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2018-06-25 Advanced

         

±(x1)±(x1)±(x1)±(x1)±(x1)" appears 2018 times=2018\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, "±(x1)\pm (x-1)" repeats 2018 times.

How many integer solutions of xx 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 ex1+x2 dx=f(x)ex+C, \int \frac{e^x}{1+x^2} \ dx = f(x)e^x + C, where CC is a constant, then ff is a rational function. That is, f(x)=P(x)Q(x),f(x) = \frac{\mathrm{P}(x)}{\mathrm{Q}(x)}, where P(x)\mathrm{P}(x) and Q(x)\mathrm{Q}(x) are real polynomials in x.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 cmr = 10 \text{ cm}, and the crown is a thin cord with a length of =37.7 cm \ell = 37.7 \text{ cm} and a mass of m=628.32 g.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 m/s2.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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