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“±(x−1)" appears 2018 times±(x−1)±(x−1)⋯±(x−1)±(x−1)=2018
On the left side of the equation, "±(x−1)" repeats 2018 times.
How many integer solutions of x are there?
Note: The "±" 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 ∫1+x2ex dx=f(x)ex+C, where C is a constant, then f is a rational function. That is, f(x)=Q(x)P(x), where P(x) and 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 cm, and the crown is a thin cord with a length of ℓ=37.7 cm and a mass of m=628.32 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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