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August, 2012

12 August
Posted in Algorithm, Article, Lua

Interval Scheduling Problem

Let’s consider the Interval Scheduling Problem, in which we have a set of n requests \{1,2,\cdots,n\}; the i^{th} request corresponds to an interval of time starting at s_i and finishing at f_i, or request $i$ spans interval $ [s_i,f_i]$. Suppose that a subset of the requests is compatible if no two of them overlap in time; find the largest compatible subset of any set of requests in a reasonable number of steps.

11 August
Posted in Article, Math Problems

Prime Congruence Class

Show that there are infinitely many primes of the form 4k - 1.

10 August
Posted in Article, Math Problems

Just the Evens

Find the sum of even positive divisors of 100000

10 August
Posted in Article, Math Problems

Sum of Divisors

Suppose we define the function \sigma: \mathbb{N} \to \mathbb{N} as the sum of the divisors of n

     $$ \sigma(n) = \sum_{d|n} d $$

Express \sigma(n) in terms of n‘s prime factorization

9 August
Posted in Article, Math Problems

That Other Little Gauss Story

Determine the product of distinct positive integer divisors of n = 420^4.

9 August
Posted in Article, Math Problems

Congruence by L.C.M

Determine the number of ordered pairs of positive integers (a,b) such that their least common multiple [a,b] = 2^35^711^{13}

9 August
Posted in Article, Math Problems

Randomly Chosen Divisors

Compute the probability that a randomly chosen positive divisor of 10^{99} is an integer multiple of 10^{88}

9 August
Posted in Article, Math Problems

Nothing in common

Let a and b be distinct integers such that a^2 + ab + b^2|ab(a+b). Show that

     $$ |a - b| > \sqrt[3]{ab} $$

8 August
Posted in Article, Math Problems

The Fib-Fib-Fib Sequence

Consider the sequences (a_n)_n, (b_n)_n, defined recursively by

     $$ \begin{tabular}{l l l l} $a_0 = 0$, & $a_1 = 2$, & $a_{n+1} = 4a_n + a_{n-1}$ & $n \ge 0$ \\ $b_0 = 0$, & $b_1 = 1$, & $b_{n+1} = a_n - b_n + b_{n-1}$ & $n \ge 0$ \end{tabular} $$

Show that (a_n)^3 = b_{3n} \fa n

7 August
Posted in Article, Math Problems

Square Sequence

The sequence f_1, f_2, \cdots satisfies

    $$ f_{m+n} + f_{m - n} = \frac{f_{2m} + f_{2n}}{2}$$

for all nonnegative integers n,m and m \ge n. If f_1 = 1, determine f_n