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
L

Physical or online

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

Loading poll details...
Do you think the AoPS print books or the online books are better?
AoPS Books
poll
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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
L

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
L

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
L

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
L

Geometric inequality

by ReticulatedPython, Apr 22, 2025, 5:12 PM

Let $A$ and $B$ be points on a plane such that $AB=n$, where $n$ is a positive integer. Let $S$ be the set of all points $P$ such that $\frac{AP^2+BP^2}{(AP)(BP)}=c$, where $c$ is a real number. The path that $S$ traces is continuous, and the value of $c$ is minimized. Prove that $c$ is rational for all positive integers $n.$
This post has been edited 2 times. Last edited by ReticulatedPython, Apr 22, 2025, 8:06 PM
geometry
inequalities
geometric inequality
L

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
L

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
L

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
L

"Where wisdom and valor fail, all that remains is faith. . . And it can overcome all." -Toa Mata Tahu

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  • The most viewed blog of all time

    by giangtruong13, Feb 24, 2025, 1:37 PM

  • wow this blog is ancient
    it started when I was an infant
    wow and like 1.5 million views

    by nmlikesmath, Oct 13, 2024, 10:09 PM

  • no ur cool

    tbh it scares me too I still dont know where those views came from

    by EpicSkills32, Oct 5, 2024, 9:44 AM

  • wowwww this blog is so cool

    by QueenArwen, Oct 4, 2024, 8:25 AM

  • it scares me how many views this blog has

    by prettyflower, Sep 29, 2024, 9:38 AM

  • bruh
    this blog started when i was like 2 months old

    by TimmyL, Sep 27, 2024, 9:18 AM

  • 1579363 visits rn, curious how many views (if any) this blog gets now

    by EpicSkills32, Sep 16, 2024, 10:47 AM

  • interesting lol

    by EpicSkills32, Jul 27, 2024, 11:27 AM

  • EpicSkills32 reaped on Jul 27, 18:44:19 and gained 10 minutes, 57 seconds

    orz orz orz

    highest raw of the game so far

    by DuoDuoling0, Jul 27, 2024, 10:47 AM

  • whoa this blog is old

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  • greetings

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  • hi$ $

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  • orzorzorz can i have contrib :)

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  • 1.5M views pro

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  • orz how did i not know about u before

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