Found problems: 4776
2025 Bulgarian Winter Tournament, 11.4
Let $A$ be a set of $2025$ non-negative integers and $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ be a function with the following two properties:
1) For every two distinct positive integers $x,y$ there exists $a\in A$, such that $x-y$ divides $f(x+a) - f(y+a)$.
2) For every positive integer $N$ there exists a positive integer $t$ such that $f(x) \neq f(y)$ whenever $x,y \in [t, t+N]$ are distinct.
Prove that there are infinitely many primes $p$ such that $p$ divides $f(x)$ for some positive integer $x$.
2006 Federal Competition For Advanced Students, Part 1, 4
Given is the function $ f\equal{} \lfloor x^2 \rfloor \plus{} \{ x \}$ for all positive reals $ x$. ( $ \lfloor x \rfloor$ denotes the largest integer less than or equal $ x$ and $ \{ x \} \equal{} x \minus{} \lfloor x \rfloor$.)
Show that there exists an arithmetic sequence of different positive rational numbers, which all have the denominator $ 3$, if they are a reduced fraction, and don’t lie in the range of the function $ f$.
2002 Flanders Math Olympiad, 2
Determine all functions $f: \mathbb{R}\rightarrow\mathbb{R}$ so that $\forall x: x\cdot f(\frac x2) - f(\frac2x) = 1$
2015 ISI Entrance Examination, 8
Find all the functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that
$$|f(x)-f(y)| = 2 |x - y| $$
2006 Junior Balkan Team Selection Tests - Romania, 2
Prove that for all positive real numbers $a,b,c$ the following inequality holds \[ \left( \frac ab + \frac bc + \frac ca \right)^2 \geq \frac 32 \cdot \left ( \frac{a+b}c + \frac{b+c}a + \frac{c+a} b \right) . \]
2003 Tuymaada Olympiad, 4
Find all continuous functions $f(x)$ defined for all $x>0$ such that for every $x$, $y > 0$
\[ f\left(x+{1\over x}\right)+f\left(y+{1\over y}\right)= f\left(x+{1\over y}\right)+f\left(y+{1\over x}\right) . \]
[i]Proposed by F. Petrov[/i]
2013 AIME Problems, 8
The domain of the function $f(x) = \text{arcsin}(\log_{m}(nx))$ is a closed interval of length $\frac{1}{2013}$, where $m$ and $n$ are positive integers and $m > 1$. Find the remainder when the smallest possible sum $m+n$ is divided by $1000$.
2017 Philippine MO, 2
Find all positive real numbers \((a,b,c) \leq 1\) which satisfy
\[ \huge \min \Bigg\{ \sqrt{\frac{ab+1}{abc}}\, \sqrt{\frac{bc+1}{abc}}, \sqrt{\frac{ac+1}{abc}} \Bigg \} = \sqrt{\frac{1-a}{a}} + \sqrt{\frac{1-b}{b}} + \sqrt{\frac{1-c}{c}}\]
2010 Tournament Of Towns, 2
Let $f(x)$ be a function such that every straight line has the same number of intersection points with the graph $y = f(x)$ and with the graph $y = x^2$. Prove that $f(x) = x^2.$
2011 Albania National Olympiad, 1
[b](a) [/b] Find the minimal distance between the points of the graph of the function $y=\ln x$ from the line $y=x$.
[b](b)[/b] Find the minimal distance between two points, one of the point is in the graph of the function $y=e^x$ and the other point in the graph of the function $y=ln x$.
2021 Alibaba Global Math Competition, 8
Let $f(z)$ be a holomorphic function in $\{\vert z\vert \le R\}$ ($0<R<\infty$). Define
\[M(r,f)=\max_{\vert z\vert=r} \vert f(z)\vert, \quad A(r,f)=\max_{\vert z\vert=r} \text{Re}\{f(z)\}.\]
Show that
\[M(r,f) \le \frac{2r}{R-r}A(R,f)+\frac{R+r}{R-r} \vert f(0)\vert, \quad \forall 0 \le r<R.\]
2016 Mathematical Talent Reward Programme, MCQ: P 12
Let $f(x)=(x-1)(x-2)(x-3)$. Consider $g(x)=min\{f(x),f'(x)\}$. Then the number of points of discontinuity are
[list=1]
[*] 0
[*] 1
[*] 2
[*] More than 2
[/list]
1983 AIME Problems, 3
What is the product of the real roots of the equation \[x^2 + 18x + 30 = 2 \sqrt{x^2 + 18x + 45}\,\,?\]
2022 Iran MO (2nd round), 2
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for any real value of $x,y$ we have:
$$f(xf(y)+f(x)+y)=xy+f(x)+f(y)$$
2022 Brazil Undergrad MO, 1
Let $0<a<1$. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ continuous at $x = 0$ such that $f(x) + f(ax) = x,\, \forall x \in \mathbb{R}$
PEN K Problems, 5
Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(f(m)+f(n))=m+n.\]
2010 Germany Team Selection Test, 3
Find all functions $f$ from the set of real numbers into the set of real numbers which satisfy for all $x$, $y$ the identity \[ f\left(xf(x+y)\right) = f\left(yf(x)\right) +x^2\]
[i]Proposed by Japan[/i]
2017 Iran MO (3rd round), 3
Let $k$ be a positive integer. Find all functions $f:\mathbb{N}\to \mathbb{N}$ satisfying the following two conditions:\\
• For infinitely many prime numbers $p$ there exists a positve integer $c$ such that $f(c)=p^k$.\\
• For all positive integers $m$ and $n$, $f(m)+f(n)$ divides $f(m+n)$.
2013 Vietnam National Olympiad, 2
Define a sequence $\{a_n\}$ as: $\left\{\begin{aligned}& a_1=1 \\ & a_{n+1}=3-\frac{a_{n}+2}{2^{a_{n}}}\ \ \text{for} \ n\geq 1.\end{aligned}\right.$
Prove that this sequence has a finite limit as $n\to+\infty$ . Also determine the limit.
2008 District Olympiad, 2
Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a countinuous and periodic function, of period $ T. $ If $ F $ is a primitive of $ f, $ show that:
[b]a)[/b] the function $ G:\mathbb{R}\longrightarrow\mathbb{R}, G(x)=F(x)-\frac{x}{T}\int_0^T f(t)dt $ is periodic.
[b]b)[/b] $ \lim_{n\to\infty}\sum_{i=1}^n\frac{F(i)}{n^2+i^2} =\frac{\ln 2}{2T}\int_0^T f(x)dx. $
2000 Baltic Way, 14
Find all positive integers $n$ such that $n$ is equal to $100$ times the number of positive divisors of $n$.
2009 Germany Team Selection Test, 2
For every $ n\in\mathbb{N}$ let $ d(n)$ denote the number of (positive) divisors of $ n$. Find all functions $ f: \mathbb{N}\to\mathbb{N}$ with the following properties: [list][*] $ d\left(f(x)\right) \equal{} x$ for all $ x\in\mathbb{N}$.
[*] $ f(xy)$ divides $ (x \minus{} 1)y^{xy \minus{} 1}f(x)$ for all $ x$, $ y\in\mathbb{N}$.[/list]
[i]Proposed by Bruno Le Floch, France[/i]
2003 District Olympiad, 3
On a board are drawn the points $A,B,C,D$. Yetti constructs the points $A^\prime,B^\prime,C^\prime,D^\prime$ in the following way: $A^\prime$ is the symmetric of $A$ with respect to $B$, $B^\prime$ is the symmetric of $B$ wrt $C$, $C^\prime$ is the symmetric of $C$ wrt $D$ and $D^\prime$ is the symmetric of $D$ wrt $A$.
Suppose that Armpist erases the points $A,B,C,D$. Can Yetti rebuild them?
$\star \, \, \star \, \, \star$
[b]Note.[/b] [i]Any similarity to real persons is purely accidental.[/i]
PEN K Problems, 20
Find all functions $f: \mathbb{Q}\to \mathbb{Q}$ such that for all $x,y \in \mathbb{Q}$: \[f(x+y)+f(x-y)=2(f(x)+f(y)).\]
2013 Turkey Team Selection Test, 2
Determine all functions $f:\mathbf{R} \rightarrow \mathbf{R}^+$ such that for all real numbers $x,y$ the following conditions hold:
$\begin{array}{rl}
i. & f(x^2) = f(x)^2 -2xf(x) \\
ii. & f(-x) = f(x-1)\\
iii. & 1<x<y \Longrightarrow f(x) < f(y).
\end{array}$