91Ó°ÊÓ

a. Define sequences \(\left(a_n\right)\) and \(\left(b_n\right)\) as follows: \(a_n\) is the result of writing down the first \(n\) odd integers in order (for example, \(a_7=135791113\) ), while \(b_n\) is the result of writing down the first \(n\) even integers in order. Evaluate \(\lim _{n \rightarrow \infty} \frac{a_n}{b_n}\). b. Now suppose we do the same thing, but we write all the odd and even integers in base \(B\) (and we interpret the fractions \(a_n / b_n\) in base \(B\) ). For example, if \(B=9\) we will now have \(a_2=13, a_7=1357101214\). Show that for any base \(B \geq 2, \lim _{n \rightarrow \infty} \frac{a_n}{b_n}\) exists. For what values of \(B\) will the limit be the same as for \(B=10\) ?

Note that the integers \(2,-3\), and 5 have the property that the difference of any two of them is an integer times the third: $$ 2-(-3)=1 \times 5, \quad(-3)-5=(-4) \times 2, \quad 5-2=(-1) \times(-3) . $$ Suppose three distinct integers \(a, b, c\) have this property. a. Show that \(a, b, c\) cannot all be positive. b. Now suppose that \(a, b, c\), in addition to having the above property, have no common factors (except \(1,-1\) ). (For example, 20, \(-30,50\) would not qualify, because although they have the above property, they have the common factor 10.) Is it true that one of the three integers has to be either \(1,2,-1\), or \(-2\) ?

Note that the three positive integers \(1,24,120\) have the property that the sum of any two of them is a different perfect square. Do there exist four positive integers such that the sum of any two of them is a perfect square and such that the six squares found in this way are all different? If so, exhibit four such positive integers; if not, show why this cannot be done.

In general, composition of functions is not commutative. For example, for the functions \(f\) and \(g\) given by \(f(x)=x+1, g(x)=2 x\), we have \(f(g(x))=2 x+1\) and \(g(f(x))=2 x+2\). Now suppose that we have three functions \(f, g, h\). Then there are six possible compositions of the three, given by \(f(g(h(x))), g(h(f(x))), \ldots\). Give an example of three continuous functions that are defined for all real \(x\) and for which exactly five of the six compositions are the same. (Reprinted with the permission of the Canadian Mathematical Society, this problem was originally published in the Mathematical Mayhem section of Crux Mathematicorum with Mathematical Mayhem, vol. 25,1999, p. 293, problem C87.)

Suppose we are given an \(m\)-gon (polygon with \(m\) sides, and including the interior for our purposes) and an \(n\)-gon in the plane. Consider their intersection; assume this intersection is itself a polygon (other possibilities would include the intersection being empty or consisting of a line segment). a. If the \(m\)-gon and the \(n\)-gon are convex, what is the maximal number of sides their intersection can have? b. Is the result from (a) still correct if only one of the polygons is assumed to be convex? (Note: A subset of the plane is convex if for every two points of the subset, every point of the line segment between them is also in the subset. In particular, a polygon is convex if each of its interior angles is less than \(\left.180^{\circ}.\right)\)

Show that \(\sum_{k=0}^n \frac{(-1)^k}{2 n+2 k+1}\left(\begin{array}{l}n \\\ k\end{array}\right)=\frac{\left(2^n(2 n) !\right)^2}{(4 n+1) !}\).

Find all real polynomials \(p(x)\), whose roots are real, for which $$ p\left(x^2-1\right)=p(x) p(-x). $$

Let \(\mathcal{L}_1\) and \(\mathcal{L}_2\) be skew lines in space (that is, straight lines which do not lie in the same plane). How many straight lines \(\mathcal{L}\) have the property that every point on \(\mathcal{L}\) has the same distance to \(\mathcal{L}_1\) as to \(\mathcal{L}_2\) ?

Note that if the edges of a regular octahedron have length 1 , then the distance between any two of its vertices is either 1 or \(\sqrt{2}\). Are there other configurations of six points in \(\mathbb{R}^3\) for which the distance between any two of the points is either 1 or \(\sqrt{2}\) ? If so, find them.