Virginia Tech Regional Mathematics Contest
From 9:00a.m. to 11:30a.m., October 30, 2010
Fill out the individual registration form
- Let d be a positive integer and let A be a
with integer entries. Suppose
I + A + A2 + ... + A100 = 0
(where I denotes the identity
d×d matrix, so I has 1's on
the main diagonal, and 0 denotes the zero matrix, which has all
entries 0). Determine the positive integers
n≤100 for which
An + An+1 + ... + A100 has determinant
- For n a positive integer, define
f1(n) = n and then for i a
positive integer, define
fi+1(n) = fi(n)fi(n).
f100(75)mod 17 (i.e. determine the remainder
f100(75) by 17, an integer between 0 and 16).
Justify your answer.
- Prove that
cos(π/7) is a root of the equation
8x3 -4x2 - 4x + 1 = 0, and find the other two roots.
ΔABC be a triangle with sides a, b, c and
corresponding angles A, B, C (so a = BC and
A = ∠BAC
etc.). Suppose that
4A + 3C = 540o. Prove that
(a - b)2(a + b) = bc2.
- Let A, B be two circles in the plane with B inside A. Assume
that A has radius 3, B has radius 1, P is a point on A, Q
is a point on B, and A and B touch so that P and Q are the
same point. Suppose that A
is kept fixed and B is rolled once round the inside of A so that
Q traces out a curve starting and finishing at P. What is the
area enclosed by this curve?
- Define a sequence by a1 = 1, a2 = 1/2, and
an+2 = an+1 - anan+1/2
for n a positive integer. Find
∑n=1∞an be a convergent series of positive
terms (so ai > 0 for all i) and set
bn = 1/(nan2) for
n≥1. Prove that
∑n=1∞n/(b1 + b2 + ... + bn)