Blow your head up 2!

There are only $4$ divisors of $298894956667:$ the number itself, $1,$ $314159,$ and $951413.$ Since $m$ and $n$ are positive integers in reverse order, they must be $314159$ and $951413.$ $m + n = 314159 + 951413 = \boxed{1265572}$

... All I can say is that the two numbers are 314159 and 951413. How to figure that out is left as an exercise to the solver (Good luck! You'll need it)

21.7 sec , counting from 0 . Slow... but it works...

I used a Python coding:

The answer for $m = \lfloor 100000\pi \rfloor$ .

Cracking a walnut with a truck.

Python :

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def re(n):   #i'm making reverse function as re(n)
    return int(str(n)[::-1])
for m in range(1,999999):
    k=re(m)
    if k*m==298894956667.0:
        print 'answer is',k+m

Python:

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def goal(n):
    m = int(str(n)[::-1])
    return m * n == 298894956667
n = 1
while not goal(n):
    n += 1
print "Answer:", n+int(str(n)[::-1])

All you need to do is to find out the factors of 298894956667.

Here is the java solution. I was going to make the solution more specific, but then I realized that 298894956667 has only 2 factors other than the number itself and 1.

for(int i = 100_000;i < 1_000_000;i++)
    if(298894956667d % i == 0)
        System.out.println(i);

Also one more thing to notice is that the factors will be between 1 million and 100 thousand.

I worked simultaneously from left and right matching till I got to the middle where I found that the squares of the digits add up to 133. Looking at both the ends 9 and 3 were obvious then 5, 4, 1 and 1 were possible choice. Of course there was lot of effort involved. Please share better methods.