If you are solving using formula, there are some corner cases.

IN
0 0
0 0
OUT
0

IN
17 0
0 17
OUT
17

And just some random test.

IN
7 9
6 8
OUT
15.165177459575631

Можно сдать тернарным поиском на Python 3.4 с параметрами :
 количество итераций -- 100
 максимальное А и максимальное B -- 10**18

Did anybody solve it with numerical algo (local optimization)?

Why A1=1 B1=2
    A2=2 B2=4
The answer is 5?

Please tell me.Thank you.

#include<iostream>
#include<iomanip>
#include<cmath>

int a1,b1,a2,b2;
double func(double x);

int main()
{
    using std::cin;
    using std::cout;
    using std::endl;

    cin >> a1 >> b1 >> a2 >> b2;
    double delta = 1e-4;
    double head = 1e-4;
    double tail = 1000000+head;
    while(1)
    {
        double mid = 0.5*(head + tail);
        double cur = func(mid);
        double up = func(mid+delta);
        double down = func(mid - delta);
        if((cur-up)*(cur-down)>=0)
        {
            cout.setf(std::ios::fixed|std::ios::showpoint);
            cout.precision(9);
            cout << cur << endl;
            return 0;
        }else
        {
            if(up>=cur)
            {
                head = mid + delta;
            }
            else
            {
                tail = mid - delta;
            }
        }
    }

}
double func(double x)
{
    return sqrt((a1+b1*x)*(a1+b1*x)/(1 + x*x) + (a2+b2*x)*(a2+b2*x) / (1 + x*x));
}

I find maximum of the maximal influence of elixir by formula:
Math.sqrt((b1 * b1 + b2 * b2 + a1 * a1 + a2 * a2 +Math.sqrt(a1 * a1 * a1 * a1 + a2 * a2 * a2 * a2 + 8 * a1 * a2 * b1 * b2 - 2 * a2 * a2 * (b1 * b1 - b2 * b2) + 2 * a1 * a1 * (a2 * a2 + b1 * b1 - b2 * b2) + (b1 * b1 + b2 * b2) * (b1 * b1 + b2 * b2))) / 2)

...and get wa. What's wrong?

 9 fractional digits.
but in case 1, output is 5.4649857042190426504511884932842

Naskol'ko ya mogu ponyat', dlya vychislenya kachestva neobhodimo znat' A i B. No... Oni tam vrode ne dany...