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.
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 .
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 .
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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 .
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.