Mock AIME 3 Pre 2005 Problems
Contents
Problem 1
Three circles are mutually externally tangent. Two of the circles have radii and . If the area of the triangle formed by connecting their centers is , then the area of the third circle is for some integer . Determine .
Problem 2
Let denote the number of digit positive integers have the property that their digits are in increasing order. Determine the remainder obtained when is divided by . (Repeated digits are allowed.)
Problem 3
A function is defined for all real numbers . For all non-zero values , we have
Let denote the sum of all of the values of for which . Compute the integer nearest to .
Problem 4
and are complex numbers such that
Compute .
Problem 5
In Zuminglish, all words consist only of the letters and . As in English, is said to be a vowel and and are consonants. A string of and is a word in Zuminglish if and only if between any two there appear at least two consonants. Let denote the number of -letter Zuminglish words. Determine the remainder obtained when is divided by .
Problem 6
Let denote the value of the sum
can be expressed as , where and are positive integers and is not divisible by the square of any prime. Determine .
Problem 7
is a cyclic quadrilateral that has an inscribed circle. The diagonals of intersect at . If and then the area of the inscribed circle of can be expressed as , where and are relatively prime positive integers. Determine .
Problem 8
Let denote the number of -tuples of real numbers such that and
Determine the remainder obtained when is divided by .
Problem 9
is an isosceles triangle with base . is a point on and is the point on the extension of past such that is right. If and , then can be expressed as , where and are relatively prime positive integers. Determine .
Problem 10
is a sequence of positive integers such that
for all integers . Compute the remainder obtained when is divided by if .
Problem 11
is an acute triangle with perimeter . is a point on . The circumcircles of triangles and intersect and at and respectively such that and . If , then the value of can be expressed as , where and are relatively prime positive integers. Compute .
Problem 12
Determine the number of integers such that and is divisible by .
Problem 13
Let denote the value of the sum
Determine the remainder obtained when is divided by .
Problem 14
Circles and are centered on opposite sides of line , and are both tangent to at . passes through , intersecting again at . Let and be the intersections of and , and and respectively. and are extended past and intersect and at and respectively. If and , then the area of triangle can be expressed as , where and are positive integers such that and are coprime and is not divisible by the square of any prime. Determine .
Problem 15
Let denote the value of the sum
The value of can be expressed as , where and are relatively prime positive integers. Compute .