

A1. The sequence a_{n} of real numbers satisfies a_{n+1} = 1/(2  a_{n}). Show that lim_{n→∞}a_{n} = 1.


A2. R is the reals. f : R → R is continuous and satisfies f(r) = f(x) f(y) for all x, y, where r = √(x^{2} + y^{2}). Show that f(x) = f(1) to the power of x^{2}.


A3. ABC is a triangle and P an interior point. Show that we cannot find a piecewise linear path K = K_{1}K_{2} ... K_{n} (where each K_{i}K_{i+1} is a straight line segment) such that: (1) none of the K_{i} do not lie on any of the lines AB, BC, CA, AP, BP, CP; (2) none of the points A, B, C, P lie on K; (3) K crosses each of AB, BC, CA, AP, BP, CP just once; (4) K does not cross itself.


A4. Take the xaxis as horizontal and the yaxis as vertical. A gun at the origin can fire at any angle into the first quadrant (x, y ≥ 0) with a fixed muzzle velocity v. Assuming the only force on the pellet after firing is gravity (acceleration g), which points in the first quadrant can the gun hit?


A5. The sequences a_{n}, b_{n}, c_{n} of positive reals satisfy: (1) a_{1} + b_{1} + c_{1} = 1; (2) a_{n+1} = a_{n}^{2} + 2b_{n}c_{n}, b_{n+1} = b_{n}^{2} + 2c_{n}a_{n}, c_{n+1} = c_{n}^{2} + 2a_{n}b_{n}. Show that each of the sequences converges and find their limits.


A6. A is the matrix
a b c
d e f
g h i
det A = 0 and the cofactor of each element is its square (for example the cofactor of b is fg  di = b^{2}). Show that all elements of A are zero.


B1. Let R be the reals. f : [1, ∞) → R is differentiable and satisfies f '(x) = 1/(x^{2} + f(x)^{2}) and f(1) = 1. Show that as x → ∞, f(x) tends to a limit which is less than 1 + π/4.


B2. R is the reals. f :(0, 1) → R is differentiable and has a bounded derivative: f '(x) <= k. Prove that : ∫_{0}^{1} f(x) dx  ∑_{1}^{n} f(i/n) /n ≤ k/n.


B3. Let O be the origin (0, 0) and C the line segment { (x, y) : x ∈ [1, 3], y = 1 }. Let K be the curve { P : for some Q ∈ C, P lies on OQ and PQ = 0.01 }. Let k be the length of the curve K. Is k greater or less than 2?


B4. p(z) = z^{2} + az + b has complex coefficients. p(z) = 1 on the unit circle z = 1. Show that a = b = 0.


B5. Let p(x) be the polynomial (x  a)(x  b)(x  c)(x  d). Assume p(x) = 0 has four distinct integral roots and that p(x) = 4 has an integral root k. Show that k is the mean of a, b, c, d.


B6. P is a variable point in space. Q is a fixed point on the zaxis. The plane normal to PQ through P cuts the xaxis at R and the yaxis at S. Find the locus of P such that PR and PS are at right angles.

