Tag Archives: Titu Andreescu

Chebyshev Polynomials

Posted on February 2, 2015 by Sumant Sumant

Chebychev Polynomials find wide application. They can be described recursively as ,  and . So if we plug   Here is a nice paper by Arthur Benjamin on Chebyshev’s Polynomial

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Polynomial with integer coefficients

Posted on February 2, 2015 by Sumant Sumant

Let be a polynomial of degree with integer coefficients and real roots, not all equal, in the interval .  Prove that if is the leading coefficient of , then

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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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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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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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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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Proof using vectors in Number theory

Posted on January 25, 2015 by Sumant Sumant

Prove that for any eight real numbers at least one of the numbers is nonnegative.

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