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2017-11-06 Advanced

         

\(ABCD\) is a quadrilateral with \(AD=BC,\) \(AB=40,\) \(CD=20,\) and \({\color{blue}m\angle A} + {\color{orange}m\angle B} = 90^\circ.\)

What is the area of quadrilateral \(ABCD?\)

Hint: Consider how copies of \(ABCD\) can be constructed into another shape.

Let \(x,\) and \(y,\) and \(z\) be real numbers satisfying \(\dfrac{x}{y+z}+\dfrac{y}{z+x}+\dfrac{z}{x+y}=1.\)

Find the maximum value of \(\dfrac{x^2}{y+z}+\dfrac{y^2}{z+x}+\dfrac{z^2}{x+y}.\)

Tom is jumping around a \(4\times4\) grid of circles with the following conditions:

  • He starts from any of the 16 circles.
  • He always jumps in a straight line from one circle to any other circle.
  • Each subsequent jump is farther than the last.
  • He doesn't visit a circle already visited.

The maximum number of circles that he can step on is \(m,\) and the number of ways of doing so is \(r.\)

Find the value of \(mr.\)

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Note: The grid is fixed on the ground, so a method obtained by rotating another is considered different.

\(128=2^7,\) but none of the other permutations of the digits of 128 form powers of 2: \[182,\ 218,\ 281,\ 812,\ 821.\]

Is there any power of 2, \(2^n,\) such that at least one of its other permutations is also a power of 2?

A uniform disk sits on a smooth floor with a rocket strapped to its perimeter. One second after the rocket ignites, how far is the disk's center from where it started, in meters?

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Details and Assumptions:

  • The mass of the disk is \(M=\SI{1}{\kilo\gram},\) its radius is \(r=\SI{0.5}{\meter},\) and the rocket provides a constant thrust of \(T = \SI{10}{\newton}\) after it ignites.
  • Neglect the mass and size of the rocket.
  • Submit your answer to to 3 decimal places.
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