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I was doing some problems for Olympiad training and encountered this: How would you prove that

$(a+b+c+d)-(a+c)(b+d)\geq 1$?

We are told that $0<a,b,c,d<1$ and the product $abcd=(1-a)(1-b)(1-c)(1-d)$.

Thanks!!

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4  
Tough indeed, try $a=b=c=d=1$. –  Did Oct 30 '11 at 17:01
    
Sorry restriction given is that a,b,c,d are between 0 and 1. And that $abcd=(1-a)(1-b(1-c)(1-d)$ –  Henry Pan Oct 30 '11 at 17:13
    
@HenryPan Out of curiosity, from where was this taken? –  mathmath8128 Oct 30 '11 at 17:49
1  
Next question: interpret this probabilistically. –  Michael Hardy Oct 30 '11 at 17:54
    
Me and my friend were in an advanced math class and he gave it to us. I belies it's froma book –  Henry Pan Oct 30 '11 at 18:04

3 Answers 3

let $x = a+c$, $y=b+d$, you have to prove

$x+y - xy \geq 1$

$(x-1)(y-1) \leq 0$

Suppose this is false, then $x-1$ and $y-1$ are both positive or both negative. But:

$x-1 > 0$ and $y-1 > 0$

means

$a+c > 1$ and $b+d > 1$

so

$a >1-c$ and $c > 1-a$ and $b> 1-d$ and $d > 1-b$

Multiplying them all you get contradiction with assumption $abcd=(1-a)(1-b)(1-c)(1-d)$. Same argument when they are both negative.

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Write $abcd = (1-a)(1-b)(1-c)(1-d)$ as $\frac{abc}{(1-a)(1-b)(1-c)} = \frac{1-d}{d}$ and solve for $d$: $$d = \frac{(1-a)(1-b)(1-c)}{(1-a)(1-b)(1-c) + abc}$$ Note that the denominator is positive under the given conditions on $a,b,c$. Now substitute that in to $F = a + b + c + d - (a+b)(c+d) - 1$ and put over the common denominator $(1-a)(1-b)(1-c) + abc$. It looks messy, but the numerator must be a polynomial in $a,b,c,d$ of total degree at most 4, and it is 0 when $b=0$ or $b=1$ or $a+c=1$. It turns out that we get $$F = \frac{b(1-b)(a+c-1)^2}{(1-a)(1-b)(1-c) + abc} \ge 0$$

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$$abcd=(1-a)(1-b)(1-c)(1-d) \Rightarrow (a+c+b+d) +abc+abd+acd+bcd =1+ab+ac+ad+bc+bd+cd$$

So you need to prove that

$$1+ ab+ac+ad+bc+bd+cd-abc-abd-acd-bcd \geq 1+ab+ad+bc+bd $$

or

$$ac+bd \geq abc+abd+acd +bcd \,.$$

This seems simpler, but still not obvious. The problem is reduced to:

$$abcd=(1-a)(1-b)(1-c)(1-d) \Rightarrow ac+bd \geq abc+abd+acd +bcd$$

with the extra restriction $0< a,b,c,d <1$.

Lagrange multipliers should solve it, but there is probably a much simpler solution

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