Let $a,b,c\ge 0$

by khanhsy, Mar 29, 2025, 11:16 AM

Let $a,b,c\ge 0$ and $ab+bc+ca>0$. Prove
$$\sum_{cyc}\dfrac{1}{a^3+bc(b+c)} \ge \dfrac{1}{a^3+b^3+c^3}+\dfrac{4}{ab(a+b)+bc(b+c)+ca(c+a)}.$$
inequalities
L

In triangle ABC prove that:

by khanhsy, Mar 29, 2025, 11:12 AM

$$\dfrac{1}{cC+aB+bA}+\dfrac{1}{bB+aC+cA}+\dfrac{1}{aA+bC+cB}\ge \dfrac{1}{aA+bB+cC}+\dfrac{4}{a(B+C)+b(C+A)+c(A+B)}.$$
inequalities
L

My problem

by hacbachvothuong, Mar 29, 2025, 10:10 AM

Let $a, b, c$ be positive real numbers such that $ab+bc+ca=3$. Prove that:
$\frac{a^2}{a^2+b+c}+\frac{b^2}{b^2+c+a}+\frac{c^2}{c^2+a+b}\ge1$
inequalities
L

Let $a,b,c$ be nonnegative real numbers. Prove that

by khanhsy, Mar 29, 2025, 9:21 AM

Let $a,b,c,$ be nonnegative real numbers such that $a+b+c=3$. Prove that:
$$\dfrac{1}{\sqrt{a^3+3bc}}+\dfrac{1}{\sqrt{b^3+3ca}}+\dfrac{1}{\sqrt{c^3+3ab}}\ge \dfrac{1}{\sqrt{a^3+b^3+c^3+abc}}+\dfrac{2\sqrt{2}}{\sqrt{(a+b)(b+c)(c+a)}}.$$
This post has been edited 3 times. Last edited by khanhsy, 4 hours ago
Reason: edit
inequalities
L

Gheorghe Țițeica 2025 Grade 9 P2

by AndreiVila, Mar 28, 2025, 9:09 PM

Let $a,b,c$ be three positive real numbers with $ab+bc+ca=4$. Find the minimum value of the expression $$E(a,b,c)=\frac{a^2+b^2}{ab}+\frac{b^2+c^2}{bc}+\frac{c^2+a^2}{ca}-(a-b)^2.$$
inequalities
L

Gheorghe Țițeica 2025 Grade 8 P2

by AndreiVila, Mar 28, 2025, 8:51 PM

Let $k\geq 2$ be a positive integer and $x_1,x_2,\dots ,x_k\in (0,1)$. Also, let $m_1,m_2,\dots ,m_k$ and $n_1,n_2,\dots ,n_k$ be integers. Define $$A=x_1^{m_1}x_2^{m_2}\dots x_k^{m_k},\quad B=x_1^{n_1}x_2^{n_2}\dots x_k^{n_k}.$$Let $$C=x_1^{\min(m_1,n_1)}x_2^{\min(m_2,n_2)}\dots x_k^{\min(m_k,n_k)}$$$$D=x_1^{\max(m_1,n_1)}x_2^{\max(m_2,n_2)}\dots x_k^{\max(m_k,n_k)}.$$Prove that $A+B\leq C+D$. When does equality hold?

Dorel Miheț
inequalities
L

An inequality on triangles sides

by nAalniaOMliO, Mar 28, 2025, 8:26 PM

Numbers $a,b,c$ are lengths of sides of some triangle. Prove the inequality$$\frac{a}{b+c-a}+\frac{b}{c+a-b}+\frac{c}{a+b-c} \geq \frac{a+b}{2c}+\frac{b+c}{2a}+\frac{c+a}{2b}$$
inequalities
algebra
L

Inspired by old results

by sqing, Mar 28, 2025, 3:10 PM

Let $ a,b,c\geq 0 $ and $a+b+c=3. $ Provethat
$$ \frac{3}{11}\leq \frac{a}{a^2+2}+\frac{b}{b^2+2}+\frac{c}{c^2+2}\leq1$$
inequalities proposed
algebra
inequalities
L

A problem with non-negative a,b,c

by KhuongTrang, Mar 4, 2025, 3:50 PM

Problem. Let $a,b,c$ be non-negative real variables with $ab+bc+ca\neq 0.$ Prove that$$\color{blue}{\sqrt{\frac{8a^{2}+\left(b-c\right)^{2}}{\left(b+c\right)^{2}}}+\sqrt{\frac{8b^{2}+\left(c-a\right)^{2}}{\left(c+a\right)^{2}}}+\sqrt{\frac{8c^{2}+\left(a-b\right)^{2}}{\left(a+b\right)^{2}}}\ge \sqrt{\frac{18(a^{2}+b^{2}+c^{2})}{ab+bc+ca}}.}$$Equality holds iff $(a,b,c)\sim(t,t,t)$ or $(a,b,c)\sim(t,t,0)$ where $t>0.$
inequalities
L

Simple inequality

by sqing, Jul 24, 2017, 3:30 AM

In $\triangle ABC$ with length-side $a,b,c$ , prove that$$\frac{a}{b+c-a}+\frac{b}{c+a-b}+\frac{c}{a+b-c}\ge\frac{b+c-a}{a}+\frac{c+a-b}{b}+\frac{a+b-c}{c}\ge3 .$$
inequalities
triangle inequality
L

scream, don't shout, for contrib

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  • hi super sophomore

    by pqr., Yesterday at 10:01 PM

  • bruh wth am i doing its just /admin never mind.. ok everyone who asked in the last like few months has been added!!

    by flec, Mar 2, 2025, 8:03 AM

  • hey, so my dumb butt hid blog info cuz i used to be petty and care about CSS gatekeeping... i recently switched laptops and forgot how to access it. so, sorry, gimme a sec

    by flec, Mar 2, 2025, 8:01 AM

  • wait contrib plz? <3

    by leyele.lee, Feb 21, 2025, 6:48 PM

  • hii i love ur blog :DD can i have contrib?

    by coolestraccoon, Feb 21, 2025, 5:42 AM

  • hi there

    by ultimate_life_form, Jan 27, 2025, 9:51 PM

  • omg flec your css is amazing and your writing is so interesting
    contrib pls?

    by cinnamon_e, Dec 27, 2024, 7:34 PM

  • Contrib?

    by rayliu985, Dec 23, 2024, 5:09 PM

  • CONTRIB?

    by Moonlight11, Dec 19, 2024, 6:47 PM

  • AAAAAAAACONTRIBAAAAAAAAAAAAA

    by Squidget, Dec 19, 2024, 1:16 AM

  • *screams for conrib.*

    by Pazzaduda, Dec 19, 2024, 12:33 AM

  • happy birthday noozie!!

    by flec, Dec 6, 2024, 4:35 PM

  • yea i'll scream with you- ahhhhhhhhhhhhhh

    by flec, Nov 25, 2024, 8:12 PM

  • can i scream even though i'm contrib already i feel very angry towards my project sekai luck

    by yume_mita, Nov 24, 2024, 11:45 PM

  • awww i love you guys

    by flec, Nov 19, 2024, 6:16 PM

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  • Joined: Sep 30, 2019
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