Found problems: 1269
2006 MOP Homework, 5
Let $\{a_n\}^{\inf}_{n=1}$ and $\{b_n\}^{\inf}_{n=1}$ be two sequences of real numbers such that $a_{n+1}=2b_n-a_n$ and $b_{n+1}=2a_n-b_n$ for every positive integer $n$. Prove that $a_n>0$ for all $n$, then $a_1=b_1$.
2010 Balkan MO Shortlist, A4
Let $n>2$ be a positive integer. Consider all numbers $S$ of the form
\begin{align*} S= a_1 a_2 + a_2 a_3 + \ldots + a_{k-1} a_k \end{align*}
with $k>1$ and $a_i$ begin positive integers such that $a_1+a_2+ \ldots + a_k=n$. Determine all the numbers that can be represented in the given form.
1989 IMO Longlists, 97
An arithmetic function is a real-valued function whose domain is the set of positive integers. Define the convolution product of two arithmetic functions $ f$ and $ g$ to be the arithmetic function $ f * g$, where \[ (f * g)(n) \equal{} \sum_{ij\equal{}n} f(i) \cdot g(j),\] and $ f^{*k} \equal{} f * f * \ldots * f$ ($ k$ times) We say that two arithmetic functions $ f$ and $ g$ are dependent if there exists a nontrivial polynomial of two variables $ P(x, y) \equal{} \sum_{i,j} a_{ij} x^i y^j$ with real coefficients such that
\[ P(f,g) \equal{} \sum_{i,j} a_{ij} f^{*i} * g^{*j} \equal{} 0,\]
and say that they are independent if they are not dependent. Let $ p$ and $ q$ be two distinct primes and set
\[ f_1(n) \equal{} \begin{cases} 1 & \text{ if } n \equal{} p, \\
0 & \text{ otherwise}. \end{cases}\]
\[ f_2(n) \equal{} \begin{cases} 1 & \text{ if } n \equal{} q, \\
0 & \text{ otherwise}. \end{cases}\]
Prove that $ f_1$ and $ f_2$ are independent.
1989 IMO Longlists, 70
Given that \[ \frac{\cos(x) \plus{} \cos(y) \plus{} \cos(z)}{\cos(x\plus{}y\plus{}z)} \equal{} \frac{\sin(x)\plus{} \sin(y) \plus{} \sin(z)}{\sin(x \plus{} y \plus{} z)} \equal{} a,\] show that \[ \cos(y\plus{}z) \plus{} \cos(z\plus{}x) \plus{} \cos(x\plus{}y) \equal{} a.\]
2005 MOP Homework, 6
Find all functions $f:\mathbb{Z} \rightarrow \mathbb{R}$ such that $f(1)=\tfrac{5}{2}$ and that \[f(x)f(y)=f(x+y)+f(x-y)\] for all integers $x$ and $y$.
2024 Euler Olympiad, Round 2, 2
Find all pairs of function $f : Q \rightarrow R$ and $g : Q \rightarrow R,$ for which equations
\begin{align*}
f(x+y) &= f(x) f(y) + g(x) g(y) \\
g(x+y) &= f(x)g(y) + g(x)f(y) + g(x)g(y)
\end{align*}
holds for all rational numbers $x$ and $y.$
[i]Proposed by Gurgen Asatryan, Armenia [/i]
2011 Poland - Second Round, 1
For $x,y\in\mathbb{R}$, solve the system of equations
\[ \begin{cases} (x-y)(x^3+y^3)=7 \\ (x+y)(x^3-y^3)=3 \end{cases} \]
2006 Bundeswettbewerb Mathematik, 2
Find all functions $f: Q^{+}\rightarrow R$ such that
$f(x)+f(y)+2xyf(xy)=\frac{f(xy)}{f(x+y)}$ for all $x,y\in Q^{+}$
2009 Rioplatense Mathematical Olympiad, Level 3, 1
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that
\[f(xy)=\max\{f(x+y),f(x) f(y)\} \]
for all real numbers $x$ and $y$.
2000 Tuymaada Olympiad, 3
Polynomial $ P(t)$ is such that for all real $ x$,
\[ P(\sin x) \plus{} P(\cos x) \equal{} 1.
\]
What can be the degree of this polynomial?
2012 South East Mathematical Olympiad, 1
Find a triple $(l, m, n)$ of positive integers $(1<l<m<n)$, such that $\sum_{k=1}^{l}k, \sum_{k=l+1}^{m}k, \sum_{k=m+1}^{n}k$ form a geometric sequence in order.
2014 Iran MO (3rd Round), 8
The polynomials $k_n(x_1, \ldots, x_n)$, where $n$ is a non-negative integer, satisfy the following conditions
\[k_0=1\]
\[k_1(x_1)=x_1\]
\[k_n(x_1, \ldots, x_n) = x_nk_{n-1}(x_1, \ldots , x_{n-1}) + (x_n^2+x_{n-1}^2)k_{n-2}(x_1,\ldots,x_{n-2})\]
Prove that for each non-negative $n$ we have $k_n(x_1,\ldots,x_n)=k_n(x_n,\ldots,x_1)$.
1998 Mediterranean Mathematics Olympiad, 2
Prove that the polynomial $z^{2n} + z^n + 1\ (n \in \mathbb{N})$ is divisible by the polynomial $z^2 + z + 1$ if and only if $n$ is not a multiple of $3$.
1991 Cono Sur Olympiad, 3
It is known that the number of real solutions of the following system if finite. Prove that this system has an even number of solutions:
$(y^2+6)(x-1)=y(x^2+1)$
$(x^2+6)(y-1)=x(y^2+1)$
1996 South africa National Olympiad, 6
The function $f$ is increasing and convex (i.e. every straight line between two points on the graph of $f$ lies above the graph) and satisfies $f(f(x))=3^x$ for all $x\in\mathbb{R}$. If $f(0)=0.5$ determine $f(0.75)$ with an error of at most $0.025$. The following are corrent to the number of digits given:
\[3^{0.25}=1.31607,\quad 3^{0.50}=1.73205,\quad 3^{0.75}=2.27951.\]
2015 IFYM, Sozopol, 7
Determine all polynomials $P(x)$ with real coefficients such that $(x+1)P(x-1)-(x-1)P(x)$ is a constant polynomial.
2010 Iran MO (2nd Round), 4
Let $P(x)=ax^3+bx^2+cx+d$ be a polynomial with real coefficients such that \[\min\{d,b+d\}> \max\{|{c}|,|{a+c}|\}\]
Prove that $P(x)$ do not have a real root in $[-1,1]$.
2008 Germany Team Selection Test, 3
Determine all functions $ f: \mathbb{R} \mapsto \mathbb{R}$ with $ x,y \in \mathbb{R}$ such that
\[ f(x \minus{} f(y)) \equal{} f(x\plus{}y) \plus{} f(y)\]
1999 Moldova Team Selection Test, 16
Define functions $f,g: \mathbb{R}\to \mathbb{R}$, $g$ is injective, satisfy:
\[f(g(x)+y)=g(f(y)+x)\]
2009 BMO TST, 4
Find all the polynomials $P(x)$ of a degree $\leq n$ with real non-negative coefficients such that $P(x) \cdot P(\frac{1}{x}) \leq [P(1)]^2$ , $ \forall x>0$.
2003 USA Team Selection Test, 4
Let $\mathbb{N}$ denote the set of positive integers. Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that \[ f(m+n)f(m-n) = f(m^2) \] for $m,n \in \mathbb{N}$.
2009 Turkey Team Selection Test, 1
For which $ p$ prime numbers, there is an integer root of the polynominal $ 1 \plus{} p \plus{} Q(x^1)\cdot\ Q(x^2)\ldots\ Q(x^{2p \minus{} 2})$ such that $ Q(x)$ is a polynominal with integer coefficients?
2010 Slovenia National Olympiad, 3
Find all functions $f: [0, +\infty) \to [0, +\infty)$ satisfying the equation
\[(y+1)f(x+y) = f\left(xf(y)\right)\]
For all non-negative real numbers $x$ and $y.$
2001 Tournament Of Towns, 1
In a certain country $10\%$ of the employees get $90\%$ of the total salary paid in this country. Supposing that the country is divided in several regions, is it possible that in every region the total salary of any 10% of the employees is no greater than $11\%$ of the total salary paid in this region?
2014 Contests, 1
Find the smallest possible value of the expression \[\left\lfloor\frac{a+b+c}{d}\right\rfloor+\left\lfloor\frac{b+c+d}{a}\right\rfloor+\left\lfloor\frac{c+d+a}{b}\right\rfloor+\left\lfloor\frac{d+a+b}{c}\right\rfloor\]
in which $a,~ b,~ c$, and $d$ vary over the set of positive integers.
(Here $\lfloor x\rfloor$ denotes the biggest integer which is smaller than or equal to $x$.)