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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
Jump to post
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
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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
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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
Jump to post
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
Jump to post
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
Jump to post
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
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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
Jump to post
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
Jump to post
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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