Mock AIME 1 Pre 2005 Problems
Contents
Problem 1
Let S denote the sum of all of the three digit positive integers with three distinct digits. Compute the remainder when S is divided by 1000.
Problem 2
2. If x^2 + y^2 - 30x - 40y + 24^2 = 0, then the largest possible value of \frac{y}{x} can be written as \frac{m}{n}, where m and n are relatively prime positive integers. Determine m + n.
Problem 3
3. A, B, C, D, and E are collinear in that order such that AB = BC = 1, CD = 2, and DE = 9. If P can be any point in space, what is the smallest possible value of AP^2 + BP^2 + CP^2 + DP^2 + EP^2?
Problem 4
4. When 1 + 7 + 7^2 + \cdots + 7^{2004} is divided by 1000, a remainder of N is obtained. Determine the value of N.
Problem 5
5. Let a and b be the two real values of x for which
\displaymode{
\sqrt[3]{x} + \sqrt[3]{20 - x} = 2
}
The smaller of the two values can be expressed as p - \sqrt{q}, where p and q are integers. Compute p + q.
Problem 6
6. 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.
Problem 7
7. Let N denote the number of permutations of the 15-character string AAAABBBBBCCCCCC such that
(1) None of the first four letter is an A. (2) None of the next five letters is a B. (3) None of the last six letters is a C.
Find the remainder when N is divided by 1000.
Problem 8
8. ABCD, a rectangle with AB = 12 and BC = 16, is the base of pyramid P, which has a height of 24. A plane parallel to ABCD is passed through P, dividing P into a frustum F and a smaller pyramid P'. Let X denote the center of the circumsphere of F, and let T denote the apex of P. If the volume of P is eight times that of P', then the value of XT can be expressed as \frac{m}{n}, where m and n are relatively prime positive integers. Compute the value of m + n.
Problem 9
9. p, q, and r are three non-zero integers such that p + q + r = 26 and
\displaymode{
\frac{1}{p} + \frac{1}{q} + \frac{1}{r} + \frac{360}{pqr} = 1
}
Compute .
Problem 10
10. 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
.
Problem 11
11. Let denote the value of the sum
Determine the remainder obtained when
is divided by
.
Problem 12
12. 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
.
Problem 13
13. 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
.
Problem 14
14. 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 My. Porter's 12 new keys on his keychain (Where rotations are considered the same and any two keys of the same type are identical.)
Problem 15
15. 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
.