Found problems: 4776
Find the greatest real number $M$ for which
\[ a^2+b^2+c^2+3abc \geq M(ab+bc+ca) \]
for all non-negative real numbers $a,b,c$ satisfying $a+b+c=4.$
Does there exists two functions $f,g :\mathbb{R}\rightarrow \mathbb{R}$ such that:
$\forall x\not =y : |f(x)-f(y)|+|g(x)-g(y)|>1$
Time allowed for this problem was 75 minutes.
Let $S$ be a finite set, and let $\mathcal{A}$ be the set of all functions from $S$ to $S$. Let $f$ be an element of $\mathcal{A}$, and let $T=f(S)$ be the image of $S$ under $f$. Suppose that $f\circ g\circ f\ne g\circ f\circ g$ for every $g$ in $\mathcal{A}$ with $g\ne f$. Show that $f(T)=T$.
The product of the positive real numbers $x, y, z$ is 1. Show that if \[ \frac{1}{x}+\frac{1}{y} + \frac{1}{z} \geq x+y+z \]then \[ \frac{1}{x^{k}}+\frac{1}{y^{k}} + \frac{1}{z^{k}} \geq x^{k}+y^{k}+z^{k} \]
for all positive integers $k$.
Prove that:
$\int_{-1}^1f^2(x)dx\ge \frac 1 2 (\int_{-1}^1f(x)dx)^2 +\frac 3 2(\int_{-1}^1xf(x)dx)^2$
Please give a proof without using even and odd functions. (the oficial proof uses those and seems to be un-natural) :D
If we write $ |x^2 \minus{} 4| < N$ for all $ x$ such that $ |x \minus{} 2| < 0.01$, the smallest value we can use for $ N$ is:
$ \textbf{(A)}\ .0301 \qquad \textbf{(B)}\ .0349 \qquad \textbf{(C)}\ .0399 \qquad \textbf{(D)}\ .0401 \qquad \textbf{(E)}\ .0499 \qquad$
Consider the sequence of numbers $(a_n)$ ($n = 1, 2, \ldots$) defined as follows: $ a_1\in (1, 2)$, $ a_{k + 1} = a_k + \frac{k}{a_k}$ ($k = 1, 2, \ldots$). Prove that there exists at most one pair of distinct positive integers $(i, j)$ such that $a_i + a_j$ is an integer.
Let $f$ be any function that maps the set of real numbers into the set of real numbers. Prove that there exist real numbers $x$ and $y$ such that \[f\left(x-f(y)\right)>yf(x)+x\]
[i]Proposed by Igor Voronovich, Belarus[/i]
Evaluate $ \int_0^1 \frac{x\minus{}e^{2x}}{x^2\minus{}e^{2x}}dx$.
Find all functions $f,g : \mathbb{R} \times \mathbb{R} \mapsto \mathbb{R}^{+}$ such that
\[ ( \sum_{j=1}^{n}a_{j}b_{j})^2 \leq (\sum_{j=1}^{n} f({a_{j},b_{j}))(\sum_{j=1}^{n} g({a_{j},b_{j})) \leq (\sum_{j=1}^{n} (a_j)^2 )(\sum_{j=1}^{n} (b_j)^2 ) }}\] for any two sets $a_j$ and $b_j$ of real numbers.
Let $f:\mathbb{R}\to \mathbb{R}$ be a differentiable function such that $\lim \limits_{x\to \infty}f'(x)=1$, then
[list=1]
[*] $f$ is increasing
[*] $f$ is unbounded
[*] $f'$ is bounded
[*] All of these
[/list]
Find all functions $f\colon \mathbb{Z}^2 \to [0, 1]$ such that for any integers $x$ and $y$,
\[f(x, y) = \frac{f(x - 1, y) + f(x, y - 1)}{2}.\]
[i]Proposed by Yang Liu and Michael Kural[/i]
Let $ ABC$ be a triangle and $ K$ and $ L$ be two points on $ (AB)$, $ (AC)$ such that $ BK \equal{} CL$ and let $ P \equal{} CK\cap BL$. Let the parallel through $ P$ to the interior angle bisector of $ \angle BAC$ intersect $ AC$ in $ M$. Prove that $ CM \equal{} AB$.
Find all function(s) $f:\mathbb R\to\mathbb R$ satisfying the equation
\[f(x+y)+f(x)f(y)=(1+y)f(x)+(1+x)f(y)+f(xy);\]
For all $x,y\in\mathbb R.$
For positive rational $x$, if $x$ is written in the form $p/q$ with $p, q$ positive relatively prime integers, define $f(x)=p+q$. For example, $f(1)=2$.
a) Prove that if $f(x)=f(mx/n)$ for rational $x$ and positive integers $m, n$, then $f(x)$ divides $|m-n|$.
b) Let $n$ be a positive integer. If all $x$ which satisfy $f(x)=f(2^nx)$ also satisfy $f(x)=2^n-1$, find all possible values of $n$.
[i]Anderson Wang.[/i]
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]
Let $f\colon\mathbb{N}\rightarrow\mathbb{N}$ be a function such that for all positive integers $x$ and $y$, the number $f(x)+y$ is a perfect square if and only if $x+f(y)$ is a perfect square. Prove that $f$ is injective.
[i]Remark.[/i] A function $f\colon\mathbb{N}\rightarrow\mathbb{N}$ is injective if for all pairs $(x,y)$ of distinct positive integers, $f(x)\neq f(y)$ holds.
[i]Ivan Novak[/i]
Determine all functions $f: \mathbb{R} \mapsto \mathbb{R}$ satisfying
\[f\left(\sqrt{2} \cdot x\right) + f\left(4 + 3 \cdot \sqrt{2} \cdot x \right) = 2 \cdot f\left(\left(2 + \sqrt{2}\right) \cdot x\right)\]
for all $x$.
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that
\[f(x + y) + y \le f(f(f(x)))\]
holds for all $x, y \in \mathbb{R}$.
Let $f: \mathbb{R}\to\mathbb{R}$ be a function such that $(f(x))^{n}$ is a polynomial for every integer $n\geq 2$. Is $f$ also a polynomial?
If $f(x)=\log \left(\frac{1+x}{1-x}\right)$ for $-1<x<1$, then $f\left(\frac{3x+x^3}{1+3x^2}\right)$ in terms of $f(x)$ is
$\textbf{(A) }-f(x)\qquad\textbf{(B) }2f(x)\qquad\textbf{(C) }3f(x)\qquad$
$\textbf{(D) }\left[f(x)\right]^2\qquad \textbf{(E) }[f(x)]^3-f(x)$
Morteza places a function $[0,1]\to [0,1]$ (that is a function with domain [0,1] and values from [0,1]) in each cell of an $n \times n$ board. Pavel wants to place a function $[0,1]\to [0,1]$ to the left of each row and below each column (i.e. to place $2n$ functions in total) so that the following condition holds for any cell in this board:
If $h$ is the function in this cell, $f$ is the function below its column, and $g$ is the function to the left of its row, then $h(x) = f(g(x))$ for all $x \in [0, 1]$.
Prove that Pavel can always fulfil his plan.
Given non-zero reals $ a$, $ b$, find all functions $ f: \mathbb{R} \longmapsto \mathbb{R}$, such that for every $ x, y \in \mathbb{R}$, $ y \neq 0$, $ f(2x) \equal{} af(x) \plus{} bx$ and $ \displaystyle f(x)f(y) \equal{} f(xy) \plus{} f \left( \frac {x}{y} \right)$.
Let $ Z$ and $ R$ denote the sets of integers and real numbers, respectively.
Let $ f: Z \rightarrow R$ be a function satisfying:
(i) $ f(n) \ge 0$ for all $ n \in Z$
(ii) $ f(mn)\equal{}f(m)f(n)$ for all $ m,n \in Z$
(iii) $ f(m\plus{}n) \le max(f(m),f(n))$ for all $ m,n \in Z$
(a) Prove that $ f(n) \le 1$ for all $ n \in Z$
(b) Find a function $ f: Z \rightarrow R$ satisfying (i), (ii),(iii) and $ 0<f(2)<1$ and $ f(2007) \equal{} 1$
Does there exist a function $f:\mathbb N\to\mathbb N$ such that
$$
f(f(n+1))=f(f(n))+2^{n-1}
$$
for any positive integer $n$? (As usual, $\mathbb N$ stands for the set of positive integers.)
[i](I. Gorodnin)[/i]