Virginia Tech Regional Mathematics Contest

- What is the remainder when
*X*^{1982}+ 1 is divided by*X*- 1? Verify your answer. - A box contains marbles, each of which is red, white or blue. The
number of blue marbles is a least half the number of white marbles
and at most on-third the number of red marbles. The number which
are white or blue is at least 55. Find the minimum possible number
of red marbles.
- Let
**a**,**b**, and**c**be vectors such that {**a**,**b**,**c**} is linearly dependent. Show that| **a**·**a****a**·**b****a**·**c****b**·**a****b**·**b****b**·**c****c**·**a****c**·**b****c**·**c**| = 0. - Prove that
*t*^{n - 1}+*t*^{1 - n}<*t*^{n}+*t*^{-n}when*t*1,*t*> 0 and*n*is a positive integer. - When asked to state the Maclaurin Series, a student writes
(incorrectly)
(*)
*f*(*x*) =*f*(*x*) +*xf'*(*x*) +*x*^{2}*f''*(*x*)/2! +*x*^{3}*f'''*(*x*)/3! + ....- (a)
- State Maclaurin's Series for
*f*(*x*) correctly. - (b)
- Replace the left-hand side of (*) by a simple closed form expression
in
*f*in such a way that the statement becomes valid (in general).

- Let
*S*be a set of positive integers and let*E*be the operation on the set of subsets of*S*defined by*EA*= {*x**A*|*x*is even}, where*A**S*. Let C*A*denote the complement of*A*in*S*.*E*C*EA*will denote*E*(C(*EA*)) etc.- (a)
- Show that
*E*C*E*C*EA*=*EA*. - (b)
- Find the maximum number of distinct subsets of
*S*that can be generated by applying the operations*E*and C to a subset*A*of*S*an arbitrary number of times in any order.

- Let
*p*(*x*) be a polynomial of the form*p*(*x*) =*ax*^{2}+*bx*+*c*, where*a*,*b*and*c*are*integers*, with the property that 1 <*p*(1) <*p*(*p*(1)) <*p*(*p*(*p*(1))). Show that*a*0. - For
*n*2, define*S*_{n}by*S*_{n}= S_{k = n}^{}*k*^{-2}.- (a)
- Prove or disprove that
1/
*n*<*S*_{n}< 1/(*n*- 1). - (b)
- Prove or disprove that
*S*_{n}< 1/(*n*- 3/4).