Math Is Fun Forum

Question:

1. If f(x)=x^6+x^5+x^4+x^3+x^2+x
Please find the quotient and remainder, when f(x^7) is divided by f(x).

2. Please evaluate, and prove, the value of the expression 1^5+2^5+…………+x^5.

3. If 1^n+2^n+3^n+…………+n^n is divisible by A, evaluate the maximum value of A, and represent your answer in terms of n, given that n is a positive integer.

Use the following information to answer the next 3 parts.

A, B and C are rivals. Today they have decided to end their problems with a game of shooting. The aim chances of A, B and C are respectively 1/3, 2/3 and 1. A will shoot first, then B, then C. Only 1 may survive.

Part I: If the three will shoot randomly,

Find the survival rates of A, B and C.

Part II: If the three will shoot, with strategy, to maximize their survival rates,

find the survival rates of A, B and C.

Part III: It is known that A has a method to further maximize his survival rate.

a) What method is it?
b) Given A uses the method in (a), find the survival rates of A, B and C.

All answers must be provided with steps. You may use a calculator.