Maximal points
Define a partial order in points on Rn based on their coordinates i.e. A>B iff all coordinates of A are greater than coordinates of B. Give an efficient algorithm to find the set of all maximal points from a given input set.
Closest pair problem
Given a set points in 2D give an algorithm to find the closest pair of points amongst them. Generalize for Rn.
I know a O(nlogn) solution for 2D. Hopefully it will generalize for Rn.
Convergence
Consider sequence a1, a2,… with a1=1, a2=2, ak = smallest integer such that there is not 3-term AP in a1, a2,…,ak. Prove that 
This actually is almost true due to Erdős conjecture.
Trees again
What is the necessary and sufficient condition for a given tuple (g1, g1,…, gk) to be the degrees of the vertices of a tree? In case they are what is the number of labeled trees possible?
IMO’08 Prob 3
Prove that there are infinitely many positive integers n such that n2+1 has a prime divisor greater than 2n + √(2n).
Bird watch
“n” birds are sitting on (0,1). Each bird sees the nearest neighbor. Find the average number of birds being seen.
Almost there
Prove
‘think i found the proof. Here’s how it goes. We construct a graph out of the given statement. If there are n-symbols, then we have 2n vertices, corresponding to the symbols and their negations. Then we connect two vertices a to b with a directed edge iff the statement contains “~a or b”. We remove all self loops. Create a matrix which stores the distances between each pair of vertices. That’s it!
Figure this out
Let P(n) denote the number of partitions of n then
![nP(n) = \Sigma_{<r,j>\in [1,n]*[1,n]} rP(n-jr)](http://l.wordpress.com/latex.php?latex=nP%28n%29+%3D+%5CSigma_%7B%3Cr%2Cj%3E%5Cin+%5B1%2Cn%5D%2A%5B1%2Cn%5D%7D+rP%28n-jr%29+&bg=eeeae8&fg=000000&s=0 "nP(n) = \Sigma_{<r,j>\in [1,n]*[1,n]} rP(n-jr)")
If you do not get the notation, the pair <r,j> vary over [1,n]x[1,n] so take n2 values.
The proof turned out to be real nice. The left hand side is the sum of all the elements of the partitions. To get the RHS count the number of times “i” occurs and the number turns out to be P(n-i) + P(n-2i) + P(n-3i) + P(n-4i) + … QED
