Problem:
For positive integer n, define Sn​ to be the minimum value of the sum
k=1∑n​(2k−1)2+ak2​​
where a1​,a2​,…,an​ are positive real numbers whose sum is 17. There is a unique positive integer n for which Sn​ is also an integer. Find this n.
Solution:
We interpret each term
tk​=(2k−1)2+ak2​​
as the length of the hypotenuse of a right triangle with legs of length 2k−1 and ak​. Put the triangles together in a "staircase" arrangement as shown in the diagram, and let A and B be the initial and terminal points of the broken path formed by the hypotenuses. The distance from A to B is
while the sum ∑k=1n​tk​ is the length of the path from A to B formed by the hypotenuses of the triangles. It follows immediately that ∑k=1n​tk​≥172+n4​, and that equality is obtained by choosing the ak​ so that the broken path is actually a straight line. Thus Sn​=172+n4​ is the minimum possible value of the given sum. When Sn​ is an integer, the equation 172=Sn2​−n4=(Sn​−n2)(Sn​+n2) implies that
