Problem Solving Marathon Week 2

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 $latex CAT$, we have $latex \angle ACT =\angle ATC$ and $latex \angle CAT = 36^\circ$. If $latex \overline{TR}$ bisects $latex \angle ATC$, then $latex \angle CRT =$

Hint 1
Triangle $latex CAT$ is an isosceles triangle

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

Hint 1
Figure out the hidden pattern of the problem.

Also Visit: Pre-Olympiad Program

Level 1

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

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

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

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

Level 2

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

Hint 1
Factorise the given expression..