

A1. For n a positive integer find f(n), the number of pairs of positive integers (a, b) such that ab/(a + b) = n.


A2. Let S be the set consisting of a square with side 1 and its interior. Show that given any three points of S, we can find two whose distance apart is at most √6  √2.


A3. Let a, b, g, d, e be arbitary reals. Show that (1  α) e^{α} + (1  β) e^{α+β} + (1  γ) e^{α+β+γ} + (1  δ) e^{α+β+γ+δ} + (1  ε) e^{α+β+γ+δ+ε} ≤ k_{4}, where k_{1} = e, k_{2} = k_{1}^{e}, k_{3} = k_{2}^{e}, k_{4} = k_{3}^{e} (so k_{4} is 10^{k} with k approx 1.66 million).


A4. Given two points P, Q on the same side of a line l, find the point X which minimises the sum of the distances from X to P, Q and l.


A5. The real polynomial p(x) is such that for any real polynomial q(x), we have p(q(x)) = q(p(x)). Find p(x).


A6. A player throws a fair die (prob 1/6 for each of 1, 2, 3, 4, 5, 6 and each throw independent) repeatedly until his total score ≥ n. Let p(n) be the probability that his final score is n. Find lim p(n).


A7. Let f(n) be the smallest integer such that any permutation on n elements, repeated f(n) times, gives the identity. Show that f(n) = p f(n  1) if n is a power of p, and f(n) = f(n  1) if n is not a prime power.


B1. Find all pairs of unequal integers m, n such that m^{n} = n^{m}.


B2. Let f(m, n) = 3m+n+(m+n)^{2}. Find ∑_{0}^{∞}∑_{0}^{∞} 2^{f(m, n)}.


B3. Fluid flowing in the plane has the velocity (y + 2x  2x^{3}  2xy^{2},  x) at (x, y). Sketch the flow lines near the origin. What happens to an individual particle as t → ∞ ?


B4. Show that if an (infinite) arithmetic progression of positive integers contains an nth power, then it contains infinitely many nth powers.


B5. Define a_{n} by a_{0} = 0, a_{n+1} = 1 + sin(a_{n}  1). Find lim (∑_{0}^{n} a_{i})/n.


B6. Let 2^{f(n)} be the highest power of 2 dividing n. Let g(n) = f(1) + f(2) + ... + f(n). Prove that ∑ exp( g(n) ) converges.


B7. Let R' be the nonnegative reals. Let f, g : R' → R be continuous. Let a : R' → R be the solution of the differential equation: a' + f a = g, a(0) = c. Show that if b : R' → R satisfies b' + f b ≥ g for all x and b(0) = c, then b(x) ≥ a(x) for all x. Show that for sufficiently small x the solution of y' + f y = y^{2}, y(0) = d, is y(x) = max ( d e^{h(x)}  ∫_{0}^{x} e^{(h(x)h(t) )} u(t)^{2} dt ), where the maximum is taken over all continuous u(t), and h(t) = ∫_{0}^{t} (f(s)  2u(s)) ds.

