## May 2019 Problem

Here is our new Problem:

In an arithmetic sequence, u7+u8+u9 > 0 and u7+u10 < 0.

For what value of n, Sn is maximum ?

Here is our new Problem:

In an arithmetic sequence, u7+u8+u9 > 0 and u7+u10 < 0.

For what value of n, Sn is maximum ?

PQR is a right triangle. It is free to move so that P is always on the x-axis and R is on the y-axis. Show that the point Q (right angle) always lies on a straight line passing through the origin.

Try our March Problem:

How many ways are there to roll a sum of 9 with 4 standard dice?

A right triangle has a hypotenuse with the length equal to 12 cm.

What is the greatest possible area of this triangle ?

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