The Quest for the National Title is On! - The Problems
04/05/2004

There’s a very exciting National Championship approaching, and I’m not talking about the NCAA Men’s Basketball Final! The competitors for the 2004 MATHCOUNTS National Competition have all been identified now that the State Competitions are complete. The following are some of the problems that these students had to tackle (successfully!) to earn their way to the National Competition this May in Washington D.C. How would you have done?

Sprint #19 (the format has been adapted for this posting): The points (2, -5), (p, -14) and (p + 2, -17) are on a straight line. The point (13, q) lies on the same line. What is the value of p + q? Express your answer as a decimal to the nearest tenth.

Target #3: A license plate consists of two letters followed by two digits: for example, MP78. Neither the digits nor the letters may be repeated, and neither the letter “O” nor the digit “0” may be used. When reading from the left to right, the letters must be in alphabetical order and the digits must be in increasing order. How many different license plate combinations are possible?

Team #9: In each blank below a single digit is inserted such that the following six three-digit numbers, in this order, form an arithmetic sequence:

1 _ _ , _ _ 9 , 2 _ 2 , _ 6 _ , 2 _ _ , _ 3 _

What is the value of the next number in the sequence?