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2017-11-06 Advanced


ABCDABCD is a quadrilateral with AD=BC,AD=BC, AB=40,AB=40, CD=20,CD=20, and mA+mB=90.{\color{#3D99F6}m\angle A} + {\color{#EC7300}m\angle B} = 90^\circ.

What is the area of quadrilateral ABCD?ABCD?

Hint: Consider how copies of ABCDABCD can be constructed into another shape.

Let x,y,x, y, and zz be real numbers satisfying xy+z+yz+x+zx+y=1.\dfrac{x}{y+z}+\dfrac{y}{z+x}+\dfrac{z}{x+y}=1.

Find the maximum value of x2y+z+y2z+x+z2x+y.\dfrac{x^2}{y+z}+\dfrac{y^2}{z+x}+\dfrac{z^2}{x+y}.

Tom is jumping around a 4×44\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,m, and the number of ways of doing so is r.r.

Find the value of

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

128=27,128=2^7, but none of the other permutations of the digits of 128 form powers of 2: 182, 218, 281, 812, 821.182,\ 218,\ 281,\ 812,\ 821.

Is there any power of 2, 2n,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=1 kg,M=\SI{1}{\kilo\gram}, its radius is r=0.5 m,r=\SI{0.5}{\meter}, and the rocket provides a constant thrust of T=10 NT = \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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