David Yang.[/i]
2009 IberoAmerican, 2
Define the succession $ a_{n}$, $ n>0$ as $ n\plus{}m$, where $ m$ is the largest integer such that $ 2^{2^{m}}\leq n2^{n}$. Find all numbers that are not in the succession.
2008 Junior Balkan Team Selection Tests - Moldova, 1
Find all integers $ (x,y,z)$, satisfying equality:
$ x^2(y \minus{} z) \plus{} y^2(z \minus{} x) \plus{} z^2(x \minus{} y) \equal{} 2$
2011 ELMO Shortlist, 7
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]
2007 Tournament Of Towns, 3
Let $f(x)$ be a polynomial of nonzero degree. Can it happen that for any real number $a$, an even number of real numbers satisfy the equation $f(x) = a$?
2012 ELMO Shortlist, 8
Find all functions $f : \mathbb{Q} \to \mathbb{R}$ such that $f(x)f(y)f(x+y) = f(xy)(f(x) + f(y))$ for all $x,y\in\mathbb{Q}$.
[i]Sammy Luo and Alex Zhu.[/i]
2009 India IMO Training Camp, 9
Let
$ f(x)\equal{}\sum_{k\equal{}1}^n a_k x^k$ and $ g(x)\equal{}\sum_{k\equal{}1}^n \frac{a_k x^k}{2^k \minus{}1}$ be two polynomials with real coefficients.
Let g(x) have $ 0,2^{n\plus{}1}$ as two of its roots. Prove That $ f(x)$ has a positive root less than $ 2^n$.
2013 China National Olympiad, 2
Find all nonempty sets $S$ of integers such that $3m-2n \in S$ for all (not necessarily distinct) $m,n \in S$.
2007 France Team Selection Test, 2
Find all functions $f: \mathbb{Z}\rightarrow\mathbb{Z}$ such that for all $x,y \in \mathbb{Z}$:
\[f(x-y+f(y))=f(x)+f(y).\]
2009 China Team Selection Test, 2
Find all complex polynomial $ P(x)$ such that for any three integers $ a,b,c$ satisfying $ a \plus{} b \plus{} c\not \equal{} 0, \frac{P(a) \plus{} P(b) \plus{} P(c)}{a \plus{} b \plus{} c}$ is an integer.
2013 European Mathematical Cup, 1
In each field of a table there is a real number. We call such $n \times n$ table [i]silly[/i] if each entry equals the product of all the numbers in the neighbouring fields.
a) Find all $2 \times 2$ silly tables.
b) Find all $3 \times 3$ silly tables.