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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 \(x, y,\) 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 \(mr.\)

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?

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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