Math Contest #2

This blog will be about the answers to selected problems Westmont's second math contest.

One of the questions was asking for a five digit number that had the following property: when a 1 is added to the right of the number, the resulting six digit number is three times greater than if the 1 was added to the left of the number. This problem is easily solved by using the following equation:

let X be the 5 digit number we are looking for, then:

10X+1=3(10^5 +X)

which, when solved, gives 42857. Checking this result with the given properties shows that it indeed is the five digit number that is needed. The equation gave the correct number because it simply stated the conditions in mathematical terms: the left side of the equation essentially shifts all the digits of X one place holder to the left, and then adds a 1 to the right, while the right side keeps the digits of X in place, while adding a 1 to the right.

Another question asked to find the number who is the sum of consecutive integers who's squares differ by 2011. Again, a simple set of equations gives the answer:

x=y+(y+1)=2y+1

(y+1)^-y^2=2y+1=2011

x=2y+1=2011

This equation also worked because it simply restated the given conditions mathematically. The number that needed to be found, x, can be expressed as y+(y+1), which simplifies to 2y+!, but this turns out the be exactly 2011. Therefore x=2011.

The first problem asked for the maximum value of the following expression:

(x^2-2x+1)^3 + (1-2x-x^2)^3

This question is made trivial after simplifying:

(x^2-2x+1)^3 + (1-2x-x^2)^3

= (x^2-2x+1)^3 + ((-1)^3)(x^2-2x+1)^3

= (x^2-2x+1)^3 - (x^2-2x+1)^3 = 0

Since the expression is 0 for all values of x, the maximum value of this expression must simply be 0.

The last problem that will be covered is the only problem whose solution is in doubt. It asked for the minimum amount of numbers required to express the number 1 as the sum of positive decimal expansions with only 0's or 8's. It is 95% likely that the solution is 7; here is one way to obtain it:

1 = .8 + .08 + .08 + .008 + .008 + .008 + .008 + .008

There is probably a way to prove mathematically that the answer is 7, or some other number, but given the thirty minute time period and the five other problems, no mathematical proof was found.