Algebraic Manipulation

by Darealzolt, Apr 30, 2025, 1:25 PM

Find the number of pairs of real numbers $a, b, c$ that satisfy the equation $a^4 + b^4 + c^4 + 1 = 4abc$ .

algebra

Very tasteful inequality

by tom-nowy, Apr 30, 2025, 10:47 AM

Let $a,b,c \in (-1,1)$ . Prove that $(a+b+c)^2+3>(ab+bc+ca)^2+3(abc)^2.$

Inequality

inequalities

Physical or online

by wimpykid, Apr 30, 2025, 6:49 AM

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Do you think the AoPS print books or the online books are better?

AoPS Books

poll

BrUMO 2025 Team Round Problem 13

by lpieleanu, Apr 27, 2025, 11:18 PM

Let $\omega$ be a circle, and let a line $\ell$ intersect $\omega$ at two points, $P$ and $Q.$ Circles $\omega_1$ and $\omega_2$ are internally tangent to $\omega$ at points $X$ and $Y,$ respectively, and both are tangent to $\ell$ at a common point $D.$ Similarly, circles $\omega_3$ and $\omega_4$ are externally tangent to $\omega$ at $X$ and $Y,$ respectively, and are tangent to $\ell$ at points $E$ and $F,$ respectively.

Given that the radius of $\omega$ is $13,$ the segment $\overline{PQ}$ has a length of $24,$ and $YD=YE,$ find the length of segment $\overline{YF}.$

This post has been edited 1 time. Last edited by lpieleanu, Apr 27, 2025, 11:44 PM
Reason: forgot \overline on last segment

BrUMO

Sequence

by lgx57, Apr 27, 2025, 12:56 PM

$a_1=1,a_{n+1}=a_n+\frac{1}{a_n}$ . Find the general term of $\{a_n\}$ .

algebra

Sequences

đề hsg toán

by akquysimpgenyabikho, Apr 27, 2025, 6:52 AM

làm ơn giúp tôi giải đề hsg

Attachments:

de-chon-hoc-sinh-gioi-tinh-toan-9-nam-2024-2025-so-gddt-lai-chau.pdf (190kb)

math

Inequalities

by sqing, Apr 26, 2025, 12:58 PM

Let $x\in(-1,1).$ Prove that
$\dfrac{1}{\sqrt{1-x^2}} + \dfrac{1}{2+ x^2} \geq \dfrac{3}{2}$ $\dfrac{2}{\sqrt{1-x^2}} + \dfrac{1}{1+x^2} \geq 3$

inequalities

algebra

Three variables inequality

by Headhunter, Apr 20, 2025, 6:58 AM

$\forall a\in R$ , $~\forall b\in R$ , $~\forall c \in R$
Prove that at least one of $(a-b)^{2}$ , $(b-c)^{2}$ , $(c-a)^{2}$ is not greater than $\frac{a^{2}+b^{2}+c^{2}}{2}$ .

I assume that all are greater than it, but can't go more.

inequalities

algebra

proof-writing

Inequlities

by sqing, Jul 19, 2024, 12:48 PM

Let $a,b,c\geq 0$ and $a^2+ab+bc+ca=3 .$ Prove that $\frac{1}{1+a^2}+ \frac{1}{1+b^2}+ \frac{1}{1+c^2} \geq \frac{3}{2}$ $\frac{1}{1+a^2}+ \frac{1}{1+b^2}+ \frac{1}{1+c^2}-bc \geq -\frac{3}{2}$

This post has been edited 1 time. Last edited by sqing, Jul 19, 2024, 12:59 PM

inequalities

algebra

Inequalities

by sqing, Jul 12, 2024, 10:41 AM

Let $a,b,c> 0$ and $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1.$ Prove that
$(1-abc) (1-a)(1-b)(1-c) \ge 208$ $(1+abc) (1-a)(1-b)(1-c) \le -224$ $(1+a^2b^2c^2) (1-a)(1-b)(1-c) \le -5840$

inequalities

algebra

Submit

Here goes first post of 2025! Great blog.
by math_holmes15, Jan 14, 2025, 8:53 AM
First post of 2024
by Yiyj1, Feb 8, 2024, 5:40 AM
First post of 2023
by HoRI_DA_GRe8, Jul 22, 2023, 7:45 AM
Nice blog ! Your isogonality lemma is really powerful !
by 554183, Oct 14, 2021, 8:55 AM
Post plss....
by samrocksnature, Apr 11, 2021, 10:12 PM
alas,this is ded
by Hamroldt, Mar 18, 2021, 4:13 PM
Thanks for the nice blog.
by Feridimo, Mar 6, 2020, 4:17 PM
I think this might be silly but ... when should we expect to have another post ?? I am very keen to see it
by gamerrk1004, Nov 4, 2019, 4:54 PM
Let's all echo what's written in the blog description - Stay Insane / 'Cause it's your labor, will and pain/ That takes you to the top of soda fountain
by Kayak, Oct 2, 2017, 7:18 PM
hey utkarsh jee is over now ... continue your elementary blog pleaseeeeeee!
by kk108, Jun 17, 2017, 11:19 AM
Congrats on becoming a contest moderator!
by Ankoganit, Mar 9, 2017, 5:22 AM
INTERSTING BLOG
by kk108, Feb 19, 2017, 2:04 PM
I have no plans for this blog right now....
No time here people !
But lets see....
I may try some combinatorics
by utkarshgupta, Feb 15, 2017, 12:47 PM
Thanks for the nice blog!
by Orkhan-Ashraf_2002, Feb 13, 2017, 6:34 PM
Revive it!!!
Best blog out there, for sure!
by rmtf1111, Jan 12, 2017, 6:02 PM
Oh you certainly will..
Bu I don't think I will
by achilles04, Feb 11, 2015, 6:24 PM
Thanks
Hope you are there too ...

And hope I get selected
by utkarshgupta, Feb 11, 2015, 11:14 AM
Nice blog ..
keep it up and you might one day become a mathematician lIke me
( just joking )
Btw don't forget us after going to imotc.
by achilles04, Feb 10, 2015, 4:49 PM
nice blog!!!!!
by pradhit, Feb 7, 2015, 11:51 AM
No ideas about the cut off but should be less than 60 according to me...
So you never know
by utkarshgupta, Feb 6, 2015, 4:06 PM
Hi utkarsh,
what do u expect the cutoff to be this year??

by kgdpsdurg, Feb 6, 2015, 12:41 PM
Hi utkarsh
How many did you clear in INMO 2015?
by moldevort, Feb 1, 2015, 3:59 PM
@ Anonymous Bunny
If we both (that should have included me) are lucky enough !
by utkarshgupta, Sep 17, 2014, 2:44 AM
Nice blog. Great job!

If I am lucky enough, hopefully we shall meet at IMOTC.
by AnonymousBunny, Sep 16, 2014, 8:08 PM
Thanx all !!

If anyone wish to contribute anything just pm !

I will add u to the contributors' list !
by utkarshgupta, Sep 16, 2014, 6:35 AM
doing good progress!! btw nice blog.
by rachitgoel, Sep 15, 2014, 4:21 PM
Nice blog.
by Kezer, Aug 5, 2014, 4:04 PM
Hi Utkarsh! Wonderful blog. Keep it up.
by YESMAths, Jul 20, 2014, 1:36 PM
I need some shouts and characters

by utkarshgupta, Jul 20, 2014, 4:22 AM