## January 2019

Dear friends and visitors of the site.

Welcome to our first 2019 Problem:

Prove that 23 is a factor of 2^{11} - 1 (without direct computation).

Dear friends and visitors of the site.

Welcome to our first 2019 Problem:

Prove that 23 is a factor of 2^{11} - 1 (without direct computation).

Dear friends and visitors of the site.

Try our December Problem.

Take numbers from 1 to 10000.

How many permutations of them exist such that neighbouring numbers differ at most by 1 ?

Dear friends and visitors of the site!
Here is our November Problem:

How many 13-digit numbers have an odd sum of digits?

Dear friends and visitors of our site!
Welcome back to Poematics.
here is our new Problem:

The front tyre of a racing bike is guaranteed for 9000 km and the rear tyre – for 3000 km.

How long can the bike run on these tyres ?

Waiting for your solutions.

Serge Hazanov

Welcome to June 2018 Problem:

Does equation

2018(x+y) =xy

have integer solutions ?

No calculator !

Find for what values of natural n

the expression n^{3 }- 7 is divisible by (n-1).

No Euclidean division!

Dear friends and visitors of the site,

this is the April 1st problem:

Prove that for any prime number m > 3, m^{4} -1 is a multiple of 48.

Dear Friends and Visitors of the site!

Try our February 2018 Problem:

Given a set of 11 natural numbers, such that the sum of any ten of them is a multiple of 7.

Show that all these numbers are multiples of 7.

Waiting for your solutions.

Serge Hazanov

Dear friends and visitors of the site,

Welcome to a New Year and new problems!

Here is the January Problem:

Vertices A and C of a solid triangle ABC (with a right angle B)

are sliding along the sides of another right angle.

What is the locus of the vertex B ?

111 x 111 = 110001

Is this possible?

What factor should be crossed out of the product

1! х 2! х 3! х ............х 16! ,

to make the remaining expression a perfect square ?

Welcome back to the site POEMATICS!

The September Problem is a warming up. It is so easy, that even I can solve it.

Go ahead!

Serge Hazanov

___________________________________________________What number is greater

sqrt(109) - sqrt(106) or sqrt (104) - sqrt (101) ?

No calculator!