Minimizing Sum Of Roots

This solution completes @Chan Lye Lee's solution.

Construct the points $O = (0,0), A = (X,2) , B = (2,Y), C = (4,3)$ . Observe that the algebraic expression is equivalent to $OA + AB + BC$ , where $A$ is a point on the line $y = 2$ and $B$ is a point on the line $x = 2$ . Normally, we would conclude that this is $\geq OC$ , but this lower bound cannot be achieved, since the points $O, A, B, C$ will not lie in order. (This is why it's important to verify that the lower bound can be achieved in order to conclude that it is indeed a minimum.)

Instead, we proceed as such:

1. Clearly, we may assume that $0 \leq X \leq 4$ since otherwise we could reduce the distance by setting $X =0$ or $X = 4$ . Similarly, we may assume that $0 \leq Y \leq 3$ . Thus, the domain of the continuous function is compact , hence a minimum exists.
2. Suppose that we know the value of $X$ , so point $A$ has been fixed. Then, we want to minimize $AB + BC$ which clearly happens when $A, B, C$ are points on a line.
3. Suppose that we know the value of $Y$ , so point $B$ has been fixed. Then, we want to minimize $OA + AB$ , which clearly happens when $O, A, B$ are points on a line.
4. Hence, the minimum occurs when $ABC$ and $OAB$ are straight lines with points in that order. This is only possible when $A = B = (2, 2)$ . We thus obtain $OA + AB + BC = \sqrt{8} + \sqrt{0} + \sqrt{5} \approx 5.0645$ .

Relevant wiki: Inequalities - Geometric Interpretation - Intermediate

Partial Solution... Let $O(0,0), A(x,2), B(2,y)$ and $C(4,3)$ . Then the expression means $|OA|+|AB|+|BC|$ , which at least $|OC|=5$ . However, $|OC|=5$ is not achievable. It is because $|OC|=5$ if $O, A, B, C$ (in this order) on a straight segment, which is not possible.

I used brute force method. It is obvious that $x$ and $y$ will not be very large, so some bounds can be placed.

Code:

alist = []
for a in range(-100,100):
    for b in range(-100,100):
    x=a/10
        y=b/10
    alist.append((x**2+4)**0.5+((x-2)**2+(y-2)**2)**0.5+(4+(3-y)**2)**0.5)
print(min(alist))