Find all functions $f$: \(\mathbb{R}\) \(\rightarrow\) \(\mathbb{R}\) such : $f(

by guramuta, May 18, 2025, 2:18 PM

Find all functions $f$: \(\mathbb{R}\) \(\rightarrow\) \(\mathbb{R}\) such :
$f(x+yf(x)) + f(xf(y)-y) = f(x) - f(y) + 2xy$
algebra
function
Function equations
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Hard math inequality

by noneofyou34, May 18, 2025, 2:00 PM

If a,b,c are positive real numbers, such that a+b+c=1. Prove that:
(b+c)(a+c)/(a+b)+ (b+a)(a+c)/(c+b)+(b+c)(a+b)/(a+c)>= Sqrt.(6(a(a+c)+b(a+b)+c(b+c)) +3
inequalities
L

Inspired by 2022 MARBLE - Mock ARML

by sqing, May 18, 2025, 1:34 PM

Let $ a,b,c\geq 0 , \frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}= 5 $ and $ \frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}=32. $ Prove that $$\frac{1}{2}>ab+bc+ca \geq \frac{49}{34}$$Let $ a,b,c\geq 0 ,ab+bc+ca = \frac{49}{34} $ and $ \frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}=32. $ Prove that $$\frac{51}{10}>\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\geq5$$Let $ a,b,c\geq 0 ,ab+bc+ca = \frac{49}{34} $ and $ \frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}=5. $ Prove that $$\frac{63}{2}<\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\leq32$$
inequalities proposed
ARML
algebra
inequalities
L

Inequality on non-nagative numbers

by TUAN2k8, May 18, 2025, 12:42 PM

Let $a,b,c$ be non-nagative real numbers such that $a+b+c=3$.
Prove that $ab+bc+ca-abc \leq \frac{9}{4}$.
algebra
inequalities
inequalities proposed
L

2-var inequality

by sqing, May 18, 2025, 6:16 AM

Let $ a,b>0, ab^2+a+2b\geq4 $. Prove that$$ \frac{a}{2a+b^2}+\frac{2}{a+2}\leq 1$$
inequalities proposed
inequalities
algebra
L

Interesting inequalities

by sqing, May 16, 2025, 4:34 AM

Let $a,b,c \geq 0 $ and $ab+bc+ca- abc =3.$ Show that
$$a+k(b+c)\geq 2\sqrt{3 k}$$Where $ k\geq 1. $
Let $a,b,c \geq 0 $ and $2(ab+bc+ca)- abc =31.$ Show that
$$a+k(b+c)\geq \sqrt{62k}$$Where $ k\geq 1. $
inequalities proposed
inequalities
algebra
L

Good divisors and special numbers.

by Nuran2010, Apr 29, 2025, 4:52 PM

$N$ is a positive integer. Call all positive divisors of $N$ which are different from $1$ and $N$ beautiful divisors.We call $N$ a special number when it has at least $2$ beautiful divisors and difference of any $2$ beautiful divisors divides $N$ as well. Find all special numbers.
number theory
TST
AZE Al-Khwarizmi IJMO TST
L

JBMO Shortlist 2023 N2

by Orestis_Lignos, Jun 28, 2024, 6:37 AM

A positive integer is called Tiranian if it can be written as $x^2+6xy+y^2$, where $x$ and $y$ are (not necessarily distinct) positive integers. The integer $36^{2023}$ is written as the sum of $k$ Tiranian integers. What is the smallest possible value of $k$?

Proposed by Miroslav Marinov, Bulgaria
This post has been edited 1 time. Last edited by Orestis_Lignos, Jul 1, 2024, 8:54 PM
Reason: Add proposer.
JBMO Shortlist
number theory
L

d1-d2 divides n for all divisors d1, d2

by a_507_bc, May 20, 2023, 4:34 PM

Determine all natural numbers $n \geq 2$ with at most four natural divisors, which have the property that for any two distinct proper divisors $d_1$ and $d_2$ of $n$, the positive integer $d_1-d_2$ divides $n$.
number theory
L

Mary and Pat play a number game, smallest initial integer for Pat not winning

by parmenides51, Sep 16, 2018, 9:17 AM

Mary and Pat play the following number game. Mary picks an initial integer greater than $2017$. She then multiplies this number by $2017$ and adds $2$ to the result. Pat will add $2019$ to this new number and it will again be Mary’s turn. Both players will continue to take alternating turns. Mary will always multiply the current number by $2017$ and add $2$ to the result when it is her turn. Pat will always add $2019$ to the current number when it is his turn. Pat wins if any of the numbers obtained by either player is divisible by $2018$. Mary wants to prevent Pat from winning the game.
Determine, with proof, the smallest initial integer Mary could choose in order to achieve this.
game strategy
game
number theory
minimum
L

Some random interesting things

avatar

Ankoganit
Shouts
Submit

  • reap 5 months
    (i came here from searching google’s “nag a ram” and somehow ended up in aops again)

    by rhydon516, Mar 23, 2025, 6:21 PM

  • 1 year passed again (not exactly but u get the point

    by alexanderhamilton124, Oct 11, 2024, 12:36 PM

  • 1 year passed again

    by HoRI_DA_GRe8, Jan 4, 2024, 4:43 PM

  • @below i get to be the second person then :)

    by kamatadu, Jan 21, 2023, 12:11 PM

  • Oh the blog is dead again.Fun fact nobody has even shouted since it's last post.

    by HoRI_DA_GRe8, Jan 4, 2023, 3:58 AM

  • monthly check finally sees its revival

    by Rickyminer, Aug 24, 2021, 5:26 PM

  • woohoo!! revive

    by DofL, Aug 12, 2021, 3:22 PM

  • yay!!!!!

    by p_square, Jul 30, 2021, 8:28 AM

  • @all proponents of revival: there you go.

    by Ankoganit, Jul 30, 2021, 8:23 AM

  • \revive \revive \revive

    by tumbleweed, Jun 4, 2021, 1:07 AM

  • Wow, this blog is amazinggg! Revive!!

    by L567, May 17, 2021, 9:04 AM

  • 'Elegant, not Elephant', 'Some properties of ferrous nitride'. Haha, creative titles I must say :). This is the reason I guessed you were one of the proposers of INMO P6 after reading 'A ninth-graders guide to polynomials.' Really love your work. :D

    by RMOAspirantFaraz, Mar 8, 2021, 1:20 PM

  • ;Puffer13

    by Puffer13, Oct 13, 2020, 5:34 PM

  • @Prabh1512 the secret is lots of caffeine

    by Ankoganit, Sep 30, 2020, 6:12 AM

  • How to get the motivation to carry out random polynomial multiplication ? TO BASH of course

    by Hamroldt, Sep 22, 2020, 4:41 AM

95 shouts
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