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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 LL and ll share the same center. Circle LL has radius 3r3r and circle ll has radius r.r. Points A,B,CA, B, C are chosen on the circumference of circle LL uniformly at random.

What is the probability that the centroid of ABC\triangle ABC lies in the interior of circle ll?

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

If EE is the intersection point of both diagonals of ABCDABCD, what is the length of ED,ED, the blue line segment in the diagram?

m3n3m2+n2mn \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 mm and nn are positive integers.

In the xyxy-coordinate system, a 1 kg\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=10 N/mk = \SI[per-mode=symbol]{10}{\newton\per\meter}, and the spring's unstretched length is 1 m\SI{1}{\meter}.

The mass is initially being held in the air horizontally at (x,y)=(1 m,0 m),(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) yy-axis?


Details and Assumptions:

  • Give your answer in meters, to 2 decimal places.
  • There is an ambient downward gravitational acceleration of 10 m/s2\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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