Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2011 Grade 8 Number Sense
Problem
Hint
1) What is k if (k2 x 6)2 = (2732) / k ?

Simplify the k terms and the constants to arrive at a value for k:



k = _____
2) A Googol is 10100. Toni has a Googol of pennies which she wants to share fairly with as many folks as possible. She splits the pile in half. Then splits both piles in half again. Then splits those piles in half. How many times can she do this splitting in half before she reaches a pile that she can't split in half evenly? (Hint: try some smaller piles first like 102)
  1. For 100, dividing by 2 successively, you get: 50,25 12.5 =
    2 divisions before one you can't split
  2. For 1000, = 500, 250, 125, 62.5 = 3 divisions.
Continue this line of logic until you notice a pattern and then extend that to the 100th power.



The number of divisions by 2 that can be done before it can't be split evenly is equal to the number of powers of 10 = _______

Problem
Hint
3) Seymore has just gotten a new job. On his first day, the boss gave him 2 choices for his salary. First Choice: Start at $30,000 and get a 5% raise at the end of every year to take effect the following year. Second Choice: Start at $23,000 and get a $5000 raise at the end of each year to take effect the following year. Which choice gives the greatest salary at the beginning of the 5th year? How much more? Round your answer to the nearest cent. Compute the salary after 4 years for each option and find the difference between the options




Choice ____ is the best by __________
4) A natural number, N, is prime if it is not evenly divisible by any prime number less than √N. How many prime numbers must we test to determine whether 2011 is prime? Take the square root of 2011 and find all the primes less than that:



= ____ primes.

5) What is the first positive integer n for which
(-4⁄3)n < -3?
  1. n must be an odd integer because all even n's result in a positive value
    (minus times minus = plus).
  2. Keep incrementing n by odd values until the inequality is met



    n = _____