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Re: Good inequality..

good and clear solution. Thanks! :)
by Katy729
Sat Jul 01, 2017 3:46 pm
 
Forum: Algebra
Topic: Good inequality..
Replies: 2
Views: 136

Beautifull divisibility

Determine all pairs $(a,b)$ of positive integers such that $a^{2}b+a+b$ is divisible by $ab^{2}+b+7$
by Katy729
Sat Jul 01, 2017 3:44 pm
 
Forum: Number Theory
Topic: Beautifull divisibility
Replies: 0
Views: 94

no solution (a,b)

Prove that there is no pair $(a,b)$ of integers such that
$a^2=b^7+7$
by Katy729
Sat Jul 01, 2017 3:40 pm
 
Forum: Number Theory
Topic: no solution (a,b)
Replies: 0
Views: 81

Yet divisibility...

Determine all ordered pairs $(a,b)$ of positive integers for which $\dfrac{b^3+1}{ab-1}$ is an integer.
by Katy729
Sat Jul 01, 2017 3:35 pm
 
Forum: Number Theory
Topic: Yet divisibility...
Replies: 0
Views: 67

a+b+c>=1/a+1/b+1/c

Let $a,b,c$ be positive real numbers, such that: $a+b+c \geq \frac{1}{a}+\frac{1}{b}+\frac{1}{c}.$

Prove that:
\[a+b+c \geq \frac{3}{a+b+c}+\frac{2}{abc}. \]
by Katy729
Sat Jul 01, 2017 3:31 pm
 
Forum: Algebra
Topic: a+b+c>=1/a+1/b+1/c
Replies: 0
Views: 61

Re: Very strange inequality..

Let $a$,$b$,$c$ be real positive numbers. Prove that
\[\left(\frac{a^3+abc}{b+c}\right)+\left(\frac{b^3+abc}{c+a}\right)+\left(\frac{c^3+abc}{a+b}\right)\ge a^2+b^2+c^2\]
by Katy729
Sat Jul 01, 2017 3:24 pm
 
Forum: Algebra
Topic: Very strange inequality..
Replies: 1
Views: 89

Inequality with a,b,c sides of a triangle

Let $a,b$ and $c$ be sides of a triangle. Prove that:
$\frac{\sqrt{b+c-a}}{\sqrt{b}+\sqrt{c}-\sqrt{a}}+\frac{\sqrt{c+a-b}}{\sqrt{c}+\sqrt{a}-\sqrt{b}}+\frac{\sqrt{a+b-c}}{\sqrt{a}+\sqrt{b}-\sqrt{c}}\leq 3$
by Katy729
Sat Jul 01, 2017 3:24 pm
 
Forum: Algebra
Topic: Inequality with a,b,c sides of a triangle
Replies: 0
Views: 53

2007 number 5 - divisibility

Let $a$ and $b$ be positive integers. Show that if $4ab-1$ divides $(4a^2-1)^2$, then $a=b$.
by Katy729
Sat Jun 24, 2017 3:31 pm
 
Forum: International Mathematical Olympiad (IMO)
Topic: 2007 number 5 - divisibility
Replies: 0
Views: 128

Divisibility... with x,y

Determine all pairs of positive integers $(x,y)$ such that
$(7^x-3^y)|(x^4+y^2)$
by Katy729
Sun Jun 18, 2017 10:50 pm
 
Forum: Number Theory
Topic: Divisibility... with x,y
Replies: 1
Views: 78

n^5+n^4

Determine all pairs of positive integers $(m,n)$ such that
$n^5+n^4=7^m-1$
by Katy729
Sun Jun 18, 2017 10:48 pm
 
Forum: Number Theory
Topic: n^5+n^4
Replies: 1
Views: 87
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