College Math number theory

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Do not try to overthink these equations

Sadigly 3

N 27 minutes ago by cj13609517288

Source: Azerbaijan Senior MO 2025 P2

Find all the positive reals $x,y,z$ satisfying the following equations: $y=\frac6{(2x-1)^2}$ $z=\frac6{(2y-1)^2}$ $x=\frac6{(2z-1)^2}$

3 replies

Sadigly

2 hours ago

cj13609517288

27 minutes ago

Another thingy inequality

giangtruong13 2

N 2 hours ago by Double07

Let $a,b,c >0$ such that: $xyz=1$ . Prove that: $\sum_{cyc} \frac{xz+xy}{1+x^3} \leq \sum_{cyc} \frac{1}{x}$

2 replies

giangtruong13

3 hours ago

Double07

2 hours ago

Kosovo MO 2010 Problem 5

Com10atorics 18

N 2 hours ago by justaguy_69

Source: Kosovo MO 2010 Problem 5

Let $x,y$ be positive real numbers such that $x+y=1$ . Prove that
$\left(1+\frac {1}{x}\right)\left(1+\frac {1}{y}\right)\geq 9$ .

18 replies

Com10atorics

Jun 7, 2021

justaguy_69

2 hours ago

Inspired by my own results

sqing 0

2 hours ago

Source: Own

Let $a,b,c$ be real numbers such that $a^2+2b^2+4c^2=1.$ Prove that
$a+2ab+4ca \geq -\frac{5}{4}\sqrt{ \frac{5}{2}}$ Let $a,b,c$ be real numbers such that $a^2+2b^2+c^2=1.$ Prove that
$a+2ab+4ca \geq -\frac{1}{24}\sqrt{2951+145\sqrt{145}}$ Let $a,b,c$ be real numbers such that $10a^2+b^2+c^2=1.$ Prove that
$a+2ab+4ca \geq -\frac{1}{80}\sqrt{3599+161\sqrt{161}}$

0 replies

sqing

2 hours ago

0 replies

help!!!!!!!!!!!!

Cobedangiu 5

N 3 hours ago by sqing

help

5 replies

Cobedangiu

Mar 23, 2025

sqing

3 hours ago

Inspired by lgx57

sqing 2

N 3 hours ago by sqing

Source: Own

Let $a,b>0, a^4+ab+b^4=10$ . Prove that
$\sqrt{10}\leq a^2+ab+b^2 \leq 6$ $2\leq a^2-ab+b^2 \leq \sqrt{10}$ $4\sqrt{10}\leq 4a^2+ab+4b^2 \leq18$ $12<4a^2-ab+4b^2 \leq14$

2 replies

sqing

4 hours ago

sqing

3 hours ago

Inspired by Bet667

sqing 3

N 4 hours ago by sqing

Source: Own

Let $a,b$ be a real numbers such that $a^3+kab+b^3\ge a^4+b^4.$ Prove that
$1-\sqrt{k+1} \leq a+b\leq 1+\sqrt{k+1}$ Where $k\geq 0.$

3 replies

sqing

5 hours ago

sqing

4 hours ago

Inspired by Bet667

sqing 3

N 4 hours ago by sqing

Source: Own

Let $a,b$ be a real numbers such that $a^2+kab+b^2\ge a^3+b^3.$ Prove that $a+b\leq k+2$ Where $k\geq 0.$

3 replies

sqing

May 6, 2025

sqing

4 hours ago

Simple inequality

sqing 22

N Today at 10:01 AM by ND_

Source: JBMO 2011 Shortlist A3

$\boxed{\text{A3}}$ If $a,b$ be positive real numbers, show that: $\displaystyle{\sqrt{\dfrac{a^2+ab+b^2}{3}}+\sqrt{ab}\leq a+b}$

22 replies

sqing

May 15, 2016

ND_

Today at 10:01 AM

Serbia national Olympiad Day 2 Problem 2

IgorM 19

N Today at 9:16 AM by IndexLibrorumProhibitorum

Source: Serbia national Olympiad Day 2 Problem 2

Let $x,y,z$ be nonnegative positive integers.
Prove $\frac{x-y}{xy+2y+1}+\frac{y-z}{zy+2z+1}+\frac{z-x}{xz+2x+1}\ge 0$

19 replies

IgorM

Mar 28, 2015

IndexLibrorumProhibitorum

Today at 9:16 AM

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