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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: 33

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: 60

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: 55

Very strange inequality..

Let $a$,$b$,$c$ be real positive numbers. Prove that
\[\left(\frac{a^3+abc}{b+c}\right)+\left(\frac{b^3+abc}{c+a}\right)+\left(\frac{c^3+abc}{a+b}\right)\ge a^2+b^2+c^2\]
[/quote]
by Katy729
Sun Jun 18, 2017 10:43 pm
 
Forum: Algebra
Topic: Very strange inequality..
Replies: 0
Views: 52

Good inequality..

Let a, b, c be real positive numbers. Prove that

$$(1+a/b)(1+b/c)(1+c/a)>=2(1+(a+b+c)/(\sqrt[3]{abc}))$$
by Katy729
Sun Jun 18, 2017 10:36 pm
 
Forum: Algebra
Topic: Good inequality..
Replies: 1
Views: 59

Particular divisibility..

Find all pairs $(a,b)$ of positive integers for which $a^2-b$ divide $b^2+a$, and $b^2-a$
Divides $a^2 + b$
by Katy729
Sun May 21, 2017 2:46 am
 
Forum: Number Theory
Topic: Particular divisibility..
Replies: 1
Views: 95

Determine n

Determine if there are integers $n$ so that $n^2-4$ has exactly $10$ divisors (positive).
by Katy729
Sun May 21, 2017 2:44 am
 
Forum: Number Theory
Topic: Determine n
Replies: 1
Views: 69

Re: Inequality with abc = 1

Good solution Atonu! :)
by Katy729
Sun May 21, 2017 2:42 am
 
Forum: Algebra
Topic: Inequality with abc = 1
Replies: 3
Views: 118

Inequality with abc = 1

Let $ x, y, z$ be positive real numbers so that $ xyz = 1$. Prove that

\[ \left( x - 1 + \frac 1y \right) \left( y - 1 + \frac 1z \right) \left( z - 1 + \frac 1x \right) \leq 1.
\]
by Katy729
Mon May 15, 2017 2:03 am
 
Forum: Algebra
Topic: Inequality with abc = 1
Replies: 3
Views: 118

A strange divisibility

Determine all pairs of positive integers $(x,y)$ such that \[ \dfrac{x^2}{2xy^2-y^3+1} \] is a positive integer.
by Katy729
Mon May 15, 2017 1:53 am
 
Forum: Number Theory
Topic: A strange divisibility
Replies: 1
Views: 115
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