Well, that about wraps it up for God

Maximal points

apurvnakade — Tue, 06 Jan 2009 10:11:51 +0000

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

apurvnakade — Sat, 03 Jan 2009 17:01:12 +0000

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

apurvnakade — Thu, 25 Dec 2008 07:14:50 +0000

Consider sequence a₁, a₂,… with a₁=1, a₂=2, a_k = smallest integer such that there is not 3-term AP in a₁, a₂,…,a_k. Prove that converges.

This actually is almost true due to Erdős conjecture.

Trees again

apurvnakade — Thu, 25 Dec 2008 07:07:38 +0000

What is the necessary and sufficient condition for a given tuple (g₁, g₁,…, g_k) 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

apurvnakade — Wed, 24 Dec 2008 12:03:26 +0000

Prove that there are infinitely many positive integers n such that n²+1 has a prime divisor greater than 2n + √(2n).

Hall’s theorem

apurvnakade — Tue, 23 Dec 2008 18:55:59 +0000

Prove it.

Trees

apurvnakade — Sat, 20 Dec 2008 16:38:37 +0000

Prove that number of labeled trees with n vertices is n^n-2.

Bird watch

apurvnakade — Thu, 18 Dec 2008 14:05:06 +0000

“n” birds are sitting on (0,1). Each bird sees the nearest neighbor. Find the average number of birds being seen.

Almost there

apurvnakade — Sun, 14 Dec 2008 12:30:04 +0000

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

apurvnakade — Tue, 09 Dec 2008 18:07:58 +0000

Let P(n) denote the number of partitions of n then

\in [1,n]*[1,n]} rP(n-jr) ' title='nP(n) = \Sigma_{\in [1,n]*[1,n]} rP(n-jr) ' class='latex' />

If you do not get the notation, the pair 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