Found problems: 4776
Find all functions $f : R \to R$ such that $f(x^2)-f(y^2) \le (f(x)+y) (x-f(y))$ for all $x, y \in R$.
Alice has two urns. Each urn contains four balls and on each ball a natural number is written. She draws one ball from each urn at random, notes the sum of the numbers written on them, and replaces the balls in the urns from which she took them. This she repeats a large number of times. Bill, on examining the numbers recorded, notices that the frequency with which each sum occurs is the same as if it were the sum of two natural numbers drawn at random from the range 1 to 4. What can he deduce about the numbers on the balls?
Let $\Lambda$ be the set of points in the plane whose coordinates are integers and let $F$ be the collection of all functions from $\Lambda$ to $\{1,-1\}.$ We call a function $f$ in $F$ [i]perfect[/i] if every function $g$ in $F$ that differs from $f$ at finitely many points satisfies the condition
\[ \sum_{0<d(P,Q)<2010} \frac{f(P)f(Q)-g(P)g(Q)}{d(P,Q)} \geq 0 \]
where $d(P,Q)$ denotes the distance between $P$ and $Q.$ Show that there exist infinitely many [i]perfect[/i] functions that are not translates of each other.
Investigate the behaviour at $t\to\infty$ of the solution of the problem
\[u_t+(u\sin x)_x=\epsilon u_{xx},\;u|_{t=0}=1,\;\epsilon\ll1\]
Determine all functions $f: \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ such that
\[ f(x,y) + f(y,z) + f(z,x) = \max \{ x,y,z \} - \min \{ x,y,z \} \] for every $x,y,z \in \mathbb{R}$
and there exists some real $a$ such that $f(x,a) = f(a,x) $ for every $x \in \mathbb{R}$.
Assume that the following generating function equation is correct, prove the following statement:
$\Pi_{i=1}^{\infty} (1+x^{3i})\Pi_{j=1}^{\infty} (1-x^{6j+3})=1$
Statement: The number of partitions of $n$ to numbers not of the form $6k+1$ or $6k-1$ is equal to the number of partitions of $n$ in which each summand appears at least twice.
(10 points)
[i]Proposed by Morteza Saghafian[/i]
Prove that there exists just one function $ f:\mathbb{N}^2\longrightarrow\mathbb{N} $ which simultaneously satisfies:
$ \text{(1)}\quad f(m,n)=f(n,m),\quad\forall m,n\in\mathbb{N} $
$ \text{(2)}\quad f(n,n)=n,\quad\forall n\in\mathbb{N} $
$ \text{(3)}\quad n>m\implies (n-m)f(m,n)=nf(m,n-m), \quad\forall m,n\in\mathbb{N} $
Let $f:[a,b]\to[a,b]$ be a continuous function. It is known that there exist $\alpha,\beta\in (a,b)$ such that $f(\alpha)=a$ and $f(\beta)=b$. Prove that the function $f\circ f$ has at least three fixed points.
In a function $f (x) = (x^2 + ax + b )/ (x^2 + cx + d)$ , the quadratics $x^2 + ax + b$ and $x^2 + cx + d$ have no common roots. Prove that the next two statements are equivalent:
(i) there is a numerical interval without any values of $f(x)$ ,
(ii) $f(x)$ can be represented in the form $f (x) = f_1 (f_2( ... f_{n-1} (f_n (x))... ))$ where each of the functions $f_j$ is o f one of the three forms $k_j x + b_j, 1/x, x^2$ .
(A Kanel)
Let $A=\{1,2,3,4\}$, and $f$ and $g$ be randomly chosen (not necessarily distinct) functions from $A$ to $A$. The probability that the range of $f$ and the range of $g$ are disjoint is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m$.
Find all functions $f: \mathbb{Q}^+ \to \mathbb{R}^+ $ with the property:
\[f(xy)=f(x+y)(f(x)+f(y)) \,,\, \forall x,y \in \mathbb{Q}^+\]
[i]Proposed by Nikolay Nikolov[/i]
Let be a finite group $ G $ having an endomorphism $ \eta $ that has exactly one fixed point.
[b]a)[/b] Demonstrate that the function $ f:G\longrightarrow G $ defined as $ f(x)=x^{-1}\cdot\eta (x) $ is bijective.
[b]b)[/b] Show that $ G $ is commutative if the composition of the function $ f $ from [b]a)[/b] with itself is the identity function.
Let $ f(x) \equal{} c_m x^m \plus{} c_{m\minus{}1} x^{m\minus{}1} \plus{}...\plus{} c_1 x \plus{} c_0$, where each $ c_i$ is a non-zero integer. Define a sequence $ \{ a_n \}$ by $ a_1 \equal{} 0$ and $ a_{n\plus{}1} \equal{} f(a_n)$ for all positive integers $ n$.
(a) Let $ i$ and $ j$ be positive integers with $ i<j$. Show that $ a_{j\plus{}1} \minus{} a_j$ is a multiple of $ a_{i\plus{}1} \minus{} a_i$.
(b) Show that $ a_{2008} \neq 0$
A function $f$ satisfies the equation
\[f\left(x\right)+f\left(1-\frac{1}{x}\right)=1+x\]
for every real number $x$ except for $x = 0$ and $x = 1$. Find a closed formula for $f$.
Let ${f : \Bbb{R} \rightarrow \Bbb{R}}$ be a continuous and strictly increasing function for which
\[ \displaystyle f^{-1}\left(\frac{f(x)+f(y)}{2}\right)(f(x)+f(y)) =(x+y)f\left(\frac{x+y}{2}\right) \]
for all ${x,y \in \Bbb{R}} ({f^{-1}}$ denotes the inverse of ${f})$. Prove that there exist real constants ${a \neq 0}$ and ${b}$ such that ${f(x)=ax+b}$ for all ${x \in \Bbb{R}}.$
[i]Proposed by Zoltán Daróczy[/i]
Calculate the $100$th derivative of the function
\[\frac{1}{x^2+3x+2}\]
at $x=0$ with $10\%$ relative error.
Let $f: \mathbb R \to \mathbb R,g: \mathbb R \to \mathbb R$ and $\varphi: \mathbb R \to \mathbb R$ be three ascendant functions such that
\[f(x) \leq g(x) \leq \varphi(x) \qquad \forall x \in \mathbb R.\]
Prove that
\[f(f(x)) \leq g(g(x)) \leq \varphi(\varphi(x)) \qquad \forall x \in \mathbb R.\]
[i]Note. The function is $k(x)$ ascendant if for every $ x,y \in D_k, x \leq {y}$ we have $g(x)\leq{g(y)}$.[/i]
Two numbers are written on each vertex of a convex $100$-gon. Prove that it is possible to remove a number from each vertex so that the remaining numbers on any two adjacent vertices are different.
[i]F. Petrov [/i]
Let $f$ be a function from the set of rational numbers to the set of real numbers. Suppose that for all rational numbers $r$ and $s$, the expression $f(r + s) - f(r) - f(s)$ is an integer. Prove that there is a positive integer $q$ and an integer $p$ such that
\[
\Bigl\lvert f\Bigl(\frac{1}{q}\Bigr) - p \Bigr\rvert \le \frac{1}{2012} \, .
\]
Let $c > 1$ be a real number. A function $f: [0 ,1 ] \to R$ is called c-friendly if $f(0) = 0, f(1) = 1$ and $|f(x) -f(y)| \le c|x - y|$ for all the numbers $x ,y \in [0,1]$. Find the maximum of the expression $|f(x) - f(y)|$ for all [i]c-friendly[/i] functions $f$ and for all the numbers $x,y \in [0,1]$.
Find all non-negative real numbers $a$ such that the equation \[ \sqrt[3]{1+x}+\sqrt[3]{1-x}=a \] has at least one real solution $x$ with $0 \leq x \leq 1$.
For all such $a$, what is $x$?
Find all real functions $f$ defined on the real numbers except zero, satisfying
$f(2019) = 1$ and $f(x)f(y)+ f\left(\frac{2019}{x}\right) f\left(\frac{2019}{y}\right) =2f(xy)$ for all $x,y \ne 0$
Given a sequence $(a_n)$ of real numbers such that the set $\{a_n\}$ is finite.
If for every $k>1$ subsequence $(a_{kn})$ is periodic, is it true that the sequence $(a_n)$ must be periodic?
For every real number $x$, let $\lfloor x\rfloor$ denote the greatest integer not exceeding $x$, and let \[f(x)=\lfloor x\rfloor(2014^{x-\lfloor x\rfloor}-1).\] The set of all numbers $x$ such that $1\leq x<2014$ and $f(x)\leq 1$ is a union of disjoint intervals. What is the sum of the lengths of those intervals?
$\textbf{(A) }1\qquad
\textbf{(B) }\dfrac{\log 2015}{\log 2014}\qquad
\textbf{(C) }\dfrac{\log 2014}{\log 2013}\qquad
\textbf{(D) }\dfrac{2014}{2013}\qquad
\textbf{(E) }2014^{\frac1{2014}}\qquad$
Let $a,b,c>0$ the sides of a right triangle. Find all real $x$ for which $a^x>b^x+c^x$, with $a$ is the longest side.