1980 VTRMC

Part of VTRMC

Subcontests

(8)

1

1980 VTRMC #8

z=x+iy

be a complex number with

x

y

rational and with

|z| = 1.

(a) Find two such complex numbers. (b) Show that

|z^{2n}-1|=2|\sin n\theta|,

z=e^{i\theta}.

|z^2n -1|

is rational for every

n.

1

1980 VTRMC #7

S

be the set of all ordered pairs of integers

(m,n)

m>0

n<0.

<

be a partial ordering on

S

defined by the statement

(m,n)<(m',n')

m\le m'

n\le n'.

(5,-10)<(8,-2).

O

be a completely ordered subset of

S,

in other words if

(a,b)\in O

(c,d) \in O,

(a,b)<(c,d)

(c,d)<(a,b).

O'

denote the collection of all such completely ordered sets.
(a) Determine whether and arbitrary

O\in O'

is finite. (b) Determine whether the carnality

|O|

O

O\in O'.

(c) Determine whether

|O|

can be countable infinite for any

O\in O'.

1

1980 VTRMC #6

Given the linear fractional transformation of

x

f_1(x) = \tfrac{2x-1}{x+1},

f_{n+1}(x) = f_1(f_n(x))

n=1,2,3,\ldots.

It can be shown that

f_{35} = f_5.

A,B,C,D

f_{28}(x) = \tfrac{Ax+B}{Cx+D}.

1

1980 VTRMC #5

x>0,

e^x < (1+x)^{1+x}.

1

1980 VTRMC #4

P(x)

be any polynomial of degree at most

3.

It can be shown that there are numbers

x_1

x_2

\textstyle\int_{-1}^1 P(x) \ dx = P(x_1) + P(x_2),

x_1

x_2

are independent of the polynomial

P.

x_1=-x_2.

x_1

x_2.

1

1980 VTRMC #3

a_n = \frac{1\cdot3\cdot5\cdot\cdots\cdot(2n-1)}{2\cdot4\cdot6\cdot\cdots\cdot2n}.

\lim_{n\to \infty}a_n

exists. (b) Show that

a_n = \frac{\left(1-\frac1{2^2}\right)\left(1-\frac1{4^2}\right)\left(1-\frac1{6^2}\right)\cdots\left(1-\frac{1}{(2n)^2}\right)}{(2n+1)a_n}.

\lim_{n\to\infty}a_n

and justify your answer

1

1980 VTRMC #2

The sum of the first

n

terms of the sequence

1,1+2,1+2+2^2,\ldots,1+2+\cdots+2^{k-1},\ldots

2^{n+R}+Sn^2+Tn+U

n>0.

R,S,T,

U.

1

1980 VTRMC #1

*

denote a binary operation on a set

S

with the property that

(w*x)*(y*z) = w * z

w,x,y,z\in S.

a*b=c,

c*c = c.

a*b=c,

a*x=c*x

x\in S.