## Tuesday, January 28, 2014

### Four positive numbers

Let $a_1,a_2,a_3,a_4$ be four positive numbers and let:
$$S_1 = a_1 + a_2 + a_3 + a_4$$
$$S_2 = \sum_{i \neq j} a_ia_j$$
$$S_3 = a_1a_2a_3 + a_1a_2a_4 + a_1a_3a_4 + a_2a_3a_4$$
$$S_4 = a_1a_2a_3a_4$$
Given that $$| \frac{S_1-S_3}{1-S_2+S_4} | < 1$$ show that there are two distinct $a_i,a_j$ such that : $$|a_i-a_j| < (\sqrt{2}-1)(1+a_ia_j)$$

Labels:
Algebra,
complex,
tangent,
transformation

## Wednesday, January 15, 2014

### Infimum of abc

Find the largest number $M$ such that for any given distinct number $a,b,c$ such that $0 < a,b,c < 1$ we always have:
$$\frac{(a+b)(b+c)(c+a)}{|(a-b)(b-c)(c-a)|} > M$$

## Solution

Clearly $M=1$ satisfies the condition. Now we prove that we can get the LHS arbitrarily close to 1.Let $a$ be an arbitrary number < 1, $b = a/(1+x)$ and $c=a/(1+x+yx)$ for some really large numbers $x,y$. The LHS now becomes: $$\frac{(2+x)(2+(y+1)x)(2+(y+2)x)}{y(y+1)x^3}$$ $$=(\frac{2}{x}+1)(\frac{2}{x(y+1)} + 1)(\frac{2}{xy} + \frac{2}{y} + 1)$$ We can set $x,y$ large enough that the expression is as close as needed to 1.

Labels:
Algebra,
infimum,
normalization,
symmetric

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