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Chapter 1: Problem 33
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Suppose you draw \(n\) parabolas in the plane. What is the largest number of (connected) regions that the plane may be divided into by those parabolas? (The parabolas can be positioned in any way; in particular, their axes need not be parallel to either the \(x\) - or the \(y\)-axis.)
Start with a circle and inscribe a regular \(n\)-gon in it, then inscribe a circle in that regular \(n\)-gon, then inscribe a regular \(n\)-gon in the new circle, then a third circle in the second \(n\)-gon, and so forth. Continuing in this way, the region (disk) inside the original circle will be divided into infinitely many smaller regions, some of which are bounded by a circle on the outside and one side of a regular \(n\)-gon on the inside (call these "type I" regions) while others are bounded by two sides of a regular \(n\)-gon on the outside and a circle on the inside ("type II" regions). Let \(f(n)\) be the fraction of the area of the original disk that is occupied by type I regions. What is the limit of \(f(n)\) as \(n\) tends to infinity?
Suppose all the integers have been colored with the three colors red, green and blue such that each integer has exactly one of those colors. Also suppose that the sum of any two (unequal or equal) green integers is blue, the sum of any two blue integers is green, the opposite of any green integer is blue, and the opposite of any blue integer is green. Finally, suppose that 1492 is red and that 2011 is green. Describe precisely which integers are red, which integers are green, and which integers are blue.
We call a sequence \(\left(x_n\right)_{n \geq 1}\) a superinteger if (i) each \(x_n\) is a nonnegative integer less than \(10^n\) and (ii) the last \(n\) digits of \(x_{n+1}\) form \(x_n\). One example of such a sequence is \(1,21,021,1021,21021,021021, \ldots\), which we abbreviate by ...21021. Note that the digit 0 is allowed (as in the example) and that (unlike in the example) there may not be a pattern to the digits. The ordinary positive integers are just those superintegers with only finitely many nonzero digits. We can do arithmetic with superintegers; for instance, if \(x\) is the superinteger above, then the product \(x y\) of \(x\) with the superinteger \(y=\ldots 66666\) is found as follows: \(1 \times 6=6\) : the last digit of \(x y\) is 6 . \(21 \times 66=1386\) : the last two digits of \(x y\) are 86 . \(021 \times 666=13986\) : the last three digits of \(x y\) are 986 . \(1021 \times 6666=6805986\) : the last four digits of \(x y\) are 5986, etc. Is it possible for two nonzero superintegers to have product \(0=\ldots 00000\) ?
The mayor of Wohascum Center has ten pairs of dress socks, ranging through ten shades of color from medium gray (1) to black (10). When he has worn all ten pairs, the socks are washed and dried together. Unfortunately, the light in the laundry room is very poor and all the socks look black there; thus, the socks get paired at random after they are removed from the drier. A pair of socks is unacceptable for wearing if the colors of the two socks differ by more than one shade. What is the probability that the socks will be paired in such a way that all ten pairs are acceptable?
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