1979 VTRMC

Part of VTRMC

Subcontests

(8)

1

1979 VTRMC #8

S

be a finite set of polynomials in two variables,

x

y

n

a positive integer, define

\Omega _ { n } ( S )

to be the collection of all expressions

p _ { 1 } p _ { 2 } \dots p _ { k } ,

p_i \in S

1\leq k \leq n

d_n(S)

indicate the maximum number of linearly independent polynomials in

\Omega _ { n } ( S )

\Omega _ { 2 } \left( \left\{ x ^ { 2 } , y \right\} \right) = \left\{ x ^ { 2 } , y , x ^ { 2 } y , x ^ { 4 } , y ^ { 2 } \right\}

d _ { 2 } \left( \left\{ x ^ { 2 } , y \right\} \right) = 5

d _ { 2 } ( \{ 1 , x , x + 1 , y \} )

. (b) Find a closed formula in

n

d _ { n } ( \{ 1 , x , y \} )

. (c) Calculate the least upper bound over all such sets of

\overline{\text{lim}} _ { n \rightarrow \infty } \frac { \log d _ { n } ( S ) } { \log n }

\overline{\text{lim}} _ { n \rightarrow \infty } a _ { n } = \lim _ { n \rightarrow \infty } ( \sup \left\{ a _ { n } , a _ { n + 1 } , \ldots \right\}

, where sup means supremum or least upper bound.)

1

1979 VTRMC #7

Let S be a finite set of non-negative integers such that

| x - y | \in S

x , y \in S

. (a) Give an example of such a set which contains ten elements. (b) If

A

S

containing more than two-thirds of the elements of

S

, prove or disprove that every element of

S

is the sum or difference of two elements from

A

1

1979 VTRMC #6

a _ { n } > 0

\sum _ { n = 1 } ^ { \infty } a _ { n }

diverges. Determine whether

\sum _ { n = 1 } ^ { \infty } a _ { n } / S _ { n } ^ { 2 }

converges, where

S _ { n } = a _ { 1 } + a _ { 2 } + \dots + a _ { n } .

1

1979 VTRMC #5

Show, for all positive integers

n = 1,2 , \dots ,

14

3 ^ { 4 n + 2 } + 5 ^ { 2 n + 1 }

1

1979 VTRMC #4

f(x)

be continuously differentiable on

(0,\infty)

\lim _ { x \rightarrow \infty } f ^ { \prime } ( x ) = 0

\lim _ { x \rightarrow \infty } f ( x ) / x = 0

1

1979 VTRMC #3

A

n\times n

nonsingular matrix with complex elements, and let

\overline{A}

be its complex conjugate. Let

B = A\overline{A}+I

I

n\times n

identity matrix. (a) Prove or disprove:

A^{-1}BA = \overline{B}

. (b) Prove or disprove: the determinant of

A\overline{A}+I

1

VTRMC 1979 #2

S

be a set which is closed under the binary operation

\circ

, with the following properties: (i) there is an element

e \in S

a \circ e = e \circ a = a

a \in S

(a \circ b) \circ (c \circ d)=(a \circ c) \circ (b \circ d)

a,b, c,d \in S

.
Prove or disprove: (a)

\circ

is associative on S (b)

\circ

is commutative on S

1

VTRMC 1979 #1

Show that the right circular cylinder of volume

V

which has the least surface area is the one whose diameter is equal to its altitude. (The top and bottom are part of the surface.)