Found problems: 4776
Let $V$ be the set of non-collinear pairs of vectors in $\mathbb{R}^3$, and $H$ be the set of lines passing through the origin. Is is true that for every continuous map $f\colon V\rightarrow H$ there exists a continuous map $g\colon V\rightarrow \mathbb{R}^3\,\backslash\,\{ 0\}$ such that $g(v)\in f(v)$ for all $v\in V$?
(translated by Miklós Maróti)
Let $ f(x)\equal{}x^2\plus{}3$ and $ y\equal{}g(x)$ be the equation of the line with the slope $ a$, which pass through the point $ (0,\ f(0))$ .
Find the maximum and minimum values of $ I(a)\equal{}3\int_{\minus{}1}^1 |f(x)\minus{}g(x)|\ dx$.
Let $A_n = \{1,2,...,n\}$. Prove or disprove:
For all integers $n \ge 2$ there exist functions $f,g : A_n \to A_n$ which satisfy $f(f(k)) = g(g(k)) = k$ for $1 \le k \le n$, and $g(f(k)) = k +1$ for $1 \le k \le n -1$.
Let $a_1,a_2,a_3,...$ be a sequence of positive real numbers such that:
(i) For all positive integers $m,n$, we have $a_{mn}=a_ma_n$
(ii) There exists a positive real number $B$ such that for all positive integers $m,n$ with $m<n$, we have $a_m < Ba_n$
Find all possible values of $\log_{2015}(a_{2015}) - \log_{2014}(a_{2014})$
The function $\mu: \mathbb{N}\to \mathbb{C}$ is defined by \[\mu(n) = \sum^{}_{k \in R_{n}}\left( \cos \frac{2k\pi}{n}+i \sin \frac{2k\pi}{n}\right),\] where $R_{n}=\{ k \in \mathbb{N}\vert 1 \le k \le n, \gcd(k, n)=1 \}$. Show that $\mu(n)$ is an integer for all positive integer $n$.
How many zeroes does the function $f(x)=2^x -1 -x^2 $ have on the real line?
Let $A$ be a set of $2025$ non-negative integers and $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ be a function with the following two properties:
1) For every two distinct positive integers $x,y$ there exists $a\in A$, such that $x-y$ divides $f(x+a) - f(y+a)$.
2) For every positive integer $N$ there exists a positive integer $t$ such that $f(x) \neq f(y)$ whenever $x,y \in [t, t+N]$ are distinct.
Prove that there are infinitely many primes $p$ such that $p$ divides $f(x)$ for some positive integer $x$.
Let be given $a < b < c$ and $f(x) =\frac{c(x - a)(x - b)}{(c - a)(c - b)}+\frac{a(x - b)(x - c)}{(a - b)(a -c)}+\frac{b(x -c)(x - a)}{(b - c)(b - a)}$.
Determine $f(2014)$.
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that
\[(x-2)f(y)+f(y+2f(x))=f(x+yf(x))\]
for all $x,y\in\mathbb{R}$.
Find all functions $f : \mathbb{Q}^{+} \rightarrow \mathbb{Q}^{+}$, where $\mathbb{Q}^{+}$ is the set of positive rationals, such that $f(x+1) = f(x) + 1$ and $f(x^3) = f(x)^3$ for all $x$.
Let $f:[0,1]\to\mathbb R$ be a continuous function such that $\int^1_0x^mf(x)dx=0$ for $m=0,1,\ldots,1999$. Prove that $f$ has at least $2000$ zeroes on the segment $[0,1]$.
Find all function $ f: R^\plus{} \rightarrow R^\plus{}$ such that for any $ x,y,z \in R^\plus{}$ such that $ x\plus{}y \ge z$ , $ f(x\plus{}y\minus{}z) \plus{}f(2\sqrt{xz})\plus{}f(2\sqrt{yz}) \equal{} f(x\plus{}y\plus{}z)$
For function $ f: \mathbb{R} \to \mathbb{R}$ given that $ f(x^2 +x +3) +2 \cdot f(x^2 - 3x + 5) = 6x^2 - 10x +17$, calculate $ f(2009)$.
Find the following limit.
$ \lim_{n\to\infty} \int_{\minus{}1}^1 |x|\left(1\plus{}x\plus{}\frac{x^2}{2}\plus{}\frac{x^3}{3}\plus{}\cdots \plus{}\frac{x^{2n}}{2n}\right)\ dx$.
Consider the continuous functions $ f:[0,\infty )\longrightarrow\mathbb{R}, g: [0,1]\longrightarrow\mathbb{R} , $ where $
f $ has a finite limit at $ \infty . $ Show that:
$$ \lim_{n \to \infty} \frac{1}{n}\int_0^n f(x) g\left( \frac{x}{n} \right) dx =\int_0^1 g(x)dx\cdot\lim_{x\to\infty} f(x) . $$
Let be a sequence of functions $ a_n:\mathbb{R}\longrightarrow\mathbb{Z} $ defined as $ a_n(x)=\sum_{i=1}^n (-1)^i\lfloor xi\rfloor . $
[b]a)[/b] Find the real numbers $ y $ such that $ \left( a_n(y) \right)_{n\ge 1} $ converges to $ 1. $
[b]b)[/b] Find the real numbers $ z $ such that $ \left( a_n(z) \right)_{n\ge 1} $ converges.
[color=darkred]Let $n$ and $m$ be two natural numbers, $m\ge n\ge 2$ . Find the number of injective functions
\[f\colon\{1,2,\ldots,n\}\to\{1,2,\ldots,m\}\]
such that there exists a unique number $i\in\{1,2,\ldots,n-1\}$ for which $f(i)>f(i+1)\, .$[/color]
The functions $f,g:[0,1]\to\mathbb R$ are given with the formulas
$$f(x)=\sqrt[4]{1-x},\enspace g(x)=f(f(x)),$$
and $c$ denotes any solution of $x=f(x)$.
(a) i. Analyze the function $f(x)$ and draw its graph. Prove that the equation $f(x)=x$ has the unique root $c$ satisfying $c\in[0.72,0.73]$.
ii. Analyze the function $f'(x)$. Let $M_1$ and $M_2$ be the points of the graph of $f(x)$ with different $x$ coordinates. What is the position of the arc $M_1M_2$ of the graph with respect to the segment $M_1M_2$?
iii. Analyze the function $g(x)$ and draw its graph. What is the position of that graph with respect to the line $y=x$? Find the tangents to the graph at points with $x$ coordinates $0$ and $1$.
iv. Prove that every sequence $\{a_n\}$ with the conditions $a_1\in(0,1)$ and
$a_{n+1}=f(a_n)$ for $n\in\mathbb N$ converges.
[hide=Official Hint]Consider the sequences $\{a_{2n-1}\},\{a_{2n}\}~(n\in\mathbb N)$ and the function $g(x)$ associated with the graph.[/hide]
(b) On the graph of the function $f(x)$ consider the points $M$ and $M'$ with $x$ coordinates $x$ and $f(x)$, where $x\ne c$.
i. Prove that the line $MM'$ intersects with the line $y=x$ at the point with $x$ coordinate
$$h(x)=x-\frac{(f(x)-x)^2}{g(x)+x-2f(x)}.$$
ii. Prove that if $x\in(0,c)$ then $h(x)\in(x,c)$.
iii. Analyze whether the sequence $\{a_n\}$ satisfying $a_1\in(0,c),a_{n+1}=h(a_n)$ for $n\in\mathbb N$ converges. Prove that the sequence $\{\tfrac{a_{n+1}-c}{a_n-c}\}$ converges and find its limit.
(c) Assume that the calculator approximates every number $b\in[-2,2]$ by number $\overline b$ having $p$ decimal digits after the decimal point. We are performing the following sequence of operations on that calculator:
1) Set $a=0.72$;
2) Calculate $\delta(a)=\overline{f(a)}-a$;
3) If $|\delta(a)|>0.5\cdot10^{-p}$, then calculate $\overline{h(a)}$ and go to the operation $2)$ using $\overline{h(a)}$ instead of $a$;
4) If $|\delta(a)|\le0.5\cdot10^{-p}$, finish the calculation.
Let $\overleftrightarrow c$ be the last of calculated values for $\overline{h(a)}$. Assuming that for each $x\in[0.72,0.73]$ we have $\left|\overline{f(x)}-f(x)\right|<\epsilon$, determine $\delta(\overleftrightarrow c)$, the accuracy (depending on $\epsilon$) of the approximation of $c$ with $\overleftrightarrow c$.
(d) Assume that the sequence $\{a_n\}$ satisfies $a_1=0.72$ and $a_{n+1}=f(a_n)$ for $n\in\mathbb N$. Find the smallest $n_0\in\mathbb N$, such that for every $n\ge n_0$ we have $|a_n-c|<10^{-6}$.
Determine all functions $ f$ from the set of positive integers to the set of positive integers such that, for all positive integers $ a$ and $ b$, there exists a non-degenerate triangle with sides of lengths
\[ a, f(b) \text{ and } f(b \plus{} f(a) \minus{} 1).\]
(A triangle is non-degenerate if its vertices are not collinear.)
[i]Proposed by Bruno Le Floch, France[/i]
If a function $f$ defined on all real numbers has at least two centers of symmetry, show that this function can be written as sum of a linear function and a periodic function.
[For every real number $x$, if there is a real number $a$ such that $f(a-x) + f(a+x) =2f(a)$, the point $(a,f(a))$ is called a center of symmetry of the function $f$.]
Let $f$ be a real function with a continuous third derivative such that $f(x)$, $f^\prime(x)$, $f^{\prime\prime}(x)$, $f^{\prime\prime\prime}(x)$ are positive for all $x$. Suppose that $f^{\prime\prime\prime}(x)\leq f(x)$ for all $x$. Show that $f^\prime(x)<2f(x)$ for all $x$.
Let the continuous functions $ f_n(x), \; n\equal{}1,2,3,...,$ be defined on the interval $ [a,b]$ such that every point of $ [a,b]$ is a root of $ f_n(x)\equal{}f_m(x)$ for some $ n \not\equal{} m$. Prove that there exists a subinterval of $ [a,b]$ on which two of the functions are equal.
Function $f(x)$. which is defined on the set of non-negative real numbers, acquires real values. It is known that $f(0)\le 0$ and the function $f(x)/x$ is increasing for $x>0$. Prove that for arbitrary $x\ge 0$ and $y\ge 0$, holds the inequality $f(x+y)\ge f(x)+ f(y)$ .
Given positive integers $m$ and $n$, prove that there is a positive integer $c$ such that the numbers $cm$ and $cn$ have the same number of occurrences of each non-zero digit when written in base ten.
A cubic function $ y \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d\ (a\neq 0)$ touches a line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha$ and intersects $ x \equal{} \beta \ (\alpha \neq \beta)$.
Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta$.