Cute inequality in equilateral triangle

by Miquel-point, Apr 6, 2025, 6:44 PM

Let $ABC$ be an equilateral triangle, $M$ be a point inside it, and $A',B',C'$ be the intersections of $AM,\; BM,\; CM$ with the sides of $ABC$ . If $A'',\; B'',\; C''$ are the midpoints of $BC$ , $CA$ , $AB$ , show that there is a triangle with sides $A'A''$ , $B'B''$ and $C'C''$ .

Laurențiu Panaitopol

inequalities

geometry

geometric inequality

Conditional upper bound for binomial coeff

by Miquel-point, Apr 6, 2025, 6:39 PM

Let $n>r\geqslant 3$ be two integers and $d$ be a positive integer such that $nd\geqslant \dbinom{n+r}{r+1}$ . Show that $(n-t)(d-t)>\dbinom{n-t+r}{r+1},$ for $t=1,2,\ldots,n-1$

Vasile Brânzănescu

algebra

binomial coefficients

inequalities

A very nice inequality

by KhuongTrang, Apr 6, 2025, 1:35 PM

Problem. Let $a,b,c\in \mathbb{R}:\ a+b+c=3.$ Prove that $\color{black}{\sqrt{5a^{2}-ab+5b^{2}}+\sqrt{5b^{2}-bc+5c^{2}}+\sqrt{5c^{2}-ca+5a^{2}}\le 2(a^2+b^2+c^2)+ab+bc+ca.}$ When does equality hold?

inequalities

3 var inquality

by sqing, Apr 6, 2025, 1:11 PM

Let $a,b,c>0$ and $\dfrac{a}{bc}+\dfrac{2b}{ca}+\dfrac{5c}{ab}\leq 12.$ Prove that $a^2+b^2+c^2\geq 1$

inequalities proposed

inequalities

inequality ( 4 var

by SunnyEvan, Apr 4, 2025, 5:19 AM

Let $a,b,c,d \in R$ , such that $a+b+c+d=4 .$ Prove that :
$a^4+b^4+c^4+d^4+3 \geq \frac{7}{4}(a^3+b^3+c^3+d^3)$ $a^4+b^4+c^4+d^4+ \frac{252}{25} \geq \frac{88}{25}(a^3+b^3+c^3+d^3)$ equality cases : ?

This post has been edited 4 times. Last edited by SunnyEvan, Apr 4, 2025, 7:01 AM

inequalities

hard problem

by Cobedangiu, Mar 27, 2025, 2:54 PM

problem

inequalities

inequality

by pennypc123456789, Mar 24, 2025, 11:17 AM

Let $x, y$ be positive real numbers satisfying $x + y = 2$ . Prove that

$3(x^{\frac{2}{3}} + y^{\frac{2}{3}}) \geq 4 + 2x^{\frac{1}{3}}y^{\frac{1}{3}}.$

This post has been edited 1 time. Last edited by pennypc123456789, Mar 24, 2025, 11:22 AM

inequalities

inequalities proposed

Conditional maximum

by giangtruong13, Mar 22, 2025, 4:26 PM

Let $a,b$ satisfy that: $1 \leq a \leq2$ and $1 \leq b \leq 2$ . Find the maximum: $A=(a+b^2+\frac{4}{a^2}+\frac{2}{b})(b+a^2+\frac{4}{b^2}+\frac{2}{a})$

This post has been edited 1 time. Last edited by giangtruong13, Mar 22, 2025, 4:26 PM

inequalities

2018 PAMO Shortlist: Inequality with condition $a^3 + b^3 + c^3 = 5abc$

by DylanN, May 6, 2019, 2:03 PM

Let $a, b, c$ be positive real numbers such that $a^3 + b^3 + c^3 = 5abc$ .

Show that
$\left( \frac{a + b}{c} \right) \left( \frac{b + c}{a} \right) \left( \frac{c + a}{b} \right) \geq 9.$

inequalities

inequalities proposed

algebra

Simple cube root inequality [Taiwan 2014 Quizzes]

by v_Enhance, Jul 18, 2014, 7:33 PM

Prove that for positive reals $a$ , $b$ , $c$ we have $3(a+b+c) \ge 8\sqrt[3]{abc} + \sqrt[3]{\frac{a^3+b^3+c^3}{3}}.$

inequalities

marvio

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