I am dumbfounded whenever i face a problem which requires coming out with a formula on your own! especially when it's related to maths. Like the Your text to link here... 1511B - Длина НОД. I started by reading the problem and re-reading and doing that in a loop and even the slightest hint of the answer did not start materializing inside my head. Tips on tackling that types of questions?..

ps:- i learned about gcd, co-prime etc but still it's literally of no help to me and i like to think that I'm not that bad at general maths either!!

Any tips or tricks for facing those kind of problems headon is appreciated!

If you have learned some theory, try to exercise purely math problems from math olympiads

Instead of searching immediately for a formula, try to understand how the problem works. Now I'll try to explain a "path" to the solution. For example, you may try to fix $$$x$$$ and check the possible values of $$$y$$$ and $$$g = \gcd(x, y)$$$. This is a bit uncomfortable, it may be better to fix $$$g$$$ instead.

$$$x$$$ and $$$y$$$ are multiples of $$$g$$$. The only additional condition required is that $$$\gcd(x / g, y / g) = 1$$$. At this point, you may find that, if the leading digits of $$$g$$$, $$$x / g$$$ and $$$y / g$$$ are small, $$$x / g$$$ should have $$$a - c + 1$$$ digits and $$$y / g$$$ should have $$$b - c + 1$$$ digits. Moreover, $$$g$$$ is basically useless, so you can set it to $$$10^{c-1}$$$. In this way, you don't have to worry about leading digits of $$$x / g$$$ and $$$y / g$$$.

The new problem is: find $$$n = x/g$$$, $$$m = y/g$$$ with a fixed number of digits and $$$\gcd(n, m) = 1$$$. There are many possible constructions: for example, $$$n = 11^{a - c}$$$, $$$m = 10^{b - c}$$$.