Problem 6

 

Let represent an arbitrary arrangement of the numbers . Prove that if n is odd then the product is even.

 

[Problem submitted by Vin Lee, LACC Associate Professor of Mathematics. Source: the 1906 Eotvos Competition, a high school mathematics competition held in Hungary since 1894.]

 

 

Solution A for Problem 6:

 

Since n is odd there is a positive integer k such that n = 2k + 1.

 

Note that , , ,, .

 

So, of the numbers , exactly are odd numbers; and of the numbers , exactly are odd. So, of the numbers exactly are odd.

 

Since the product has exactly n factors and contains odd numbers, at least one of these factors is the difference of two odd numbers; that is, at least one factor is even. Therefore, the product is even.

 

 

Solution B for Problem 6:

 

Let us fill each blank in following with a number from the set {1, 2, 3, 4,.n}.

(-1) (-2) (-3) (-4) (-5) (-6).. (-n)

In order to make the final product odd, each odd number must be filled in one of the blanks underlined above. When n =2k+1, there are only k underlined-blanks but k+1 odd numbers. Therefore an odd final product is impossible.