AoPS Academy
is given. Let 
be set of all 
segments of real line of type ![$[i, j]$](http://latex.artofproblemsolving.com/a/6/4/a644edd67caa59eaded765c417b6d9822f782ec2.png)
, where 
and 
are integers, 
. A subset 
is said to be valuable if the intersection of any two segments from 
is either empty, or is a segment of nonzero length belonging to . Find the number of valuable subsets of set .
rows of coins. There is 1 coin in the first row, 2 coins in the second row, 3 coins in the third row, and so on, up to coins in the th row. Initially, all of the coins show heads (H). Carley plays a game in which, on each turn, she chooses three mutually adjacent coins and flips these three coins over. To win the game, all of the coins must be showing tails (T) after a sequence of turns. An example game with 4 rows of coins after a sequence of two turns is shown. for which it is possible to win the game starting with rows of coins.
satisfying 
, prove that![\[ \frac{a^4+5b^4}{a(a+2b)} + \frac{b^4+5c^4}{b(b+2c)} + \frac{c^4+5a^4}{c(c+2a)} \geq 1- ab-bc-ca \]](http://latex.artofproblemsolving.com/f/e/8/fe85cf52b883c214806ab837a2eb25cebb511d7b.png)
and 
to triangles 
and 
. By the properties of rotation and the given condition, it is easy to see that 
and 
and 
, 
, and 
. The latter 3 conditions give 
, so 
. Since any broken line starting at A and ending at C is longer than 
itself, the problem is solved.
to 
such that coincides with , and define 
similarly. Then 
, and thus 
by triangle inequality.
denote the reflections of 
and 
respectively. We remark that 
and that 
are perpendicular to lines 
respectively. It follows that 
,
.![\[BM+MN+BN = XM+MN+NY \geq XY = AC\]](http://latex.artofproblemsolving.com/5/0/1/50154f69553d05a6b5fcd6d1d59f0c650e40b068.png)
and thus the perimeter of 
is not less than 