2006 SMT/Algebra Problems
Contents
[hide]Problem 1
A finite sequence of positive integers for are defined so that and for . How many of these integers are divisible by ?
Problem 2
Find the minimum value of for real numbers .
Problem 3
A Gaussian prime is a Gaussian integer (where and are integers) with no Guassian integer factors of smaller absolute value. Factor into Gaussian primes with positive real parts. is a symbol with the property that .
Problem 4
Simplify:
Problem 5
Jerry is bored one day, so he makes an array of Cocoa pebbles. He makes equal rows with the pebbles remaining in a box. When Kramer drops by and eats one, Jerry yells at him until Kramer realizes he can make equal rows with the remaining pebbles. After Kramer eats another, he finds he can make equal rows with the remaining pebbles. Find the smallest number of pebbles that were in the box in the beginning.
Problem 6
Let be real numbers satisfying:
Determine all possible values of .
Problem 7
Find all solutions to , where and are nonzero digits, and is an integer. ( and are not necessarily distinct.)
Problem 8
Evaluate:
Problem 9
Principal Skinner is thinking of two integers and and bets Superintendent Chalmers that he will not be able to determine these integers with a single piece of information. Chalmers asks Skinner the numerical value of . From the value of this expression alone, he miraculously determines both and . What is the value of the above expression?
Problem 10
Evaluate: for all .