29th Annual
Virginia Tech Regional Mathematics Contest
From 9:00a.m. to 11:30a.m., October 27, 2007
Fill out the individual registration form
 Evaluate
∫_{0}^{x}(dθ)/(2 + tanθ),
where
0≤x≤π/2. Use your result to show that
∫_{0}^{π/4}(dθ)/(2 + tanθ) = (π + ln(9/8))/10.
 Given that
e^{x} = 1/0! + x/1! + x^{2}/2! + ... + x^{n}/n! + ...
find, in terms of e, the exact values of
 (a)

1/1! + 2/3! + 3/5! + ... + n/(2n  1)! + ... and
 (b)

1/3! + 2/5! + 3/7! + ... + n/(2n + 1)! + ...
 Solve the initial value problem
dy/dx = y ln y + ye^{x}, y(0) = 1
(i.e. find y in terms of x).
 In the diagram below, P, Q, R are points on BC, CA, AB
respectively such that the lines AP, BQ, CR are concurrent at
X. Also PR bisects
∠BRC, i.e.
∠BRP = ∠PRC. Prove that
∠PRQ = 90^{o}.
 Find the third digit after the decimal point of
(2 + √5)^{100}((1 + √2)^{100} + (1 + √2)^{100}).
For example, the third digit after the decimal point of
π = 3.14159... is 1.
 Let n be a positive integer, let A, B be square
symmetric
n×n matrices with real entries (so if a_{ij} are
the entries of A, the a_{ij} are real numbers and
a_{ij} = a_{ji}). Suppose there are
n×n matrices X, Y
(with complex entries) such that
det(AX + BY)≠ 0. Prove that
det(A^{2} + B^{2})≠ 0 (det
indicates the determinant).
 Determine whether the series
∑_{n=2}^{∞}n^{(1+(ln(ln n))2)} is convergent or divergent
(ln denotes natural log).
Peter Linnell
20071027