Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2011 Grade 7 Number Sense
Problem
Solution
1) A palindromic number is one which is the same read backwards and forwards. For example, 121 and 3443 are palindromic numbers. What is the next palindromic number after 79311397?
  1. This number has 8 digits, of which the last 4 repeat the first 4 in reverse order.
  2. Therefore the first 3 digits (and the last 3) must remain the same while the 4th (and 5th) numbers increment.
  3. Since the 4th and 5th digits are 1's, the next palindromic number must increment this to a 2, making the next palindromic number
    79322397
2) A number with exactly five factors can have at most how many different prime factors?
  1. If a number has exactly 5 factors, then there are only 3 factors that are not 1 and the number itself.
  2. Square numbers have an ODD number of factors. This is because factors are always in pairs, but in a square, one of the factors is multiplied by itself and it is only counted once. (the square root). Therefore the numbers that have 5 factors will be squares.
  3. The numbers that have 5 factors will be "squares of squares" or 4th powers of some basal number.
  4. The only numbers which have 3 factors, other than 1 and the number itself, are 16, 81, 625 and 2401, which are 24, 34, 54 and 74:
    factors of 16: 1,2,4,8,16
    factors of 81: 1,3,9,27,81
    factors of 625: 1,5,25,125,625
    factors of 2401: 1,7,49,343,2401
  5. Each of these is the 4th power of a prime number. Therefore numbers that have 5 factors can have at most     1 prime factor
3) Maria has a $50 gift certificate to go out to dinner with her best friend. They decide to spend as much of the $50 as possible. They also decide to order the same thing and narrow it down to three options. What should they choose and what will be the total they spend? Add on 25% for tax and tip.
  a) Chicken cordon bleu with drinks - $14.95
  b) Prime rib with drinks - $21.95
  c) Lobster tail with drinks - $19.95
  1. (a) Chicken cordon bleu: ($14.95 x 2) x 1.25 = $37.38
  2. (b) Prime rib: ($21.95 x 2) x 1.25 = $54.88
  3. (c) Lobster tail: ($19.95 x 2) x 1.25 = $49.88
  4. They should choose (c) $49.88

Problem
Solution
4) Insert parentheses into the expression so that the result is as large as possible. What is that result?
  1. Since there are divisions in this expression, you should look for a way to find a divisor with a maximum denominator.
  2. The only way to do that is to divide
    1⁄2 by (2⁄3 x 3⁄4) making that second expression
        1⁄2
  3. This makes the total expression:
        1⁄2 ÷ (2⁄3 x 3⁄4) ÷ 4⁄5
    which, by the way, is 5⁄4
5) A company has a 1 billion dollar loan from a bank. The first year the loan is interest free. At the start of the next year the company must pay back 1% of the loan plus 0.1% interest on the new outstanding balance. What is the company's first interest payment to the bank expressed in scientific notation?
  1. After that first payment of 1% at the beginning of the second year, the remaining balance is .99 x 1x109 = 9.9x108
  2. The interest payment is
    0.1% of 9.9x108 = .001 x 9.9x108 =
    1x10-3 x 9.9x108 = 9.9x105