Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2017 Grade 8 Algebra
Problem
Hint
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
       
       
     
  3. See if you can find some combination of the above numbers that gets you the correct result.
  4. The expression for the operator ╬ is _______________

  5. 8 ╬ 5 = _______
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. Write the expression:
        x2 + ______________ = ______________

  2. Put this equation in a x2 + b x + c = 0 form =
    ________________________
  3. Use the quadratic formula so solve for x :




    x = _________

Problem
Hint
3)
  1. Add the first 2 expressions together:


    _______________

  2. Simplify the 3rd expression and solve for d:



    d = __________

  3. ∛a+c  - _____ 2 = ______
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. At each game her added score is n. I will do the second game for you:
  1. Second game:
        (17 + n)/2 = 19
        n = ______
  2. Third game:
        _____________________
        n = ___
  3. Fourth game:
        _____________________
        n = _____
  4. Fifth game:
        _____________________
        n = ____
  5. Sixth game:
        ______________________
        n = ____
  6. Seventh game:
        __________________________

        n = ______
5) What is the smallest integer value for x that makes this statement true?
    4x - 75 < 57 < 2x - 3
  1. Look at the second expression:
    2x - 3 must be greater than ___ , so x must be __________