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2017-05-15 Advanced


You go to a carnival and decide to play the Cover the Spot game.

The rules are simple: you must cover the largest circular spot possible using 5 identical circular disks. Which configuration can cover a larger spot?

A) All 5 disks pass through the center of the spot.
B) Exactly 3 disks pass through the center of the spot.

Circles \(L\) and \(l\) share the same center. Circle \(L\) has radius \(3r\) and circle \(l\) has radius \(r.\) Points \(A, B, C\) are chosen on the circumference of circle \(L\) uniformly at random.

What is the probability that the centroid of \(\triangle ABC\) lies in the interior of circle \(l\)?

A cyclic quadrilateral \(ABCD\) is constructed within a circle such that \(AB = 3, BC = 6,\) and \(\triangle ACD\) is equilateral, as shown to the right.

If \(E\) is the intersection point of both diagonals of \(ABCD\), what is the length of \(ED,\) the blue line segment in the diagram?

\[ \frac{m^3-n^3}{m^2+n^2-mn}\]

Find the sum of all prime numbers less than 900 that can be expressed as the above fraction where \(m\) and \(n\) are positive integers.

In the \(xy\)-coordinate system, a \(\SI{1}{\kilo\gram}\) mass is attached to one end of a massless ideal spring. The other end of the spring is fixed at the origin. The spring constant is \(k = \SI[per-mode=symbol]{10}{\newton\per\meter}\), and the spring's unstretched length is \(\SI{1}{\meter}\).

The mass is initially being held in the air horizontally at \((x,y) = (\SI{1}{\meter}, \SI{0}{\meter}),\) and then is released so that gravity pulls it downwards. How far is the mass from the origin right at the moment it first crosses the (vertical) \(y\)-axis?

Details and Assumptions:

  • Give your answer in meters, to 2 decimal places.
  • There is an ambient downward gravitational acceleration of \(\SI[per-mode=symbol]{10}{\meter\per\second\squared}\).
  • For the sake of this problem, assume that the spring can be stretched to great lengths.

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