Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2017 Grade 8 Algebra
Problem
Solution
1) If
    3 ╬ 5 = 17,

    2 ╬ 7 = 19,

    7 ╬ 2 = 9 and

    6 ╬ 5 = 29 then

what is 8 ╬ 5 = ? 		Call the first number a and the second number b.
  1. Since many of the results are prime, there must be additions or subtractions to get the results. No multiplication results in a prime!
  2. There also must be a multiplication because of the magnitude of the results, so set up a table of ab and a-b:
      a     b     ab   a - b
    35 15 -2
    27 14 -5
    72 14 5
    65 30 1
  3. Notice that if you take ab and the negative of a-b you get the correct result: 3x5 -(3 - 5) = 15 + 2 = 17
  4. The expression for the operator ╬ is ab + (b - a)
  5. 8 ╬ 5 = 8x5 + (5-8) = 40 - 3 = 37
2) I'm thinking of a number. If you square my number and add 16, you get the same answer as adding 20 to my number and then multiplying by 5. There are two possibilities for my number. Find them both. Let x be the number.
  1. The expression is
        x2 + 16 = 5(x + 20)
  2. Simplifying:
        x2 - 5x - 84 = 0
  3. Using the quadratic formula :

        x = 5 ± √ 25 +336      = (5 ± 19) / 2 = 12 or -7
                      2

Problem
Solution
3)
  1. Add the first 2 expressions together:
        a + b       =  155
             -b + c =   61
        a        + c = 216
  2. Simplify the 3rd expression by dividing by 3:
            1       =     1  
      (d + 13)       3x6

    So, d + 13 = 18 and d = -5
  3. ∛216  - (-5)2 = 6 - 25 = -19
4) Payton scores 17 points in her first basketball game. Each game after that her average points per game increases by 2. If this pattern holds, how many points will she score in the 7th game? After every game her average goes up by 2 points, so you average all her scores.
  1. Second game:
        (17 + n)/2 = 19
        n = 38 - 17 = 21
  2. Third game:
        (17+21+n)/3 = 21
        n = 63 - 38 = 25
  3. Fourth game:
        (17+21+25+n)/4 = 23
        n = 92 - 63 = 29
  4. Fifth game:
        (17+21+25+29+n)/5 = 25
        n =125 - 92 = 33
  5. Sixth game:
        (17+21+25+29+33+n)/6 = 27
        n = 162 - 125 = 37
  6. Seventh game:
        (17+21+25+29+33+37+n)/7 = 29
        n = 203 - 162 = 41
5) What is the smallest integer value for x that makes this statement true?
    4x - 75 < 57 < 2x - 3
  1. 2x - 3 must be greater than 57, so x must be greater than 30 = 31
  2. 4(31) - 75 = 124 - 75 = 49, so x = 31