Every week, we will select three notable or interesting problems, marked with E,M,H (“easy”, “medium”, “hard”) for the relative difficulty. Easy problems will be around TMO or easier than TMO; medium problems will be around Oct Camp / easy IMO, and hard problems will be around medium / hard IMO.
E2 [@Konigsberg on AoPS]
In a convex pentagon, show that we can choose 
diagonals such that their lengths can form a triangle.
M2 [China TST 2007 Quiz]
Let 
be the incenter of triangle 
Let 
be the midpoints of 
respectively. Points 
lie on 
respectively such that 
The line perpendicular to 
through 
intersects the line perpendicular to 
through 
at 
Prove that 
H2 [Google CodeJam 2011]
Goro wants to sort a list of 
distinct numbers in an increasing order. In each round, Goro can fix some elements of the list. All non-fixed elements of the list will then be permuted randomly (with each permutation having equal probability.) Given and the initial list, determine the expected number of rounds Goro will need to sort the list, under Goro’s best strategy.
Solutions will be available next week.
The InfinityDots Project 