**C**heenta has planed to initiate a problem solving Marathon with existing students. Here we are providing the problems and hints of “**Problem Solving Marathon Week 1**“. The Set comprises three levels of questions as following-Level 0- for Class III-V; Level 1- for Class VI-VIII; Level 2- for the class IX-XII. You can post your alternative idea/solution in here.

## Level 0

(Q.1)Which of the following is equal to ?

**Hint 1**

Calculate

**Hint 2**

Try to use the fact

(Q.2)Let and be the following sums of arithmetic sequences: What is the value of ?

**Hint 1**

Observe that how many terms are there, in and . If there are nos. of terms then pair up like term, term, term, term

**Hint 2**

If you add the elements of every pair, then you will get same result of every pair.

Also Visit: Pre-Olympiad Program

## Level 1

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

**Hint 1**

can be written as

**Hint 2**

Try to find the necessary condition for

(Q.2)Two geometric sequences and have the same

common ratio, with , , and . Find .

Example of Geometric Sequence , here common ratio is .

**Hint 1**

Try to find the th term of geometric sequence

**Analyze the example**

is an example of geometric sequence. Here common ratio is . Now term . term term , term term , term term .

## Level 2

(Q.1)Let , , be real numbers such that .

Prove that

**Hint 1**

Note that

**Hint 2**

Here you can use the idea of Cauchy Schwarz Inequality