

A1. ABC is a triangle. P, Q, R are points on the sides BC, CA, AB. Show that one of the triangles AQR, BRP, CPQ has area no greater than PQR. If BP ≤ PC, CQ ≤ QA, AR ≤ RB, show that the area of PQR is at least 1/4 of the area of ABC.


A2. a_{n} = ±1/n and a_{n+8} > 0 iff a_{n} > 0. Show that if four of a_{1}, a_{2}, ... , a_{8} are positive, then ∑ a_{n} converges. Is the converse true?


A3. n is a positive integer. Prove that [√(4n + 1) ] = [ min (k + n/k) ], where the minimum is taken over all positive integers k.


A4. How many real roots does 2^{x} = 1 + x^{2} have?


A5. An object's equations of motion are: x' = yz, y' = zx, z' = xy. Its coordinates at time t = 0 are (x_{0}, y_{0}, z_{0}). If two of these coordinates are zero, show that the object is stationary for all t. If (x_{0}, y_{0}, z_{0}) = (1, 1, 0), show that at time t, (x, y, z) = (sec t, sec t, tan t). If (x_{0}, y_{0}, z_{0}) = (1, 1, 1), show that at time t, (x, y, z) = (1/(1 + t), 1/(1 + t), 1/(1 + t) ). If two of the coordinates x_{0}, y_{0}, z_{0} are nonzero, show that the object's distance from the origin d → ∞ at some finite time in the past or future.


A6. Show that there are no seven lines in the plane such that there are at least six points on just three lines and at least four points on just two lines.


B1. S is a finite collection of integers, not necessarily distinct. If any element of S is removed, then the remaining integers can be divided into two collections with the same size and the same sum. Show that all elements of S are equal.


B2. The real and imaginary parts of z are rational, and z has unit modulus. Show that z^{2n}  1 is rational for any integer n.


B3. The prime p has the property that n^{2}  n + p is prime for all positive integers less than p. Show that there is exactly one integer triple (a, b, c) such that b^{2}  4ac = 1  4p, 0 < a ≤ c, a ≤ b < a.


B4. f is defined on the closed interval [0, 1], f(0) = 0, and f has a continuous derivative with values in (0, 1]. By considering the inverse f^{ 1} or otherwise, show that ( ∫_{0}^{1} f(x) dx )^{2} ≥ ∫_{0}^{1} f(x)^{3} dx. Give an example where we have equality.


B5. If x is a solution of the quadratic ax^{2} + bx + c = 0, show that, for any n, we can find polynomials p and q with rational coefficients such that x = p(x^{n}, a, b, c) / q(x^{n}, a, b, c). Hence or otherwise find polynomials r, s with rational coefficients so that x = r (x^{3}, x + 1/x) / s(x^{3}, x + 1/x).


B6. Show that sin^{2}x sin 2x has two maxima in the interval [0, 2π], at π/3 and 4π/3. Let f(x) = the absolute value of sin^{2}x sin^{3}4x sin^{3}8x ... sin^{3}2^{n1}x sin 2^{n}x. Show that f(π/3) ≥ f(x). Let g(x) = sin^{2}x sin^{2}4x sin^{2}8x ... sin^{2}2^{n}x. Show that g(x) ≤ 3^{n}/4^{n}.

