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$\underbrace{\pm (x-1) \pm (x-1) \cdots \pm (x-1) \pm (x-1)}_{“\pm (x-1)" \text{ appears 2018 times}} = 2018$
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.
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Can the hour, minute, and second hands—when extended to the circumference—ever cut a properly functioning circular clock into three equal areas?
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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.
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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:
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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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