,
,
,![\[ \frac {1}{a^3 + b^3 + abc} + \frac {1}{b^3 + c^3 + abc} + \frac {1}{c^3 + a^3 + abc} \leq \frac {1}{abc}
\]](http://latex.artofproblemsolving.com/f/a/9/fa964e29de91f0e2f3cb32e335a52d4521bc38fd.png)
.![\[ \frac {1}{a^{3}\left(b + c\right)} + \frac {1}{b^{3}\left(c + a\right)} + \frac {1}{c^{3}\left(a + b\right)}\geq \frac {3}{2}.
\]](http://latex.artofproblemsolving.com/b/2/9/b295b0b2a765e41a639657e13b724c60323fa208.png)
.
min
be a triangle. Let 
be the feet of the altitudes from 
respectively. Let 
be the projections of 
.
.
Y by
You find yourself on a flight to Bosnia and Herzegovina. You are seated next to a strange man wearing a hat—similar to the one worn by Indiana Jones. The man spends most of the flight rambling about some pyramids.
However, as you approach your destination, he tells you the following:
"I have thought of a
number. I will allow you to ask me questions. A question consists of you saying a three-digit number, to which I respond 'yes' if at least two of its digits are correct, or 'no' otherwise. (A digit is correct if it matches the corresponding digit in my number.)
You can ask as many questions as you like, but they come at a price. The game ends when you have correctly identified two digits of my number. If
you guess correctly, I will sell you my precious pyramid. You will have to pay a number of Bosnian marks equal to the number of questions you asked.
What is the maximum number of Bosnian marks you will have to pay using the best strategy?"
However, as you approach your destination, he tells you the following:
"I have thought of a
number. I will allow you to ask me questions. A question consists of you saying a three-digit number, to which I respond 'yes' if at least two of its digits are correct, or 'no' otherwise. (A digit is correct if it matches the corresponding digit in my number.)You can ask as many questions as you like, but they come at a price. The game ends when you have correctly identified two digits of my number. If
you guess correctly, I will sell you my precious pyramid. You will have to pay a number of Bosnian marks equal to the number of questions you asked.
What is the maximum number of Bosnian marks you will have to pay using the best strategy?"
