Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2016 Grade 8 Algebra
Problem
Solution
1) If 35 = 243 and xy = 625, what is the value of yx?
  1. The square root of 625 = 25 and the square root of 25 = 5, so 54 = 625.
  2. 45 = 1024
2) A bucket has 10 balls numbered 1,1,2,3,4,4,4,5,6, and 6. If one ball is drawn at random, write an inequality that relates the chance of getting an even number to the chance of getting a multiple of 5? Express in lowest terms.
  1. The probability of getting an even number is 6⁄10
  2. The probability of getting a multiple of 5 is 1⁄10
  3. So the inequality is
    1⁄10 < 3⁄5
3) If a=4 in the equations below, what is the value of (d+c+b)a?
  1. Substituting 4 for a in the first expression we get 42 - 4 = b; b = 12.
    Substituting this in the second equation:
  2. 12 - c = 9; c = 3.
    ... and in the 3rd:
  3. 4-12+3-d = 0, so -5 - d = 0; d = -5
  4. (d+c+b)4 = (-5+3+12)4 = 104 = 10,000

Problem
Solution
4) How many integers for x satisfy
(x - 11)(x + 3) < 0?
  1. In the first expression, any x below 11 results in a negative number
  2. In the second expression, any x above -3 results in a negative number.
  3. The range for x is -2 to +10 = 13 integers (including zero)
5) You inherit an unusual stamp collection containing hundreds of 6 and 7-cent stamps. What is the greatest amount of postage that could not be made using a combination of these stamps? (For example, 25 cents postage could be made with three 6-cent stamps and one 7-cent stamp) Here is the table of 6-cent and 7-cent stamp combinations:
# 6-cent stamps
↓ 		# 7-cent stamps
0 1 2 3 4 5 6
0 0 7 14 21 28 35 42
1 6 13 20 27 34 41 48
2 12 19 26 33 40 47 54
3 18 25 32 39 46 53 60
4 24 31 38 45 52 59 66
5 30 37 44 51 58 65 72
6 36 43 50 58 65 72 79
7 42 49 56 63 70 77 84
Notice that you can make any sum up above a certain number of 6-cent and 7-cent stamps. If you trace that back along the diagonals (for example 35, down and left 34, 33,32 ...) you will soon come to a number that cannot be made with 6-cent and 7-cent stamps = 29 cents