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. The next palindromic number is _______________
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. Examine some of these numbers to discover how many primes they have.
  5. Numbers that have 5 factors can have at most ____ prime factor(s).
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
 Compute the total cost of each dinner for both people, then add the tax and tip. The highest one whose total is below $50 will be the selection.

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. This makes the total expression:


    __________________________
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 _____________

  2. The interest payment is _______ on this balance.