MathDB

Let

S

be the set of all lattice points

(x, y)

in the plane satisfying

|x|+|y|\le 10

. Let

P_1,P_2,\ldots,P_{2013}

be a sequence of 2013 (not necessarily distinct) points such that for every point

Q

S

, there exists at least one index

i

such that

1\le i\le 2013

and

P_i = Q

. Suppose that the minimum possible value of

|P_1P_2|+|P_2P_3|+\cdots+|P_{2012}P_{2013}|

can be expressed in the form

a+b\sqrt{c}

, where

a,b,c

are positive integers and

c

is not divisible by the square of any prime. Find

a+b+c

. (A lattice point is a point with all integer coordinates.) [hide="Clarifications"]
[*]

k = 2013

, i.e. the problem should read, ``... there exists at least one index

i

such that

1\le i\le 2013

...''. An earlier version of the test read

1 \le i \le k

.
Anderson Wang

Problem Statement