Computing functions

by BBNoDollar, May 18, 2025, 5:25 PM

Let $f : [0, \infty) \to [0, \infty)$ , $f(x) = \dfrac{ax + b}{cx + d}$ , with $a, d \in (0, \infty)$ , $b, c \in [0, \infty)$ . Prove that there exists $n \in \mathbb{N}^*$ such that for every $x \geq 0$
$f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.$ (For $n \in \mathbb{N}^*$ and $x \geq 0$ , the notation $f_n(x)$ represents $\underbrace{(f \circ f \circ \dots \circ f)}_{n \text{ times}}(x)$ . )

function

algebra

Functional Inequality Implies Uniform Sign

by peace09, Jul 17, 2024, 12:00 PM

Let $\mathbb{R}$ be the set of real numbers. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that $f(x+y)f(x-y)\geqslant f(x)^2-f(y)^2$ for every $x,y\in\mathbb{R}$ . Assume that the inequality is strict for some $x_0,y_0\in\mathbb{R}$ .

Prove that either $f(x)\geqslant 0$ for every $x\in\mathbb{R}$ or $f(x)\leqslant 0$ for every $x\in\mathbb{R}$ .

This post has been edited 2 times. Last edited by peace09, Jul 17, 2024, 12:27 PM

algebra

function

IMO Shortlist

3^x+4xy=5^y diophantine

by parmenides51, Dec 3, 2023, 8:20 AM

Find all ordered pairs of natural numbers $(x,y)$ such that $3^x+4xy=5^y.$
Proposed by i3435

number theory

Diophantine equation

diophantine

Oh no! Inequality again?

by mathisreaI, Jul 13, 2022, 2:52 AM

Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that for each $x \in \mathbb{R}^+$ , there is exactly one $y \in \mathbb{R}^+$ satisfying $xf(y)+yf(x) \leq 2$

IMO

Functional inequality

Floor double summation

by CyclicISLscelesTrapezoid, Jul 12, 2022, 12:52 PM

Which positive integers $n$ make the equation $\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}$ true?

Grand finale of 2021 Iberoamerican MO

by jbaca, Oct 20, 2021, 11:08 PM

Consider a $n$ -sided regular polygon, $n \geq 4$ , and let $V$ be a subset of $r$ vertices of the polygon. Show that if $r(r-3) \geq n$ , then there exist at least two congruent triangles whose vertices belong to $V$ .

geometry

congruent triangles

Constructing two sets from conditions on their intersection, union and product

by jbaca, Oct 20, 2021, 11:02 PM

For a finite set $C$ of integer numbers, we define $S(C)$ as the sum of the elements of $C$ . Find two non-empty sets $A$ and $B$ whose intersection is empty, whose union is the set $\{1,2,\ldots, 2021\}$ and such that the product $S(A)S(B)$ is a perfect square.

This post has been edited 1 time. Last edited by jbaca, Oct 20, 2021, 11:54 PM
Reason: Typo

sets of integers

Intersection

union

Sets with Polynomials

by insertionsort, Jul 20, 2021, 9:06 PM

Let $\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\mathcal{B}$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as
$\begin{align*} (x + y + z)P(x, y, z) + (xy + yz + zx)Q(x, y, z) + xyzR(x, y, z) \end{align*}$ with $P, Q, R \in \mathcal{A}$ . Find the smallest non-negative integer $n$ such that $x^i y^j z^k \in \mathcal{B}$ for all non-negative integers $i, j, k$ satisfying $i + j + k \geq n$ .

Mmmmmm...Tasty!

by whatshisbucket, Jun 26, 2017, 7:03 AM

An integer $n>2$ is called tasty if for every ordered pair of positive integers $(a,b)$ with $a+b=n,$ at least one of $\frac{a}{b}$ and $\frac{b}{a}$ is a terminating decimal. Do there exist infinitely many tasty integers?

Proposed by Vincent Huang

number theory

ELMO 2017

Elmo

IMO Shortlist 2010 - Problem N1

by Amir Hossein, Jul 17, 2011, 2:46 AM

Find the least positive integer $n$ for which there exists a set $\{s_1, s_2, \ldots , s_n\}$ consisting of $n$ distinct positive integers such that
$\left( 1 - \frac{1}{s_1} \right) \left( 1 - \frac{1}{s_2} \right) \cdots \left( 1 - \frac{1}{s_n} \right) = \frac{51}{2010}.$

Proposed by Daniel Brown, Canada

Submit

how do you have so many posts
by krithikrokcs, Jul 14, 2023, 6:20 PM
lol⠀⠀⠀⠀⠀
by math_explorer, Jan 20, 2021, 8:43 AM
woah ancient blog
by suvamkonar, Jan 20, 2021, 4:14 AM
https://artofproblemsolving.com/community/c47h361466
by math_explorer, Jun 10, 2020, 1:20 AM
when did the first greed control game start?
by piphi, May 30, 2020, 1:08 AM
ok..........
by asdf334, Sep 10, 2019, 3:48 PM
There is one existing way to obtain contributorship documented on this blog. See if you can find it.
by math_explorer, Sep 10, 2019, 2:03 PM
SO MANY VIEWS!!!
PLEASE CONTRIB

by asdf334, Sep 10, 2019, 1:58 PM
Hullo bye
by AnArtist, Jan 15, 2019, 8:59 AM
Hullo bye
by tastymath75025, Nov 22, 2018, 9:08 PM
Hullo bye
by Kayak, Jul 22, 2018, 1:29 PM
It's sad; the blog is still active but not really ;-;
by GeneralCobra19, Sep 21, 2017, 1:09 AM
dope css
by zxcv1337, Mar 27, 2017, 4:44 AM
nice blog ^_^
by chezbgone, Mar 28, 2016, 5:18 AM
shouts make blogs happier
by briantix, Mar 18, 2016, 9:58 PM
I like your CSS. Can you send me your shoutbox block? I'll credit you.
by bluecarneal, Feb 21, 2011, 8:24 PM
Hey contrib?
by Dsquared, Dec 16, 2010, 3:14 PM
$\sum_{n=0}^{\infty}2^n=-1$
by AwesomeToad, Dec 13, 2010, 2:12 PM
Delete the shouts...

If you wish something more mathematical, here's some digits.
1679350278936041982752
Enjoy. (no offense)
by chaotic_iak, Dec 3, 2010, 1:00 PM
Now I feel guilty my shoutbox is so nonmathematical.
by math_explorer, Dec 3, 2010, 12:36 PM
Congratulations on winning Achievement Unlocked: AoPS Style! I'm looking to your participation in Achievement Unlocked 2: Reunlock the Achievements!
by chaotic_iak, Dec 3, 2010, 12:32 PM
El_ectric won game 11
by halowarfare17, Nov 17, 2010, 2:36 PM
Did you win reaper or did electric?
by dragon96, Nov 11, 2010, 2:14 AM
Very nice blog
by Amir Hossein, Oct 28, 2010, 2:58 PM
Changed your avatar?
by asf, Oct 15, 2010, 3:47 AM
Gosh, post count over 9000.
by math_explorer, Sep 20, 2010, 1:44 PM
Nice blog I read pretty much whenever you make a new post, but I don't comment.
by AIME15, Sep 19, 2010, 2:38 PM
I see, that part of the CSS needs a lot of tweaking.
by math_explorer, Sep 12, 2010, 9:20 AM
Hello. Shout?
by asf, Sep 11, 2010, 5:18 PM

91 shouts

math_exploration

by BBNoDollar, May 18, 2025, 5:25 PM

by peace09, Jul 17, 2024, 12:00 PM

by parmenides51, Dec 3, 2023, 8:20 AM

by mathisreaI, Jul 13, 2022, 2:52 AM

by CyclicISLscelesTrapezoid, Jul 12, 2022, 12:52 PM

by jbaca, Oct 20, 2021, 11:08 PM

by jbaca, Oct 20, 2021, 11:02 PM

by insertionsort, Jul 20, 2021, 9:06 PM

by whatshisbucket, Jun 26, 2017, 7:03 AM

by Amir Hossein, Jul 17, 2011, 2:46 AM