

A1. Let f(n) = ∑_{1}^{n} [r/2]. Show that f(m + n)  f(m  n) = mn for m > n > 0.


A2. A triangle has sides a, b, c. The radius of the inscribed circle is r and s = (a + b + c)/2. Show that 1/(s  a)^{2} + 1/(s  b)^{2} + 1/(s  c)^{2} ≥ 1/r^{2}.


A3. Define the sequence {a_{n}} by a_{1} ∈ (0, 1), and a_{n+1} = a_{n}(1  a_{n}). Show that lim_{n→∞}n a_{n} = 1.


A4. Delete all the squares from the sequence 1, 2, 3, ... . Show that the nth number remaining is n + m, where m is the nearest integer to √n.


A5. Let S be the set of continuous realvalued functions on the reals. φ :S → S is a linear map such that if f, g ∈ S and f(x) = g(x) on an open interval (a, b), then φf = φg on (a, b). Prove that for some h ∈ S, (φf)(x) = h(x)f(x) for all f and x.


A6. Let a_{n} = √(1 + 2 √(1 + 3 √(1 + 4 √(1 + 5 √( ... + (n  1) √(1 + n) ... ) ) ) ) ). Prove lim a_{n} = 3.


B1. A convex polygon does not extend outside a square side 1. Prove that the sum of the squares of its sides is at most 4.


B2. Prove that at least one integer in any set of ten consecutive integers is relatively prime to the others in the set.


B3. a_{n} is a sequence of positive reals such that ∑ 1/a_{n} converges. Let s_{n} = ∑_{1}^{n} a_{i}. Prove that ∑ n^{2}a_{n}/s_{n}^{2} converges.


B4. Given a set of (mn + 1) unequal positive integers, prove that we can either (1) find m + 1 integers b_{i}in the set such that b_{i} does not divide b_{j} for any unequal i, j, or (2) find n+1 integers a_{i} in the set such that a_{i} divides a_{i+1} for i = 1, 2, ... , n.


B5. Given n points in the plane, no three collinear, prove that we can label them P_{i} so that P_{1}P_{2}P_{3} ... P_{n} is a simple closed polygon (with no edge intersecting any other edge except at its endpoints).


B6. y = f(x) is a solution of y'' + e^{x}y = 0. Prove that f(x) is bounded.

