Mock AIME 1 Pre 2005 Problems
Let denote the sum of all of the three digit positive integers with three distinct digits. Compute the remainder when is divided by .
If , then the largest possible value of can be written as , where and are relatively prime positive integers. Determine .
and are collinear in that order such that and . If can be any point in space, what is the smallest possible value of ?
When is divided by , a remainder of is obtained. Determine the value of .
Let and be the two real values of for which The smaller of the two values can be expressed as , where and are integers. Compute .
A paperboy delivers newspapers to 10 houses along Main Street. Wishing to save effort, he doesn't always deliver to every house, but to avoid being fired he never misses three consecutive houses. Compute the number of ways the paperboy could deliver papers in this manner.
Let denote the number of permutations of the -character string such that
- None of the first four letters is an .
- None of the next five letters is a .
- None of the last six letters is a .
Find the remainder when is divided by .
, a rectangle with and , is the base of pyramid , which has a height of . A plane parallel to is passed through , dividing into a frustum and a smaller pyramid . Let denote the center of the circumsphere of , and let denote the apex of . If the volume of is eight times that of , then the value of can be expressed as , where and are relatively prime positive integers. Compute the value of .
and are three non-zero integers such that and Compute .
is a regular heptagon inscribed in a unit circle centered at . is the line tangent to the circumcircle of at , and is a point on such that triangle is isosceles. Let denote the value of . Determine the value of .
Let denote the value of the sum Determine the remainder obtained when is divided by .
is a rectangular sheet of paper. and are points on and respectively such that . If is folded over , maps to on and maps to such that . If and , then the area of can be expressed as square units, where and are integers and is not divisible by the square of any prime. Compute .
A sequence obeys the recurrence for any integers . Additionally, and . Let
can be expressed as for two relatively prime positive integers and . Determine the value of .
Wally's Key Company makes and sells two types of keys. Mr. Porter buys a total of 12 keys from Wally's. Determine the number of possible arrangements of Mr. Porter's 12 new keys on his keychain (rotations are considered the same and any two keys of the same type are identical).
Note: The problem is meant to be interpreted so that if you cannot produce one arrangement from another by rotation, then the two arrangements are different, even if you can produce one from the other from a combination of rotation and reflection.
Triangle has an inradius of and a circumradius of . If , then the area of triangle can be expressed as , where and are positive integers such that and are relatively prime and is not divisible by the square of any prime. Compute .