Inequalities

by sqing, Mar 21, 2025, 3:58 PM

Let $a,b,c\geq 0$ and $a+b+c\geq 2+abc .$ Prove that
$a^2+b^2+c^2- \frac{2}{5}abc-\frac{1}{2}a^2b^2c^2\geq \frac{48}{25}$ $a^2+b^2+c^2- \frac{3}{5}abc-\frac{1}{2}a^2b^2c^2\geq \frac{91}{50}$ $a^2+b^2+c^2- \frac{4}{5}abc-\frac{1}{2}a^2b^2c^2\geq \frac{42}{25}$ $a^2+b^2+c^2- \frac{8}{5}abc-\frac{1}{2}a^2b^2c^2\geq \frac{3(7\sqrt{21}-27)}{25}$ $a^2+b^2+c^2- \frac{9}{5}abc-\frac{1}{2}a^2b^2c^2\geq \frac{8}{261+41 \sqrt{41}}$

inequalities

Polynomial with roots in geometric progression

by red_dog, Mar 21, 2025, 9:54 AM

Let $f\in\mathbb{C}[X], \ f=aX^3+bX^2+cX+d, \ a,b,c,d\in\mathbb{R}^*$ a polynomial whose roots $x_1,x_2,x_3$ are in geometric progression with ration $q\in(0,\infty)$ . Find $S_n=x_1^n+x_2^n+x_3^n$ .

a+b+c=3 ine

by jokehim, Mar 18, 2025, 9:48 AM

Problem. Given non-negative real numbers $a,b,c$ satisfying $a+b+c=3.$ Prove that $\color{black}{\frac{a\left(b+c\right)}{bc+3}+\frac{b\left(c+a\right)}{ca+3}+\frac{c\left(a+b\right)}{ab+3}\le \frac{3}{2}.}$ Proposed by Phan Ngoc Chau

inequalities

Inequalities

by sqing, Mar 10, 2025, 2:21 AM

Let $a,b>0$ and $\frac{1}{a}+\frac{1}{b}=1.$ Prove that
$(a^2-a+1)(b^2-b+1) \geq 9$ $(a^2-a+b+1)(b^2-b+a+1) \geq 25$ Let $a,b>0$ and $\frac{1}{a}+\frac{1}{b}=\frac{2}{3}.$ Prove that
$(a+8)(a^2-a+b+2)(b^2-b+5)\geq1331$ $(a+10)(a^2-a+b+4)(b^2-b+7)\geq2197$

This post has been edited 2 times. Last edited by sqing, Mar 10, 2025, 3:15 AM

inequalities

Inequalities

by sqing, Mar 8, 2025, 12:43 PM

Let $a,b$ be real numbers such that $a + b \geq |ab + 1|.$ Prove that $a^3 + b^3 \geq |a^3 b^3 + 1|$ Let $a,b$ be real numbers such that $2(a + b ) \geq |ab + 1|.$ Prove that $26( a^3 + b^3) \geq |a^3 b^3 + 1|$ Let $a,b$ be real numbers such that $4(a + b) \geq 3|ab + 1|.$ Prove that $148(a^3 + b^3) \geq27 |a^3 b^3 + 1|$

This post has been edited 1 time. Last edited by sqing, Mar 8, 2025, 12:54 PM

inequalities

Functional Equation

by AnhQuang_67, Jan 7, 2025, 10:03 AM

Find all function $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying:
$3f(\dfrac{x-1}{3x+2})-5f(\dfrac{1-x}{x-2})=\dfrac{8}{x-1}, \forall x \notin \{0,\dfrac{-2}{3},1,2\}$

function

algebra

functional equation

IOQM P5 2024

by SomeonecoolLovesMaths, Sep 8, 2024, 9:29 AM

Let $a = \frac{x}{y} +\frac{y}{z} +\frac{z}{x}$ , let $b = \frac{x}{z} +\frac{y}{x} +\frac{z}{y}$ and let $c = \left(\frac{x}{y} +\frac{y}{z} \right)\left(\frac{y}{z} +\frac{z}{x} \right)\left(\frac{z}{x} +\frac{x}{y} \right)$ . The value of $|ab-c|$ is:

This post has been edited 1 time. Last edited by SomeonecoolLovesMaths, Sep 8, 2024, 12:52 PM

Good Functional equation question

by vexploresmathysics, Jul 1, 2024, 4:40 PM

If f : R^+ --> R^+ satisfying f(f(x)/y ) = yf ( y ) + (f(x)). Then the value of α such that Sigma K = 1 to n [ 1 / f(K) ] = 420

This post has been edited 1 time. Last edited by vexploresmathysics, Jul 2, 2024, 4:22 AM
Reason: Mistake in question

algebra

functional equation

2019 Chile Classification / Qualifying NMO Juniors XXXI

by parmenides51, Oct 11, 2021, 8:52 PM

p1. Consider the sequence of positive integers $2, 3, 5, 6, 7, 8, 10, 11 ...$ . which are not perfect squares. Calculate the $2019$ -th term of the sequence.

p2. In a triangle $ABC$ , let $D$ be the midpoint of side $BC$ and $E$ be the midpoint of segment $AD$ . Lines $AC$ and $BE$ intersect at $F$ . Show that $3AF = AC$ .

p3. Find all positive integers $n$ such that $n! + 2019$ is a square perfect.

p4. In a party, there is a certain group of people, none of whom has more than $3$ friends in this. However, if two people are not friends at least they have a friend in this party. What is the largest possible number of people in the party?

This post has been edited 3 times. Last edited by parmenides51, Sep 4, 2022, 4:14 PM

FB = BK , circumcircle and altitude related (In the World of Mathematics 516)

by parmenides51, Apr 19, 2020, 2:09 AM

Let $BT$ be the altitude and $H$ be the intersection point of the altitudes of triangle $ABC$ . Point $N$ is symmetric to $H$ with respect to $BC$ . The circumcircle of triangle $ATN$ intersects $BC$ at points $F$ and $K$ . Prove that $FB = BK$ .

(V. Starodub, Kyiv)

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SSD's Musings and Dreams

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by jokehim, Mar 18, 2025, 9:48 AM

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