Problems of the Week (or General Period of Time) 2

Here're some problems from the Math Contests book I got for winning a contest–– see if you can do them!

1. (Easy) In the ordered sequence of positive integers, {1, 2, 2, 3, 3, 3, 4, 4, 4, 4....}, each positive integer n occurs in a block of n terms. For what value of k is the sum of the reciprocals of the first k terms equal to 1000?

2. (Easy) My area code is a positive three-digit number. Add 7 to it and the result is divisible by y. Add 8 to it and the result is divisible by 8. Add 9 to it and the result is divisible by 9. What is my area code?

3. (Easy-Moderate) Al and Bob each took some money from their piggy bank to get ice cream. Al was 24 cents short of a cone and Bob was 2 cents short. They pooled their money just to find out that they were still short. How much money did Bob take out of his bank? (sorry the names are boring, the original ones were Rufus and Dufus. I thought those would be stupid enough to take away the point of the problem...)

4. (Easy-Moderate) The factorial 23! can be written as the product of n consecutive integers such that 1< n <23. What are the possible values of n?

5. (Easy-Moderate) On his birthday, Brian was fourteen and his father was 41. What's the next time their ages will be the 'reverse' of each other? (Don't trial and error!)

6. (Moderate-Hard) Both x and y are positive numbers less than 2. Every positive number less than 2 is equally likely to be x and ditto y. What is the probability that x and y differ by less than one?

 7.  (Moderate) Form any positive integer n with less than 10 digits. Form the three digit number x where the hundreds digit is equal to the number of even digits in that number, the tens is equal to the number of odd digits, and the ones digit is equal to the sum of the other two. Then replace the original number with the new one. If this process is repeated infinitely, what number remains?

 8. (Moderate-Hard) What are the real values of x that satisfy   abs(2 - abs(1- abs(x))) = 1?

9. (Hard) An equiangular hexagon with side-lengths 9, 11, 10, 6, 14, and 7 (going clockwise) can be inscribed in an equilateral triangle with side-length 30. What is the side length of the other possible triangles?

10. (Hard) Two externally tangent circles have radii 2 and 3 units. If there is a segment whose endpoints are the center of the small circle and the point at which it is tangent to the larger circle, what is the segment's length?

The answers will be posted tomorrow evening!

Stay coolio,
John