Found problems: 4776
2010 Laurențiu Panaitopol, Tulcea, 2
Let be a nonnegative integer $ n $ such that $ \sqrt n $ is not integer. Show that the function
$$ f:\{ a+b\sqrt n | a,b\in\{ 0\}\cup\mathbb{N} , a^2-nb^2=1 \}\longrightarrow\{ 0\}\cup\mathbb{N} , f(x) =\lfloor x \rfloor $$
is injective and non-surjective.
2005 India IMO Training Camp, 2
Find all functions $ f: \mathbb{N^{*}}\to \mathbb{N^{*}}$ satisfying
\[ \left(f^{2}\left(m\right)+f\left(n\right)\right) \mid \left(m^{2}+n\right)^{2}\]
for any two positive integers $ m$ and $ n$.
[i]Remark.[/i] The abbreviation $ \mathbb{N^{*}}$ stands for the set of all positive integers:
$ \mathbb{N^{*}}=\left\{1,2,3,...\right\}$.
By $ f^{2}\left(m\right)$, we mean $ \left(f\left(m\right)\right)^{2}$ (and not $ f\left(f\left(m\right)\right)$).
[i]Proposed by Mohsen Jamali, Iran[/i]
2006 China Team Selection Test, 2
Given three positive real numbers $ x$, $ y$, $ z$ such that $ x \plus{} y \plus{} z \equal{} 1$, prove that
$ \frac {xy}{\sqrt {xy \plus{} yz}} \plus{} \frac {yz}{\sqrt {yz \plus{} zx}} \plus{} \frac {zx}{\sqrt {zx \plus{} xy}} \le \frac {\sqrt {2}}{2}$.
2008 Singapore Senior Math Olympiad, 3
Let there's a function $ f: \mathbb{R}\rightarrow\mathbb{R}$
Find all functions $ f$ that satisfies:
a) $ f(2u)\equal{}f(u\plus{}v)f(v\minus{}u)\plus{}f(u\minus{}v)f(\minus{}u\minus{}v)$
b) $ f(u)\geq0$
2019 Pan-African Shortlist, A3
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that
$$
f\left(x^2\right) - yf(y) = f(x + y) (f(x) - y)
$$
for all real numbers $x$ and $y$.
1993 Polish MO Finals, 2
Find all real-valued functions $f$ on the reals such that $f(-x) = -f(x)$, $f(x+1) = f(x) + 1$ for all $x$, and $f\left(\dfrac{1}{x}\right) = \dfrac{f(x)}{x^2}$ for $x \not = 0$.
2012 China Team Selection Test, 2
Given two integers $m,n$ which are greater than $1$. $r,s$ are two given positive real numbers such that $r<s$. For all $a_{ij}\ge 0$ which are not all zeroes,find the maximal value of the expression
\[f=\frac{(\sum_{j=1}^{n}(\sum_{i=1}^{m}a_{ij}^s)^{\frac{r}{s}})^{\frac{1}{r}}}{(\sum_{i=1}^{m})\sum_{j=1}^{n}a_{ij}^r)^{\frac{s}{r}})^{\frac{1}{s}}}.\]
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.
KoMaL A Problems 2021/2022, A. 821
[b]a)[/b] Is it possible to find a function $f:\mathbb N^2\to\mathbb N$ such that for every function $g:\mathbb N\to\mathbb N$ and positive integer $M$ there exists $n\in\mathbb N$ such that set $\left\{k\in \mathbb N : f(n,k)=g(k)\right\}$ has at least $M$ elements?
[b]b)[/b] Is it possible to find a function $f:\mathbb N^2\to\mathbb N$ such that for every function $g:\mathbb N\to\mathbb N$ there exists $n\in \mathbb N$ such that set $\left\{k\in\mathbb N : f(n,k)=g(k)\right\}$ has an infinite number of elements?
2009 India IMO Training Camp, 6
Prove The Following identity:
$ \sum_{j \equal{} 0}^n \left ({3n \plus{} 2 \minus{} j \choose j}2^j \minus{} {3n \plus{} 1 \minus{} j \choose j \minus{} 1}2^{j \minus{} 1}\right ) \equal{} 2^{3n}$.
The Second term on left hand side is to be regarded zero for j=0.
1969 IMO Longlists, 8
Find all functions $f$ defined for all $x$ that satisfy the condition $xf(y) + yf(x) = (x + y)f(x)f(y),$ for all $x$ and $y.$ Prove that exactly two of them are continuous.
2014 Contests, 4
(a) Let $a,x,y$ be positive integers. Prove: if $x\ne y$, the also
\[ax+\gcd(a,x)+\text{lcm}(a,x)\ne ay+\gcd(a,y)+\text{lcm}(a,y).\]
(b) Show that there are no two positive integers $a$ and $b$ such that
\[ab+\gcd(a,b)+\text{lcm}(a,b)=2014.\]
1998 Romania Team Selection Test, 4
Consider in the plane a finite set of segments such that the sum of their lengths is less than $\sqrt{2}$. Prove that there exists an infinite unit square grid covering the plane such that the lines defining the grid do not intersect any of the segments.
[i]Vasile Pop[/i]
1978 IMO Longlists, 6
Prove that for all $X > 1$, there exists a triangle whose sides have lengths $P_1(X) = X^4+X^3+2X^2+X+1, P_2(X) = 2X^3+X^2+2X+1$, and $P_3(X) = X^4-1$. Prove that all these triangles have the same greatest angle and calculate it.
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} $
2015 China Team Selection Test, 4
Let $n$ be a positive integer, let $f_1(x),\ldots,f_n(x)$ be $n$ bounded real functions, and let $a_1,\ldots,a_n$ be $n$ distinct reals.
Show that there exists a real number $x$ such that $\sum^n_{i=1}f_i(x)-\sum^n_{i=1}f_i(x-a_i)<1$.
2013 Macedonian Team Selection Test, Problem 3
Denote by $\mathbb{Z}^{*}$ the set of all nonzero integers and denote by $\mathbb{N}_{0}$ the set of all nonnegative integers. Find all functions $f:\mathbb{Z}^{*} \rightarrow \mathbb{N}_{0}$ such that:
$(1)$ For all $a,b \in \mathbb{Z}^{*}$ such that $a+b \in \mathbb{Z}^{*}$ we have $f(a+b) \geq $ [b]min[/b] $\left \{ f(a),f(b) \right \}$.
$(2)$ For all $a, b \in \mathbb{Z}^{*}$ we have $f(ab) = f(a)+f(b)$.
2011 Hanoi Open Mathematics Competitions, 9
For every pair of positive integers $(x, y)$ we de fine $f(x,y)$ as follows:
$f(x,1) = x$
$f(x,y) = 0$ if $y > x$
$f(x +1,y) = y[f(x,y)+ f(x, y-1)]$
Evaluate $f(5, 5)$.
2013 HMNT, 10
How many functions $\{f : 1,2, \cdots, 2013\} \rightarrow \{1,2, \cdots, 2013\}$ satisfy $f(j) < f(i) + j - i$ for all integers $i,j$ such that $1 \leq i < j \leq 2013$ ?
1996 Moldova Team Selection Test, 9
Let $x_1,x_2,...,x_n \in [0;1]$ prove that
$x_1(1-x_2)+x_2(1-x_3)+...+x_{n-1}(1-x_n)+x_n(1-x_1) \leq [\frac{n}{2}]$
2013 Vietnam Team Selection Test, 3
Given a number $n\in\mathbb{Z}^+$ and let $S$ denotes the set $\{0,1,2,...,2n+1\}$. Consider the function $f:\mathbb{Z}\times S\to [0,1]$ satisfying two following conditions simultaneously:
i) $f(x,0)=f(x,2n+1)=0\forall x\in\mathbb{Z}$;
ii) $f(x-1,y)+f(x+1,y)+f(x,y-1)+f(x,y+1)=1$ for all $x\in\mathbb{Z}$ and $y\in\{1,2,3,...,2n\}$.
Let $F$ be the set of such functions. For each $f\in F$, let $v(f)$ be the set of values of $f$.
a) Proof that $|F|=\infty$.
b) Proof that for each $f\in F$ then $|v(f)|<\infty$.
c) Find the maximum value of $|v(f)|$ for $f\in F$.
2009 Indonesia TST, 3
Let $ x,y,z$ be real numbers. Find the minimum value of $ x^2\plus{}y^2\plus{}z^2$ if $ x^3\plus{}y^3\plus{}z^3\minus{}3xyz\equal{}1$.
2017 Philippine MO, 1
Given \(n \in \mathbb{N}\), let \(\sigma (n)\) denote the sum of the divisors of \(n\) and \(\phi (n)\) denote the number of integers \(n \geq m\) for which \(\gcd(m,n) = 1\). Show that for all \(n \in \mathbb{N}\),
\[\large \frac{1}{\sigma (n)} + \frac{1}{\phi (n)} \geq \frac{2}{n}\]
and determine when equality holds.
2014 USAMTS Problems, 3a:
A group of people is lined up in [i]almost-order[/i] if, whenever person $A$ is to the left of person $B$ in the line, $A$ is not more than $8$ centimeters taller than $B$. For example, five people with heights $160, 165, 170, 175$, and $180$ centimeters could line up in almost-order with heights (from left-to-right) of $160, 170, 165, 180, 175$ centimeters.
(a) How many different ways are there to line up $10$ people in [i]almost-order[/i] if their heights are $140, 145, 150, 155,$ $160,$ $165,$ $170,$ $175,$ $180$, and $185$ centimeters?
1969 IMO Longlists, 61
$(SWE 4)$ Let $a_0, a_1, a_2, \cdots$ be determined with $a_0 = 0, a_{n+1} = 2a_n + 2^n$. Prove that if $n$ is power of $2$, then so is $a_n$