Alan

Subtract the first equation from the second one, term by term, to get a nice simple expression for x + y.  Rearrange this to get y in terms of x and substitute the result back into one of the two original equations.

Use the fact that

\((x+y+z)^2=x^2+y^2+z^2+2(xy +xz+yz)\)

3 s.f. means 3 significant figures.  

Divide \(3x^2+4x-12=0\) by 3:  \(x^2+\frac{4}{3}x-4=0\)   ...(1)

Quadratic with roots r and s:  \((x-r)(x-s)=0\)  or  \(x^2-(r+s)x+rs=0\)    ...(2)

Compare (1) and (2)  to see that \(r+s=-\frac{4}{3}\)    ...(3)

Note that \(r^2 +s^2 + 2rs = (r+s)^2 \)

Now it should be easy to answer the question.

I've just run a Monte-Carlo simulation with 20000 trials and get a probability of 0.32125 (this will vary slightly each time it's calculated because of the use of random numbers of course).  This is close to 9/28 (≈0.32143).

Use conservation of momentum:  m1*v1+m2*v2 = 0  (assuming they were at rest to start with).

So:   30*v1 + 25*1 = 0;

Find v1 (the velocity of the girl) from this.

If abc represents the number 100a + 10b +c, then a, b and c must be integers.

Set a + 5b + 15c = 100a + 10b + c

Manipulate this to get c in terms of a and b, then see what the minimum values of a and b must be for c to be the smallest possible positive integer.

Here's a helping hand:

The following graphs should help:

Notice that the numerator of each term cancels with the denominator of the next term, so that overall you are left with \(\frac{a}{2}=9\)

Hence a = 18.  The denominator of each term is just 1 less than the numerator, so b = a-1 = 17.