Virginia Tech Regional Mathematics Contest
From 9:00a.m. to 11:30a.m., October 27, 2007
Fill out the individual registration form
∫0x(dθ)/(2 + tanθ),
0≤x≤π/2. Use your result to show that
∫0π/4(dθ)/(2 + tanθ) = (π + ln(9/8))/10.
- Given that
ex = 1/0! + x/1! + x2/2! + ... + xn/n! + ...
find, in terms of e, the exact values of
1/1! + 2/3! + 3/5! + ... + n/(2n - 1)! + ... and
1/3! + 2/5! + 3/7! + ... + n/(2n + 1)! + ...
- Solve the initial value problem
dy/dx = y ln y + yex, 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
∠BRP = ∠PRC. Prove that
∠PRQ = 90o.
- 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
n×n matrices with real entries (so if aij are
the entries of A, the aij are real numbers and
aij = aji). Suppose there are
n×n matrices X, Y
(with complex entries) such that
det(AX + BY)≠ 0. Prove that
det(A2 + B2)≠ 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).