

A1. Let the largest power of 5 dividing 1^{1}2^{2}3^{3} ... n^{n} be 5^{f(n)}. What is lim_{n→∞} f(n)/n^{2} ?


A2. We can label the squares of an 8 x 8 chess board from from 1 to 64 in 64! different ways. For each way we find D, the largest difference between the labels of two squares which are adjacent (orthogonally or diagonally). What is the smallest possible D?


A3. Evaluate: lim_{k→∞} e^{k} ∫_{R} (e^{x}  e^{y}) / (x  y) dx dy, where R is the rectangle 0 ≤ x, y, ≤ k.


A4. A particle moves in a straight line inside a square side 1. It is reflected from the sides, but absorbed by the four corners. It starts from an arbitrary point P inside the square. Let c(k) be the number of possible starting directions from which it reaches a corner after traveling a distance k or less. Find the smallest constant a_{2}, such that from some constants a_{1} and a_{0}, c(k) ≤ a_{2}k^{2} + a_{1}k + a_{0} for all P and all k.


A5. p(x) is a real polynomial with at least n distinct real roots greater than 1. [To be precise we can find at least n distinct values a_{i} > 1 such that p(a_{i}) = 0. It is possible that one or more of the a_{i} is a multiple root, and it is possible that there are other roots.] Put q(x) = (x^{2} + 1) p(x) p'(x) + x p(x)^{2} + x p'(x)^{2}. Must q(x) have at least 2n  1 distinct real roots?


A6. A, B, C are lattice points in the plane. The triangle ABC contains exactly one lattice point, X, in its interior. The line AX meets BC at E. What is the largest possible value of AX/XE?


B1. Evaluate lim_{n→∞} 1/n^{5} ∑ (5 r^{4}  18 r^{2} s^{2} + 5 s^{4}), where the sum is over all r, s satisfying 0 < r, s ≤ n.


B2. What is the minimum value of (a  1)^{2} + (b/a  1)^{2} + (c/b  1)^{2} + (4/c  1)^{2}, over all real numbers a, b, c satisfying 1 ≤ a ≤ b ≤ c ≤ 4.


B3. Prove that infinitely many positive integers n have the property that for any prime p dividing n^{2} + 3, we can find an integer m such that (1) p divides m^{2} + 3, and (2) m^{2} < n.


B4. A is a set of 5 x 7 real matrices closed under scalar multiplication and addition. It contains matrices of ranks 0, 1, 2, 4 and 5. Does it necessarily contain a matrix of rank 3?


B5. f(n) is the number of 1s in the base 2 representation of n. Let k = ∑ f(n) / (n + n^{2}), where the sum is taken over all positive integers. Is e^{k} rational?


B6. Let P be a convex polygon each of whose sides touches a circle C of radius 1. Let A be the set of points which are a distance 1 or less from P. If (x, y) is a point of A, let f(x, y) be the number of points in which a unit circle center (x, y) intersects P (so certainly f(x, y) ≥ 1). What is sup 1/A ∫_{A} f(x, y) dx dy, where the sup is taken over all possible polygons P?

