2018-07-30 Intermediate

         

When a cannon is on the ground, the cannonball lands the farthest from the cannon if it's shot at a \(45^\circ\) angle.

Now, when the cannon is on a cliff, what shooting angle \(\theta\) makes the cannonball land the farthest?

Details and Assumptions:

  • Neglect air resistance.
  • The cannon shoots the cannonball at the same speed regardless of the angle of projection.

Can \(\sqrt2, \sqrt3, \sqrt5\) all be numbers in an arithmetic progression?

Note: The three numbers do not need to be consecutive in the progression.

A cube with side length 1 is sliced such that the cut is a regular hexagon (in red).

What is the side length of this regular hexagon (to 4 decimal places)?

\[\large \sqrt[3]{x+\sqrt[3]{x+\sqrt[3]{x+\cdots}}}\ =\ \sqrt[3]{x\ \sqrt[3]{x\ \sqrt[3]{x\ldots}}}\]

The non-zero solution \(x\) of the equation above can be written as \(\frac{\sqrt{a}+\sqrt{b}}{2},\) where \(a\) and \(b\) are positive integers.

What is \(a+b?\)

The infinite Atwood’s machine shown has a string passing over each pulley, with one end attached to a mass and the other end attached to another pulley.

The masses are held fixed and then released simultaneously, and the acceleration of the top mass is \(\lambda g.\)

What is \(\lambda?\)

Details and Assumptions:

  • All the masses are equal to \(M\).
  • All the pulleys and strings are massless.
  • \(g\) is the acceleration due to gravity.
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