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
$(1-abc) (1-a)(1-b)(1-c) \ge 208$ $(1+abc) (1-a)(1-b)(1-c) \le -224$ $(1+a^2b^2c^2) (1-a)(1-b)(1-c) \le -5840$