Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2012 Grade 8 Number Sense
Problem
Solution
1) Joey is playing a game where winning means getting the biggest product. He rolls a 6 sided die 5 times getting the numbers 1, 3, 4, 5, 6. He creates a 3 digit number and a 2 digit number using each digit once. He wins if the product of his two numbers is bigger than the products of his competitors. What should his 3 digit number be?
  1. The product of a 3 digit number and a 2 digit number are to be maximized, so it is clear that the units digits of both numbers should be the smallest 2 numbers, namely 1 and 3
  2. So, we need the placement of 3 numbers in the 2 remaining digits of the 3-digit number and the 1 remaining digit of the 2-digit number.
  3. That leaves only 6 possible combinations of the remaining 3 numbers (4,5 and 6), so they can all be investigated:
    45x6 = 270
    46x5 = 230
    54x6 = 324 *
    56x4 = 224
    65x4 = 260
    64x5 = 320
  4. So 54a X 6b is the best combination, so now all we have to do is figure out where the 1 and 3 go:
    541x63 = 34083
    543x61 = 33123
  5. Therefore, the largest product is 541 x 63
2) Initially on Michael's hands there are 3.9 x 106 bacteria. He sprays his hands with a hand sanitizer, killing 99.5% of them. He then washes his hands twice, removing 10% of the remaining bacteria each time. How many bacteria are left? Express your answer in scientific notation. 3.9x106 x 0.005 x.9 x .9 = 0.015795x106 = 1.5795x104

Problem
Solution
3) The expression represented by the boxes is the quotient of a 2-digit number raised to an exponent by a 2-digit number raised to an exponent. Each box represents a digit from the list 1, 2, 3, 4, 5, and 6 with no repeats. To make the resulting quotient as small as possible what is the second 2-digit number represented by the fourth and fifth box?
  1. To make the quotient as small as possible the exponent in the denominator should be the largest value, namely 6
  2. The other 2 digits, the 4th and 5th should be as large as possible: 5 and 4.
  3. This makes the whole division 231 ÷ 546 = 9.28x10-10
  4. The second 2-digit number is 54
4) What is the difference between the average of all integers from -100 to 101 and the average of all integers from -500 to 501?
  1. The sum of all integers from -100 to +101 is:
    Sum = (-100 + 101) x 202/2 = 101
  2. Their average = 101/202 = 1⁄2
  3. The sum of all integers from -500 to 501 is:
    Sum = (-500 + 501) x 202/2 = 101
  4. Their average = 101/202 = 1⁄2 also
  5. Their difference is 0
5) What is the smallest whole number k > 0 so that
  1. Since both expressions are taken to the power of k, we can simplify the expression to:
    (k/10) > 2
  2. The value of k for which this is true is 21