3.1 SYMMETRY 71It is worth exploring the concept of cyclic permutation in more detail. Given ann-variable expression f(x\ ,X 2 ,'" ,xn), we will denote the cyclic sum by
�f(X\,X 2 ,. .. ,Xn) :=
af(x\ ,X 2 ,··· ,xn) + f(X 2 ,X 3 ,'" ,xn,xd + ... + f(Xn,Xl,'" ,xII-d·
For example, if our variables are x, y and z, then
� x^3 = � + y^3 + z^3 and � xz^2 = xz^2 + yx^2 + zi.
a a
Let us use this notation to factor a symmetric cubic in three variables.Example 3.1.12 Factor a^3 + b^3 + e^3 - 3abe.
Solution: We hope for the best and proceed naively, making sure that our guessesstay symmetric. The simplest guess for a factor would be a + b + e, so let us try it.
MUltiplying a + b + e by a^2 + b^2 + e^2 would give us the cubic terms, with some error
terms. Specifically, we havea^3 +b^3 +c^3 - 3abc = (a+b+c)(a^2 +b^2 + (^2 ) -�(a^2 h+h^2 a) - 3ahc
a= (a+b+c)(a^2 +b^2 +e^2 ) -�(a^2 b+h^2 a+abe)
a= (a +b + c)(a^2 + b^2 + e^2 ) -� (ab(a+ b+ e))
a� (a+h+c) (i' +b^2 +c"-�ab)
= (a + b + e) (a^2 + b^2 + e^2 - ab - he - ac).
Notice how the La notation saves time and, once you get used to it, reduces the
chance for an error.Problems and Exercises
3.1.13 Find the length of the shortest path from the
point (3,5) to the point (8,2) that touches the x-axis
and also touches the y-axis.
3.1.14 In Example 1.2.1 on page 4, we saw that the
product of four consecutive integers is always one less
than a square. What can you say about the product
of four consecutive terms of an arbitrary arithmetical
progression, e.g., 3. 8. 13. 18?
3.1.15 Find (and prove) a nice formula for the product
of the divisors of any integer. For example, if n = 12,
the product of its divisors is
1 ·2·3·4·6· 12 = 1728.You may want use the d(n) function (defined in the
solution to the Locker problem on page 68) in your
formula.
3.1.16 How many subsets of the set { I ,2,3,4 .... ,30}
have the property that the sum of the elements of the
subset is greater than 232?
3.1.17 (Putnam 1998) Given a point (a,b) with 0 <
b < a, determine the minimum perimeter of a triangle
with one vertex at (a,b), one on the x-axis, and one
on the line y = x. You may assume that a triangle of
minimum perimeter exists.