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 $latex {1 + \frac{1}{1+\frac{1}{1+1}}}$?

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

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

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

Also Visit: Pre-Olympiad Program

Level 1

(Q.1)Find all positive integers $latex n$ such that $latex n^2+1$ is divisible by $latex n+1$.

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

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

Hint 1
Try to find the $latex n$th term of geometric sequence

Level 2

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

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