Monthly Archives: January 2015

2n-1 problem

Posted on January 31, 2015 by Sumant Sumant

The problem says that if there are elements then the minimal number of elements in a set which guarantees that there will be exactly elements divisible by is

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3n Problem

Posted on January 31, 2015 by Sumant Sumant

I found this in Mathematical miniatures by Titu Andreescu. Suppose we have integers divided into three sets of each. Then one can alway pick an element from each of the three sets such that one can be expressed as the … Continue reading →

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Among n integer one can choose several (possibly one) whose sum is divided by n

Posted on January 29, 2015 by Sumant Sumant

This is a classic problem in number theory which can be easily solved by Pigeon Hole principle. Let be the sum of first integers. Thus we have different sums. Suppose none of these sums are divisible by . However any number … Continue reading →

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Sundaram’s sieve and 2n+1 prime

Posted on January 28, 2015 by Sumant Sumant

If the number is present in sieve then not prime, else it is. The sieve is Let represents the term in the above sieve. So along the x-axis we have and along y-axis we have . The common difference along … Continue reading →

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Existence of a real number such that when added to seq of real gives all irrationals

Posted on January 27, 2015 by Sumant Sumant

Let be arbitrary real numbers. Prove that there exists a real number such that each of the numbers is irrational. It’s easy to see that if all are rational numbers then any irrational number will do the job because the … Continue reading →

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Pythagorean Triplets

Posted on January 27, 2015 by Sumant Sumant

We all know the basic triplets or or some even . However what is the general recipe to concoct all these. Observe these two expansions and . Subtracting we get which is same as  or . Now choose different value … Continue reading →

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Inductive Construction

Posted on January 27, 2015 by Sumant Sumant

Given numbers is it possible to arrange the numbers in a row so that the arithmetic mean of any two of these numbers is not equal to some number between them? Let’s begin with two numbers Three numbers or  or … Continue reading →

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Cyclic product of positive real numbers each less than 1 is less than 1/4

Posted on January 27, 2015 by Sumant Sumant

Let be non-negative real numbers that add up to or Prove that . Well its easy to see that it is true for Assume that they are all arranged about a circle, so that can be regarded as a pair … Continue reading →

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Game of Nim

Posted on January 27, 2015 by Sumant Sumant

Nim is played with stacks of coins. One can pick any number of coins as one wish and the person who picks the last set of coins win.

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n! as sum of n divisors, n bigger than 3

Posted on January 27, 2015 by Sumant Sumant

Prove that, for each , the number can be represented as the sum of distinct divisors of itself. Note for it does not work because the divisors but it does work for where the divisors are . Similarly for the … Continue reading →

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