2018-06-25 Advanced

         

\[\underbrace{\pm (x-1) \pm (x-1) \cdots \pm (x-1) \pm (x-1)}_{“\pm (x-1)" \text{ appears 2018 times}} = 2018\]

On the left side of the equation, "\(\pm (x-1)\)" repeats 2018 times.

How many integer solutions of \(x\) are there?

Note: The "\(\pm\)" symbols are independent of one another.

Can the hour, minute, and second hands—when extended to the circumference—ever cut a properly functioning circular clock into three equal areas?

True or False?

If \[ \int \frac{e^x}{1+x^2} \ dx = f(x)e^x + C,\] where \(C\) is a constant, then \(f\) is a rational function. That is, \(f(x) = \frac{\mathrm{P}(x)}{\mathrm{Q}(x)},\) where \(\mathrm{P}(x)\) and \(\mathrm{Q}(x)\) are real polynomials in \(x.\)


Note: This problem was adapted from a question in the 2016 STEP III exam.

The blacksmiths of the kingdom of Mechania are wondering how much tension is holding a crown together as it rests on the head of their sovereign. The king of Mechania has a frictionless, perfectly spherical head with radius \(r = 10 \text{ cm}\), and the crown is a thin cord with a length of \( \ell = 37.7 \text{ cm}\) and a mass of \(m = 628.32 \text{ g}.\)

Compute the tension in the crown (in Newtons).

Assumptions:

  • The crown has no thickness and rests perfectly level (that is, it makes a circle which is contained in a plane parallel to the horizontal).
  • The gravitational constant of Mechania is \(g = 10 \text{ m/s}^2. \)

We know a unit sphere will look like a unit circle when viewed from any direction.

Now, we have a solid which has the silhouette of a unit circle when viewed from the top, front, or side (from perpendicular directions).

Find the maximum volume of this solid.

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