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We are having a full fledged Problem Solving Marathon. We are receiving wonderful responses from the end of our students which is making the session more and more alluring day by day. Here we are providing the problems and hints of “Problem Solving Marathon Week 2“. 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] In triangle $CAT$, we have $\angle ACT =\angle ATC$ and $\angle CAT = 36^\circ$. If $\overline{TR}$ bisects $\angle ATC$, then $\angle CRT =$

Hint 1
Triangle $CAT$ is an isosceles triangle

Hint 2
Try to find the value of $\angle ATC$

[Q.2] What is the product of $\frac{3}{2}\times\frac{4}{3}\times\frac{5}{4}\times\cdots\times\frac{2019}{2018}$?

Hint 1
Figure out the hidden pattern of the problem.

Hint 2
You can make a prototype of that problem i.e, try to calculate product of first three or four terms.

## Level 1

[Q.1] How many $7$-digit palindromes (numbers that read the same backward as forward) can be formed using the digits $2$, $2$, $3$, $3$, $5$, $5$, $5$?

Hint 1
First symbolize the number i.e., the number looks like “$\textbf{mnoponm}$“.

Hint 2
Now $\textbf{p}$ cannot be $2$ or $3$.

[Q.2] How many pairs of positive integers $(a,b)$ are there such that $a$ and $b$ have no common factors greater than 1 and: $\frac{a}{b} + \frac{14b}{9a}$ is an integer?

Hint 1
put $x=\frac{a}{b}$

Hint 2
Discriminant of a Quadratic equation plays a significant role here..

## Level 2

[Q.1] Prove that if $m$, $n$ are integers, then the expression $E = m^5 + 3m^4n - 5m^3n^2 - 15m^2n^3 + 4mn^4 + 12n^5$ cannot take the value $33$.

Hint 1
Factorise the given expression..

Hint 2
Then try to find the pairwise different divisors $33$