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Problem Solving Marathon Week 1

Cheenta 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 {1 + \frac{1}{1+\frac{1}{1+1}}}?

Hint 1
Calculate {1+\frac{1}{1+1}}


Hint 2
Try to use the fact {\frac{\frac{a}{b}}{\frac{c}{d}} =\frac{a \times d}{b \times c}}


(Q.2)Let {X} and {Y} be the following sums of arithmetic sequences: 
 What is the value of {Y - X}?

Hint 1
Observe that how many terms are there, in X and Y. If there are n nos. of terms then pair up like (1^{st} term,n^{th} term),(2^{nd} term,{n-1}^{th} 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 n such that n^2+1 is divisible by n+1.

Hint 1
n^2+1 can be written as n(n+1)-(n-1)


Hint 2
Try to find the necessary condition for (n+1)|(n-1)


(Q.2)Two geometric sequences a_1, a_2, a_3, \ldots and b_1, b_2, b_3, \ldots have the same
common ratio, with a_1 = 27, b_1=99, and a_{15}=b_{11}. Find a_9.
Example of Geometric Sequence 2,4,8,16, here common ratio is 2.

Hint 1
Try to find the nth term of geometric sequence


Analyze the example
2,4.8,16 is an example of geometric sequence. Here common ratio is 2. Now 1^{st} term =2. 2^{nd} term =1^{st} term\cdot 2, 3^{rd} term =1^{st} term\cdot 2^2, 4^{th} term =1^{st} term\cdot 2^3.

Level 2

(Q.1)Let m, n, p be real numbers such that m^2 + n^2 + p^2 - 2mnp = 1.
Prove that (1+m)(1+n)(1+p) \leq 4 + 4mnp

Hint 1
Note that (m+n+p)^2 = m^2 + n^2 + p^2 + 2(mn+np+pmx)


Hint 2
Here you can use the idea of Cauchy Schwarz Inequality

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