Found problems: 4776
1995 Turkey MO (2nd round), 6
Find all surjective functions $f: \mathbb{N}\rightarrow \mathbb{N}$ such that for all $m,n\in \mathbb{N}$ \[f(m)\mid f(n) \mbox{ if and only if }m\mid n.\]
2002 Moldova National Olympiad, 3
Let $ a,b> 0$ such that $ a\ne b$. Prove that:
$ \sqrt {ab} < \dfrac{a \minus{} b}{\ln a \minus{} \ln b} < \dfrac{a \plus{} b}{2}$
2010 District Olympiad, 2
Consider the matrix $ A,B\in \mathcal l{M}_3(\mathbb{C})$ with $ A=-^tA$ and $ B=^tB$. Prove that if the polinomial function defined by
\[ f(x)=\det(A+xB)\]
has a multiple root, then $ \det(A+B)=\det B$.
2011 Spain Mathematical Olympiad, 2
Each rational number is painted either white or red. Call such a coloring of the rationals [i]sanferminera[/i] if for any distinct rationals numbers $x$ and $y$ satisfying one of the following three conditions: [list=1][*]$xy=1$,
[*]$x+y=0$,
[*]$x+y=1$,[/list]we have $x$ and $y$ painted different colors. How many sanferminera colorings are there?
2004 France Team Selection Test, 1
Let $n$ be a positive integer, and $a_1,...,a_n, b_1,..., b_n$ be $2n$ positive real numbers such that
$a_1 + ... + a_n = b_1 + ... + b_n = 1$.
Find the minimal value of
$ \frac {a_1^2} {a_1 + b_1} + \frac {a_2^2} {a_2 + b_2} + ...+ \frac {a_n^2} {a_n + b_n}$.
2015 Romania National Olympiad, 1
Find all differentiable functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that verify the conditions:
$ \text{(i)}\quad\forall x\in\mathbb{Z} \quad f'(x) =0 $
$ \text{(ii)}\quad\forall x\in\mathbb{R}\quad f'(x)=0\implies f(x)=0 $
2007 F = Ma, 3
The coordinate of an object is given as a function of time by $x = 8t - 3t^2$, where $x$ is in meters and $t$ is in seconds. Its average velocity over the interval from $ t = 1$ to $t = 2 \text{ s}$ is
$ \textbf{(A)}\ -2\text{ m/s}\qquad\textbf{(B)}\ -1\text{ m/s}\qquad\textbf{(C)}\ -0.5\text{ m/s}\qquad\textbf{(D)}\ 0.5\text{ m/s}\qquad\textbf{(E)}\ 1\text{ m/s} $
2016 Nordic, 3
Find all $a\in\mathbb R$ for which there exists a function $f\colon\mathbb R\rightarrow\mathbb R$, such that
(i) $f(f(x))=f(x)+x$, for all $x\in\mathbb R$,
(ii) $f(f(x)-x)=f(x)+ax$, for all $x\in\mathbb R$.
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 $x,y\in\mathbb{R}$.
2007 Junior Balkan Team Selection Tests - Romania, 1
Find the positive integers $n$ with $n \geq 4$ such that $[\sqrt{n}]+1$ divides $n-1$ and $[\sqrt{n}]-1$ divides $n+1$.
[hide="Remark"]This problem can be solved in a similar way with the one given at [url=http://www.mathlinks.ro/Forum/resources.php?c=1&cid=97&year=2006]Cono Sur Olympiad 2006[/url], problem 5.[/hide]
1969 IMO, 2
Let $f(x)=\cos(a_1+x)+{1\over2}\cos(a_2+x)+{1\over4}\cos(a_3+x)+\ldots+{1\over2^{n-1}}\cos(a_n+x)$, where $a_i$ are real constants and $x$ is a real variable. If $f(x_1)=f(x_2)=0$, prove that $x_1-x_2$ is a multiple of $\pi$.
2014 Contests, 3
Let $a\# b$ be defined as $ab-a-3$. For example, $4\#5=20-4-3=13$ Compute $(2\#0)\#(1\#4)$.
2019 Belarus Team Selection Test, 1.1
Does there exist a function $f:\mathbb N\to\mathbb N$ such that
$$
f(f(n+1))=f(f(n))+2^{n-1}
$$
for any positive integer $n$? (As usual, $\mathbb N$ stands for the set of positive integers.)
[i](I. Gorodnin)[/i]
1997 Iran MO (3rd Round), 1
Find all strictly ascending functions $f$ such that for all $x\in \mathbb R$,
\[f(1-x)=1-f(f(x)).\]
1995 IMC, 10
a) Prove that for every $\epsilon>0$ there is a positive integer $n$ and real
numbers $\lambda_{1},\dots,\lambda_{n}$ such that
$$\max_{x\in [-1,1]}|x-\sum_{k=1}^{n}\lambda_{k}x^{2k+1}|<\epsilon.$$
b) Prove that for every odd continuous function $f$ on $[-1,1]$ and for every $\epsilon>0$ there is a positive integer $n$ and real numbers $\mu_{1},\dots,\mu_{n}$ such that
$$\max_{x\in [-1,1]}|f(x)-\sum_{k=1}^{n}\mu_{k}x^{2k+1}|<\epsilon.$$
2008 Vietnam Team Selection Test, 3
Consider the set $ M = \{1,2, \ldots ,2008\}$. Paint every number in the set $ M$ with one of the three colors blue, yellow, red such that each color is utilized to paint at least one number. Define two sets:
$ S_1=\{(x,y,z)\in M^3\ \mid\ x,y,z\text{ have the same color and }2008 | (x + y + z)\}$;
$ S_2=\{(x,y,z)\in M^3\ \mid\ x,y,z\text{ have three pairwisely different colors and }2008 | (x + y + z)\}$.
Prove that $ 2|S_1| > |S_2|$ (where $ |X|$ denotes the number of elements in a set $ X$).
1989 IMO Longlists, 15
A sequence $ a_1, a_2, a_3, \ldots$ is defined recursively by $ a_1 \equal{} 1$ and $ a_{2^k\plus{}j} \equal{} \minus{}a_j$ $ (j \equal{} 1, 2, \ldots, 2^k).$ Prove that this sequence is not periodic.
1950 Miklós Schweitzer, 6
Consider an arc of a planar curve; let the radius of curvature at any point of the arc be a differentiable function of the arc length and its derivative be everywhere different from zero; moreover, let the total curvature be less than $ \frac{\pi}{2}$. Let $ P_1,P_2,P_3,P_4,P_5$ and $ P_6$ be any points on this arc, subject to the only condition that the radius of curvature at $ P_k$ is greater than at $ P_j$ if $ j<k$.
Prove that the radius of the circle passing through the points $ P_1,P_3$ and $ P_5$ is less than the radius of the circle through $ P_2,P_4$ and $ P_6$
1967 IMO Shortlist, 3
The function $\varphi(x,y,z)$ defined for all triples $(x,y,z)$ of real numbers, is such that there are two functions $f$ and $g$ defined for all pairs of real numbers, such that
\[\varphi(x,y,z) = f(x+y,z) = g(x,y+z)\]
for all real numbers $x,y$ and $z.$ Show that there is a function $h$ of one real variable, such that
\[\varphi(x,y,z) = h(x+y+z)\]
for all real numbers $x,y$ and $z.$
2011 Moldova Team Selection Test, 2
Let $x_1, x_2, \ldots, x_n$ be real positive numbers such that $x_1\cdot x_2\cdots x_n=1$. Prove the inequality
$\frac1{x_1(x_1+1)}+\frac1{x_2(x_2+1)}+\cdots+\frac1{x_n(x_n+1)}\geq\frac n2$
2023 Irish Math Olympiad, P4
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ with the property that
$$f(x)f(y) = (xy - 1)^2f\left(\frac{x + y - 1}{xy - 1}\right)$$
for all real numbers $x, y$ with $xy \neq 1$.
1970 Miklós Schweitzer, 9
Construct a continuous function $ f(x)$, periodic with period $ 2 \pi$, such that the Fourier series of $ f(x)$ is divergent at $ x\equal{}0$, but the Fourier series of $ f^2(x)$ is uniformly convergent on $ [0,2 \pi].$
[i]P. Turan[/i]
2014 SEEMOUS, Problem 1
Let $n$ be a nonzero natural number and $f:\mathbb R\to\mathbb R\setminus\{0\}$ be a function such that $f(2014)=1-f(2013)$. Let $x_1,x_2,x_3,\ldots,x_n$ be real numbers not equal to each other. If
$$\begin{vmatrix}1+f(x_1)&f(x_2)&f(x_3)&\cdots&f(x_n)\\f(x_1)&1+f(x_2)&f(x_3)&\cdots&f(x_n)\\f(x_1)&f(x_2)&1+f(x_3)&\cdots&f(x_n)\\\vdots&\vdots&\vdots&\ddots&\vdots\\f(x_1)&f(x_2)&f(x_3)&\cdots&1+f(x_n)\end{vmatrix}=0,$$prove that $f$ is not continuous.
2015 Greece Team Selection Test, 4
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ which satisfy $yf(x)+f(y) \geq f(xy)$
2010 Today's Calculation Of Integral, 541
Find the functions $ f(x),\ g(x)$ satisfying the following equations.
(1) $ f'(x) \equal{} 2f(x) \plus{} 10,\ f(0) \equal{} 0$
(2) $ \int_0^x u^3g(u)du \equal{} x^4 \plus{} g(x)$