Very easy inequality

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
$(a+8)(a^2-a+b+2)(b^2-b+5)\geq1331$ $(a+10)(a^2-a+b+4)(b^2-b+7)\geq2197$

29 replies

sqing

Mar 10, 2025

SomeonecoolLovesMaths

5 hours ago

2019 Chile Classification / Qualifying NMO Juniors XXXI

parmenides51 6

N 5 hours ago by bhontu

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?

6 replies

parmenides51

Oct 11, 2021

bhontu

5 hours ago

Inequalities

sqing 12

N 6 hours ago by sqing

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 $148(a^3 + b^3) \geq27 |a^3 b^3 + 1|$

12 replies

sqing

Mar 8, 2025

sqing

6 hours ago

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

parmenides51 3

N Today at 12:09 PM by AshAuktober

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)

3 replies

parmenides51

Apr 19, 2020

AshAuktober

Today at 12:09 PM

Polynomial with roots in geometric progression

red_dog 0

Today at 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 $S_n=x_1^n+x_2^n+x_3^n$ .

0 replies

red_dog

Today at 9:54 AM

0 replies

Good Functional equation question

vexploresmathysics 1

N Today at 9:30 AM by jasperE3

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

1 reply

vexploresmathysics

Jul 1, 2024

jasperE3

Today at 9:30 AM

Functional Equation

AnhQuang_67 2

N Today at 9:03 AM by jasperE3

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

2 replies

AnhQuang_67

Jan 7, 2025

jasperE3

Today at 9:03 AM

a+b+c=3 ine

jokehim 4

N Today at 8:26 AM by lbh_qys

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

4 replies

jokehim

Mar 18, 2025

lbh_qys

Today at 8:26 AM

IOQM P5 2024

SomeonecoolLovesMaths 13

N Today at 8:10 AM by quasar_lord

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:

13 replies

SomeonecoolLovesMaths

Sep 8, 2024

quasar_lord

Today at 8:10 AM

IOQM P4 2024

SomeonecoolLovesMaths 8

N Today at 8:04 AM by quasar_lord

Let $ABCD$ be a quadrilateral with $\angle ADC = 70^{\circ}$ , $\angle ACD = 70^{\circ}$ , $\angle ACB = 10^{\circ}$ and $\angle BAD = 110^{\circ}$ . The measure of $\angle CAB$ (in degrees) is: