

A1. Let N be the set {1, 2, ... , n}, where n is an odd integer. Let f : N x N → N satisfy: (1) f(r, s) = f(s, r) for all r, s; (2) {f(r, s) : s ∈ N} = N for each r. Show that {f(r, r) : r ∈ N} = N.


A2. Given any five points in the interior of a square side 1, show that two of the points are a distance apart less than k = 1/√2. Is this result true for a smaller k?


A3. Let S be the set of all curves satisfying y' + a(x) y = b(x), where a(x) and b(x) are never zero. Show that if C ∈ S, then the tangent at the point x = k on C passes through a point P_{k} which is independent of C.


A4. A uniform rod length 2a is suspended in midair with one end resting against a smooth vertical wall at X and the other end attached by a string length 2b to a point on the wall above X. For what angles between the rod and the string is equilibrium possible?


A5. R is the reals. f : (0, 1) → R satisfies lim_{x→0} f(x) = 0, and f(x)  f(x/2) = o(x) as x→0. Show that f(x) = o(x) as x→0.


A6. The real sequence a_{n} satisfies a_{n} = ∑_{n+1}^{∞} a_{k}^{2}. Show ∑ a_{n} does not converge unless all a_{n} are zero.


A7. Prove that the equation m^{2} + 3mn  2n^{2} = 122 has no integral solutions.


B1. Show that for any positive integer r, we can find integers m, n such that m^{2}  n^{2} = r^{3}.


B2. Let a_{n} = ∑_{1}^{n} (1)^{i+1}/i. Assume that lim_{n→∞} a_{n} = k. Rearrange the terms by taking two positive terms, then one negative term, then another two positive terms, then another negative term and so on. Let b_{n} be the sum of the first n terms of the rearranged series. Assume that lim_{n→∞} b_{n} = h. Show that b_{3n} = a_{4n} + a_{2n}/2, and hence that h ≠ k.


B3. Let S be a finite collection of closed intervals on the real line such that any two have a point in common. Prove that the intersection of all the intervals is nonempty.


B4. Let F be a point, and L and D lines, in the plane. Show how to construct the point of intersection (if any) between L and the parabola with focus F and directrix D.


B5. Let R be the reals. Let f : (1, 1) → R be a function with a derivative at 0. Let a_{n} be a sequence in (1, 0) tending to zero and b_{n} a sequence in (0, 1) tending to zero. Show that lim_{n→∞} (f(b_{n})  f(a_{n})/(b_{n}  a_{n}) = f '(0).


B6. If x is a positive rational, show that we can find distinct positive integers a_{1}, a_{2}, ... , a_{n} such that x = ∑ 1/a_{i}.


B7. Let α be a positive real. Let a_{n} = S_{1}^{n} (α/n + i/n)^{n}. Show that lim a_{n} ∈ (e^{α}, e^{α+1}).

