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This problem trains for: AMC-10, AMC-12, AIME.

An ant emerges at the top of a conical anthill with aperture 90° and starts moving south at a speed of 2 anters per eyeblink. After 2 eyeblinks another ant emerges and starts moving east at a speed of 3 anters per eyeblink. After another 2 eyeblinks a third ant emerges and starts moving west at a speed of 4 anters per eyeblink. After another 2 eyeblinks the radius of the circle that the three ants are on is a number of the form m/n√p where m and n are coprime and p does not have any factor that is a perfect square. What is p?

Denote the positions of the three ants after 6 eyeblinks from start by A, B and C. We have to find the radius of the circumcircle of the triangle ABC.

Because the ants move in directions that are orthogonal, it makes sense to solve this problem using coordinates. Let us select the origin of the time at the time when the first ant emerged and count time from there.

Let us consider the anthill to be a cone positioned so that the z axis passes through its vertex V. The origin of the system of coordinates is at the center of the base circle of the cone:

figure

After 6 eyeblinks the ants A and B are both 12 anters away from V, while ant C is 8 anters away from V. We choose to place the origin of the system of coordinates in the same plane as A and B and we denote with θ half of the cone's aperture:

figure

The coordinates of the ants at time t=6 are:

equation

equation

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The radius of the circumcircle is:

equation

The vectors AB, BC, AC are written in components:

equation

equation

equation

and their magnitudes are:

equation

equation

equation

The area of the triangle can be computed using the cross-product of any two of the vectors forming the triangle ABC. One of the cross-products we can use is:

equation

The area of the triangle is:

equation

The radius of the circumcircle is:

equation

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