Found problems: 4776
2010 Pan African, 3
Does there exist a function $f:\mathbb{Z}\to\mathbb{Z}$ such that $f(x+f(y))=f(x)-y$ for all integers $x$ and $y$?
2007 Moldova National Olympiad, 12.8
Find all continuous functions $f\colon [0;1] \to R$ such that
\[\int_{0}^{1}f(x)dx = 2\int_{0}^{1}(f(x^{4}))^{2}dx+\frac{2}{7}\]
2022 European Mathematical Cup, 2
Find all pairs $(x,y)$ of positive real numbers such that $xy$ is an integer and $x+y = \lfloor x^2 - y^2 \rfloor$.
1989 IMO Longlists, 5
Let $ n > 1$ be a fixed integer. Define functions $ f_0(x) \equal{} 0,$ $ f_1(x) \equal{} 1 \minus{} \cos(x),$ and for $ k > 0,$ \[ f_{k\plus{}1}(x) \equal{} f_k(x) \cdot \cos(x) \minus{} f_{k\minus{}1}(x).\] If $ F(x) \equal{} \sum^n_{r\equal{}1} f_r(x),$ prove that
[b](a)[/b] $ 0 < F(x) < 1$ for $ 0 < x < \frac{\pi}{n\plus{}1},$ and
[b](b)[/b] $ F(x) > 1$ for $ \frac{\pi}{n\plus{}1} < x < \frac{\pi}{n}.$
2021 Iran Team Selection Test, 2
Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for any two positive integers $m,n$ we have :
$$f(n)+1400m^2|n^2+f(f(m))$$
2014 China Team Selection Test, 3
Let the function $f:N^*\to N^*$ such that
[b](1)[/b] $(f(m),f(n))\le (m,n)^{2014} , \forall m,n\in N^*$;
[b](2)[/b] $n\le f(n)\le n+2014 , \forall n\in N^*$
Show that: there exists the positive integers $N$ such that $ f(n)=n $, for each integer $n \ge N$.
(High School Affiliated to Nanjing Normal University )
Kvant 2021, M2661
An infinite table whose rows and columns are numbered with positive integers, is given. For a sequence of functions
$f_1(x), f_2(x), \ldots $ let us place the number $f_i(j)$ into the cell $(i,j)$ of the table (for all $i, j\in \mathbb{N}$).
A sequence $f_1(x), f_2(x), \ldots $ is said to be {\it nice}, if all the numbers in the table are positive integers, and each positive integer appears exactly once. Determine if there exists a nice sequence of functions $f_1(x), f_2(x), \ldots $, such that each $f_i(x)$ is a polynomial of degree 101 with integer coefficients and its leading coefficient equals to 1.
2010 Contests, 2
Find all the continuous functions $f : \mathbb{R} \mapsto\mathbb{R}$ such that $\forall x,y \in \mathbb{R}$,
$(1+f(x)f(y))f(x+y)=f(x)+f(y)$.
2016 Iran Team Selection Test, 6
Let $\mathbb{Z}_{>0}$ denote the set of positive integers. For any positive integer $k$, a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ is called [i]$k$-good[/i] if $\gcd(f(m) + n, f(n) + m) \le k$ for all $m \neq n$. Find all $k$ such that there exists a $k$-good function.
[i]Proposed by James Rickards, Canada[/i]
2010 Today's Calculation Of Integral, 654
A function $f(x)$ defined in $x\geq 0$ satisfies $\lim_{x\to\infty} \frac{f(x)}{x}=1$.
Find $\int_0^{\infty} \{f(x)-f'(x)\}e^{-x}dx$.
[i]1997 Hokkaido University entrance exam/Science[/i]
2023 4th Memorial "Aleksandar Blazhevski-Cane", P6
Denote by $\mathbb{N}$ the set of positive integers. Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ such that:
[b]•[/b] For all positive integers $a> 2023^{2023}$ it holds that $f(a) \leq a$.
[b]•[/b] $\frac{a^2f(b)+b^2f(a)}{f(a)+f(b)}$ is a positive integer for all $a,b \in \mathbb{N}$.
[i]Proposed by Nikola Velov[/i]
2016 Mathematical Talent Reward Programme, MCQ: P 9
$f$ be a function satisfying $2f(x)+3f(-x)=x^2+5x$. Find $f(7)$
[list=1]
[*] $-\frac{105}{4}$
[*] $-\frac{126}{5}$
[*] $-\frac{120}{7}$
[*] $-\frac{132}{7}$
[/list]
2001 Bundeswettbewerb Mathematik, 4
A square $ R$ of sidelength $ 250$ lies inside a square $ Q$ of sidelength $ 500$. Prove that: One can always find two points $ A$ and $ B$ on the perimeter of $ Q$ such that the segment $ AB$ has no common point with the square $ R$, and the length of this segment $ AB$ is greater than $ 521$.
2014 District Olympiad, 4
Find all functions $f:\mathbb{Q}\to \mathbb{Q}$ such that
\[ f(x+3f(y))=f(x)+f(y)+2y \quad \forall x,y\in \mathbb{Q}\]
2016 Belarus Team Selection Test, 1
Find all functions $f:\mathbb{R}\to \mathbb{R},g:\mathbb{R}\to \mathbb{R}$ such that
$$f(x-2f(y))= xf(y)-yf(x)+g(x)$$ for all real $x,y$
2023-24 IOQM India, 21
For $n \in \mathbb{N}$, consider non-negative valued functions $f$ on $\{1,2, \cdots , n\}$ satisfying $f(i) \geqslant f(j)$ for $i>j$ and $\sum_{i=1}^{n} (i+ f(i))=2023.$ Choose $n$ such that $\sum_{i=1}^{n} f(i)$ is at least. How many such functions exist in that case?
1957 AMC 12/AHSME, 10
The graph of $ y \equal{} 2x^2 \plus{} 4x \plus{} 3$ has its:
$ \textbf{(A)}\ \text{lowest point at } {(\minus{}1,9)}\qquad
\textbf{(B)}\ \text{lowest point at } {(1,1)}\qquad \\
\textbf{(C)}\ \text{lowest point at } {(\minus{}1,1)}\qquad
\textbf{(D)}\ \text{highest point at } {(\minus{}1,9)}\qquad \\
\textbf{(E)}\ \text{highest point at } {(\minus{}1,1)}$
Today's calculation of integrals, 766
Let $f(x)$ be a continuous function defined on $0\leq x\leq \pi$ and satisfies $f(0)=1$ and
\[\left\{\int_0^{\pi} (\sin x+\cos x)f(x)dx\right\}^2=\pi \int_0^{\pi}\{f(x)\}^2dx.\]
Evaluate $\int_0^{\pi} \{f(x)\}^3dx.$
2005 Today's Calculation Of Integral, 45
Find the function $f(x)$ which satisfies the following integral equation.
\[f(x)=\int_0^x t(\sin t-\cos t)dt+\int_0^{\frac{\pi}{2}} e^t f(t)dt\]
1991 Arnold's Trivium, 76
Investigate the behaviour at $t\to\infty$ of the solution of the problem
\[u_t+(u\sin x)_x=\epsilon u_{xx},\;u|_{t=0}=1,\;\epsilon\ll1\]
2013 Putnam, 2
Let $S$ be the set of all positive integers that are [i]not[/i] perfect squares. For $n$ in $S,$ consider choices of integers $a_1,a_2,\dots, a_r$ such that $n<a_1<a_2<\cdots<a_r$ and $n\cdot a_1\cdot a_2\cdots a_r$ is a perfect square, and let $f(n)$ be the minimum of $a_r$ over all such choices. For example, $2\cdot 3\cdot 6$ is a perfect square, while $2\cdot 3,2\cdot 4, 2\cdot 5, 2\cdot 3\cdot 4,$ $2\cdot 3\cdot 5, 2\cdot 4\cdot 5,$ and $2\cdot 3\cdot 4\cdot 5$ are not, and so $f(2)=6.$ Show that the function $f$ from $S$ to the integers is one-to-one.
Dumbest FE I ever created, 7.
Find all function $f : \mathbb{R} \to \mathbb{R}$ such that for all $x,y \in \mathbb{R}$ . $$f(x+f(y))+f(x+y)=2x+f(y)+f(f(y))$$ . [hide=Original]$$f(x+f(y))+f(x+y)=2x+f(y)+y$$[/hide]
2008 ISI B.Stat Entrance Exam, 10
Two subsets $A$ and $B$ of the $(x,y)$-plane are said to be [i]equivalent[/i] if there exists a function $f: A\to B$ which is both one-to-one and onto.
(i) Show that any two line segments in the plane are equivalent.
(ii) Show that any two circles in the plane are equivalent.
2009 ISI B.Math Entrance Exam, 3
Let $1,2,3,4,5,6,7,8,9,11,12,\cdots$ be the sequence of all positive integers which do not contain the digit zero. Write $\{a_n\}$ for this sequence. By comparing with a geometric series, show that $\sum_{k=1}^n \frac{1}{a_k} < 90$.
2008 Croatia Team Selection Test, 2
For which $ n\in \mathbb{N}$ do there exist rational numbers $ a,b$ which are not integers such that both $ a \plus{} b$ and $ a^n \plus{} b^n$ are integers?