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2018-11-05 Intermediate

         

This star has five spokes that are each 11 unit long, and the angles between adjacent spokes are all the same.

How many copies of this star would you need to completely cover a bigger star with spokes of length 1.1?1.1?


Inspiration

Pierre in Paris (48.9 N,2.3 E)(48.9 ^\circ \text{ N}, 2.3 ^\circ \text{ E}) and Sadie in Sydney (33.9 S,151.2 E)(33.9 ^\circ \text{ S}, 151.2 ^\circ \text{ E}) each perform the following experiment:

  • Set a pendulum swinging exactly along the East-West direction.
  • As Earth rotates about its axis, the plane of oscillation of the pendulum slowly rotates. Eventually, the pendulum will be momentarily swinging along the North-South direction.
  • Measure the amount of time it takes for this to happen for the first time.

Whose pendulum will take a greater amount of time?

A stuntman falls from the top of a building of height HH. On his way down, he bounces off of an awning at a height of hh in a perfectly elastic collision. Immediately after the bounce, the stuntman moves horizontally.

For what value of hH\frac hH will the man hit the safety cushion (at ground level) the farthest from the building?

Assume that air resistance is negligible.


Inspiration

144=1221444=382\begin{aligned} 144 &= 12^2 \\ 1444 &= 38^2 \end{aligned}

Are there any other numbers 14444 144 \ldots 44 (starting with 1 and followed by all 4s) that are perfect squares?

The function f(x)f(x) is defined for real constants a,a, b,b, c,c, and d:d:

f(x)=x4+ax3+bx2+cx+d.f(x)=x^4+ax^3+bx^2+cx+d.

The graph of the function is shown below.

Given that the tangent lines at (1,f(1))\big(1,f(1)\big) and at (5,f(5))\big(5,f(5)\big) coincide, what is the value of a+b?a+b?

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