2012 AIME II Problems
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Find the number of ordered pairs of positive integer solutions to the equation .
Two geometric sequences and have the same common ratio, with , , and . Find .
At a certain university, the division of mathematical sciences consists of the departments of mathematics, statistics, and computer science. There are two male and two female professors in each department. A committee of six professors is to contain three men and three women and must also contain two professors from each of the three departments. Find the number of possible committees that can be formed subject to these requirements.
Ana, Bob, and Cao bike at constant rates of meters per second, meters per second, and meters per second, respectively. They all begin biking at the same time from the northeast corner of a rectangular field whose longer side runs due west. Ana starts biking along the edge of the field, initially heading west, Bob starts biking along the edge of the field, initially heading south, and Cao bikes in a straight line across the field to a point on the south edge of the field. Cao arrives at point at the same time that Ana and Bob arrive at for the first time. The ratio of the field's length to the field's width to the distance from point to the southeast corner of the field can be represented as , where , , and are positive integers with and relatively prime. Find .
In the accompanying figure, the outer square has side length . A second square of side length is constructed inside with the same center as and with sides parallel to those of . From each midpoint of a side of , segments are drawn to the two closest vertices of . The result is a four-pointed starlike figure inscribed in . The star figure is cut out and then folded to form a pyramid with base . Find the volume of this pyramid.
Let be the complex number with and such that the distance between and is maximized, and let . Find .
Let be the increasing sequence of positive integers whose binary representation has exactly ones. Let be the 1000th number in . Find the remainder when is divided by .
The complex numbers and satisfy the system Find the smallest possible value of .
Let and be real numbers such that and . The value of can be expressed in the form , where and are relatively prime positive integers. Find .
Find the number of positive integers less than for which there exists a positive real number such that .
Note: is the greatest integer less than or equal to .
Let , and for , define . The value of that satisfies can be expressed in the form , where and are relatively prime positive integers. Find .
For a positive integer , define the positive integer to be -safe if differs in absolute value by more than from all multiples of . For example, the set of -safe numbers is . Find the number of positive integers less than or equal to which are simultaneously -safe, -safe, and -safe.
Equilateral has side length . There are four distinct triangles , , , and , each congruent to , with . Find .
In a group of nine people each person shakes hands with exactly two of the other people from the group. Let be the number of ways this handshaking can occur. Consider two handshaking arrangements different if and only if at least two people who shake hands under one arrangement do not shake hands under the other arrangement. Find the remainder when is divided by .
Triangle is inscribed in circle with , , and . The bisector of angle meets side at and circle at a second point . Let be the circle with diameter . Circles and meet at and a second point . Then , where and are relatively prime positive integers. Find .
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