Problem Solving Marathon Week1 Solution is the effortless attempt from Cheenta’s existing student as well as from the end of mentor. Question, rules and hints are given here.

## Level 0

**Q.1** Which of the following is equal to ?

This solution is proposed by Swetaabh Mishra from Thousand Flowers.

**Answer of (Q.2) ** This solution is using hints.

If we pair up the elements of it will look like . Now sum of the each pair is . Number of pair . so will be equal to Number of pair , similarly will be .

So, will be .

## Level 1

**Q.1** Find all positive integers such that is divisible by .

Since, can be written as

We can say that if then . At a glance, it’s look like impossible to get any positive integer. If then it is possible. So there is only one such positive integer .

Also Visit: Cheenta Olympiad Program

**Q.2** Two geometric sequences and have the same common ratio, with , , and . Find .

Example of Geometric Sequence , here common ratio is .

This solution is proposed by Saikrish Kailash from Thousand Flowers.

Two geometric sequence have same common ratio, let it is .

Now . From the given condition . Which is equivalent to . Now put the value of in .

## Level 2

**Q.1** Let , , be real numbers such that . Prove that .

This solution is proposed by Sampreety Pillai from Early Bird Math Olypmiad Group.

We konw which imply . similarly , . Also , and . Adding these six inequalities, we get . by the hypothesis .

So . Add both side of last inequality.

Also you can do another type of proof using Cauchy-Schwarz inequality. For that click here.