Difference between revisions of "Mock AIME 2 2006-2007 Problems"
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== Problem 6 ==
== Problem 6 ==
If <math>\displaystyle \tan 15^\circ \tan 25^\circ \tan 35^\circ =\tan \theta</math> and <math>\displaystyle 0^\circ \le \theta \le
If <math>\displaystyle \tan 15^\circ \tan 25^\circ \tan 35^\circ =\tan \theta</math> and <math>\displaystyle 0^\circ \le \theta \le ^\circ, </math> find <math>\displaystyle \theta.</math>
Revision as of 12:52, 25 July 2006
A positive integer is called a dragon if it can be partitioned into four positive integers and such that Find the smallest dragon.
The set consists of all integers from to inclusive. For how many elements in is an integer?
Let be the sum of all positive integers such that is a perfect square. Find the remainder when is divided by
Let be the smallest positive integer for which there exist positive real numbers and such that . Compute .
Given that and find
If and find
A right circular cone of base radius cm and slant height cm is given. is a point on the circumference of the base and the shortest path from around the cone and back is drawn (see diagram). If the minimum distance from the vertex to this path is where and are relatively prime positive integers, find
The positive integers satisfy and for . Find the last three digits of .
In right triangle Cevians and are drawn to and respectively such that and If where and are relatively prime and has no perfect square divisors excluding find
Find the number of solutions, in degrees, to the equation where
Find the sum of the squares of the roots, real or complex, of the system of simultaneous equations
In quadrilateral and If , and the area of is where are relatively prime positive integers, find
Note*: and refer to the areas of triangles and
In his spare time, Richard Rusczyk shuffles a standard deck of 52 playing cards. He then turns the cards up one by one from the top of the deck until the third ace appears. If the expected (average) number of cards Richard will turn up is where and are relatively prime positive integers, find
In triangle ABC, and Given that , and intersect at and are an angle bisector, median, and altitude of the triangle, respectively, compute the length of
A cube is composed of unit cubes. The faces of unit cubes are colored red. An arrangement of the cubes is if there is exactly red unit cube in every rectangular box composed of unit cubes. Determine the number of colorings.