Please solve this as soon as possible with clarity Document Preview: 1 A1 Let f be a real-valued function on the plane such that for every square ABCD in the plane, f(A) + f(B) + f(C) + f(D) = 0. Does it follow that f(P) = 0 for all points P in the plane? A2 Functions f; g; h are dierentiable on some open interval around 0 and satisfy the equations and initial conditions f0 = 2f2gh + 1 gh; f(0) = 1; g0 = fg2h + 4 fh; g(0) = 1; h0 = 3fgh2 + 1 fg ; h(0) = 1: Find an explicit formula for f(x), valid in some open interval around 0. A3 Let dn be the determinant of the n n matrix whose entries, from left to right and then from top to bottom, are cos 1; cos 2; : : : ; cos n2. (For example, d3 =cos 1 cos 2 cos 3 cos 4 cos 5 cos 6 cos 7 cos 8 cos 9: The argument of cos is always in radians, not degrees.) Evaluate limn!1 dn. B2 A game involves jumping to the right on the real number line. If a and b are real numbers and b > a, the cost of jumping from a to b is b3 ?? ab2. For what real numbers c can one travel from 0 to 1 in a nite number of jumps with total cost exactly c? B3 Call a subset S of f1; 2; : : : ; ng mediocre if it has the following property: Whenever a and b are elements of S whose average is an integer, that average is also an element of S. Let A(n) be the number of mediocre subests of f1; 2; : : : ; ng. [For instance, every subset of f1; 2; 3g except f1; 3g is mediocre, so A(3) = 7.] Find all positive integers n such that A(n + 2) ?? 2A(n + 1) + A(n) = 1. B4 Say that a polynomial with real coecients in two variables, x; y, is balanced if the average value of the polynomial on each circle centered at the origin is 0. The balanced polynomials of degree at most 2009 form a vector space V over R. Find the dimension of V . B5 Let f : (1;1) ! R be a dierentiable function such that f0(x) = x2 ?? (f(x))2 x2((f(x))2 + 1) for all x > 1. Prove that limx!1 f(x) = 1. B6 Prove that for every positive integer n, there is a sequence of integers a0; a1; : : : ; a2009 with a0 = 0 and a2009 = n such that each term after a0 is either an earlier Attachments: 2009.pdf