Found problems: 4776
2010 Victor Vâlcovici, 2
Let be a finite set $ S. $ Determine the number of functions $ f:S\rightarrow S $ that satisfy $ f\circ f=f. $
2008 Korean National Olympiad, 7
Prove that the only function $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying the following is $f(x)=x$.
(i) $\forall x \not= 0$, $f(x) = x^2f(\frac{1}{x})$.
(ii) $\forall x, y$, $f(x+y) = f(x)+f(y)$.
(iii) $f(1)=1$.
2008 China Second Round Olympiad, 3
For all $k=1,2,\ldots,2008$,$a_k>0$.Prove that iff $\sum_{k=1}^{2008}a_k>1$,there exists a function $f:N\rightarrow R$ satisfying
(1)$0=f(0)<f(1)<f(2)<\ldots$;
(2)$f(n)$ has a finite limit when $n$ approaches infinity;
(3)$f(n)-f({n-1})=\sum_{k=1}^{2008}a_kf({n+k})-\sum_{k=0}^{2007}a_{k+1}f({n+k})$,for all $n=1,2,3,\ldots$.
2014 PUMaC Algebra A, 3
A function $f$ has its domain equal to the set of integers $0$, $1$, $\ldots$, $11$, and $f(n)\geq 0$ for all such $n$, and $f$ satisfies
[list]
[*]$f(0)=0$
[*]$f(6)=1$
[*]If $x\geq 0$, $y\geq 0$, and $x+y\leq 11$, then $f(x+y)=\tfrac{f(x)+f(y)}{1-f(x)f(y)}$.[/list]
Find $f(2)^2+f(10)^2$.
1990 China Team Selection Test, 2
Find all functions $f,g,h: \mathbb{R} \mapsto \mathbb{R}$ such that $f(x) - g(y) = (x-y) \cdot h(x+y)$ for $x,y \in \mathbb{R}.$
2013 IMO Shortlist, A3
Let $\mathbb Q_{>0}$ be the set of all positive rational numbers. Let $f:\mathbb Q_{>0}\to\mathbb R$ be a function satisfying the following three conditions:
(i) for all $x,y\in\mathbb Q_{>0}$, we have $f(x)f(y)\geq f(xy)$;
(ii) for all $x,y\in\mathbb Q_{>0}$, we have $f(x+y)\geq f(x)+f(y)$;
(iii) there exists a rational number $a>1$ such that $f(a)=a$.
Prove that $f(x)=x$ for all $x\in\mathbb Q_{>0}$.
[i]Proposed by Bulgaria[/i]
2017 Mathematical Talent Reward Programme, SAQ: P 5
Let $\mathbb{N}$ be the set of all natural numbers. Let $f:\mathbb{N} \to \mathbb{N}$ be a bijective function. Show that there exists three numbers $a$, $b$, $c$ in arithmatic progression such that $f(a)<f(b)<f(c)$
2016 IMC, 2
Today, Ivan the Confessor prefers continuous functions $f:[0,1]\to\mathbb{R}$ satisfying $f(x)+f(y)\geq |x-y|$ for all pairs $x,y\in [0,1]$. Find the minimum of $\int_0^1 f$ over all preferred functions.
(Proposed by Fedor Petrov, St. Petersburg State University)
2014 USAMO, 1
Let $a$, $b$, $c$, $d$ be real numbers such that $b-d \ge 5$ and all zeros $x_1, x_2, x_3,$ and $x_4$ of the polynomial $P(x)=x^4+ax^3+bx^2+cx+d$ are real. Find the smallest value the product $(x_1^2+1)(x_2^2+1)(x_3^2+1)(x_4^2+1)$ can take.
2000 Hungary-Israel Binational, 1
Let $A$ and $B$ be two subsets of $S = \{1, 2, . . . , 2000\}$ with $|A| \cdot |B| \geq 3999$. For a set $X$ , let $X-X$ denotes the set $\{s-t | s, t \in X, s \not = t\}$. Prove that $(A-A) \cap (B-B)$ is nonempty.
2012 IMC, 4
Let $n \ge 2$ be an integer. Find all real numbers $a$ such that there exist real numbers $x_1,x_2,\dots,x_n$ satisfying
\[x_1(1-x_2)=x_2(1-x_3)=\dots=x_n(1-x_1)=a.\]
[i]Proposed by Walther Janous and Gerhard Kirchner, Innsbruck.[/i]
2014-2015 SDML (High School), 2
What is the maximum value of the function $$\frac{1}{\left|x+1\right|+\left|x+2\right|+\left|x-3\right|}?$$
$\text{(A) }\frac{1}{3}\qquad\text{(B) }\frac{1}{4}\qquad\text{(C) }\frac{1}{5}\qquad\text{(D) }\frac{1}{6}\qquad\text{(E) }\frac{1}{7}$
2007 Today's Calculation Of Integral, 245
A sextic funtion $ y \equal{} ax^6 \plus{} bx^5 \plus{} cx^4 \plus{} dx^3 \plus{} ex^2 \plus{} fx \plus{} g\ (a\neq 0)$ touches the line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha ,\ \beta ,\ \gamma \ (\alpha < \beta < \gamma ).$
Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta ,\gamma .$
created by kunny
2018 Taiwan TST Round 1, 5
Find all functions $ f: \mathbb{N} \to \mathbb{Z} $ satisfying $$ n \mid f\left(m\right) \Longleftrightarrow m \mid \sum\limits_{d \mid n}{f\left(d\right)} $$ holds for all positive integers $ m,n $
2018 China Team Selection Test, 2
An integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition.
[quote]For example, 4 can be partitioned in five distinct ways:
4
3 + 1
2 + 2
2 + 1 + 1
1 + 1 + 1 + 1[/quote]
The number of partitions of n is given by the partition function $p\left ( n \right )$. So $p\left ( 4 \right ) = 5$ .
Determine all the positive integers so that $p\left ( n \right )+p\left ( n+4 \right )=p\left ( n+2 \right )+p\left ( n+3 \right )$.
1996 Abels Math Contest (Norwegian MO), 2
Prove that $[\sqrt{n}+\sqrt{n+1}]=[\sqrt{4n+1}]$ for all $n \in N$.
1994 China National Olympiad, 3
Find all functions $f:[1,\infty )\rightarrow [1,\infty)$ satisfying the following conditions:
(1) $f(x)\le 2(x+1)$;
(2) $f(x+1)=\dfrac{1}{x}[(f(x))^2-1]$ .
1993 Turkey MO (2nd round), 3
$n\in{Z^{+}}$ and $A={1,\ldots ,n}$. $f: N\rightarrow N$ and $\sigma: N\rightarrow N$ are two permutations, if there is one $k\in A$ such that $(f\circ\sigma)(1),\ldots ,(f\circ\sigma)(k)$ is increasing and $(f\circ\sigma)(k),\ldots ,(f\circ\sigma)(n)$ is decreasing sequences we say that $f$ is good for $\sigma$. $S_\sigma$ shows the set of good functions for $\sigma$.
a) Prove that, $S_\sigma$ has got $2^{n-1}$ elements for every $\sigma$ permutation.
b)$n\geq 4$, prove that there are permutations $\sigma$ and $\tau$ such that, $S_{\sigma}\cap S_{\tau}=\phi$
.
2020 LIMIT Category 2, 2
The number of functions $g:\mathbb{R}^4\to\mathbb{R}$ such that, $\forall a,b,c,d,e,f\in\mathbb{R}$ :
(i) $g(1,0,0,1)=1$
(ii) $g(ea,b,ec,d)=eg(a,b,c,d)$
(iii) $g(a+e, b, c+f, d)= g(a,b,c,d)+g(e,b,f,d)$
(iv) $g(a,b,c,d)+g(b,a,d,c)=0$
is :
(A)$1$
(B)$0$
(C)$\text{infinitely many}$
(D)$\text{None of these}$
[Hide=Hint(given in question)]
Think of matrices[/hide]
2007 Romania National Olympiad, 2
Consider the triangle $ ABC$ with $ m(\angle BAC \equal{} 90^\circ)$ and $ AC \equal{} 2AB$. Let $ P$ and $ Q$ be the midpoints of $ AB$ and $ AC$,respectively. Let $ M$ and $ N$ be two points found on the side $ BC$ such that $ CM \equal{} BN \equal{} x$. It is also known that $ 2S[MNPQ] \equal{} S[ABC]$. Determine $ x$ in function of $ AB$.
PEN A Problems, 88
Find all positive integers $n$ such that $9^{n}-1$ is divisible by $7^n$.
PEN K Problems, 8
Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(f(f(n)))+6f(n)=3f(f(n))+4n+2001.\]
2020 Lusophon Mathematical Olympiad, 6
Prove that $\lfloor{\sqrt{9n+7}}\rfloor=\lfloor{\sqrt{n}+\sqrt{n+1}+\sqrt{n+2}}\rfloor$ for all postive integer $n$.
1998 Harvard-MIT Mathematics Tournament, 9
Suppose $f(x)$ is a rational function such that $3f\left(\dfrac{1}{x}\right)+\dfrac{2f(x)}{x}=x^2$ for $x\neq 0$. Find $f(-2)$.
2022 Romania National Olympiad, P1
Let $f:[0,1]\to(0,1)$ be a surjective function.
[list=a]
[*]Prove that $f$ has at least one point of discontinuity.
[*]Given that $f$ admits a limit in any point of the interval $[0,1],$ show that is has at least two points of discontinuity.
[/list][i]Mihai Piticari and Sorin Rădulescu[/i]