MathDB

1

T=sum_{i=1}^n f(i)* (fi-1)/2 and n is odd, find min(T)

(x_1,x_2,\ldots, x_n)

of 0's and 1's, let

A

be the number of triples

(x_i,x_j,x_k)

with

i<j<k

such that

(x_i,x_j,x_k)

equals

(0,1,0)

or

(1,0,1)

. For

1\leq i \leq n

, let

d_i

denote the number of

j

for which either

j < i

and

x_j = x_i

or else

j > i

and

x_j\neq x_i

.
(a) Prove that

A = \binom n3 - \sum_{i=1}^n\binom{d_i}2.

(Of course,

\textstyle\binom ab = \tfrac{a!}{b!(a-b)!}

.) [5 points]
(b) Given an odd number

n

, what is the maximum possible value of

A

? [15 points]

4

1

Value of |PY|/|QY| such that |PQ| is maximized

Three circles

C_i

are given in the plane:

C_1

has diameter

AB

of length

1

;

C_2

is concentric and has diameter

k

(

1 < k < 3

);

C_3

has center

A

and diameter

2k

. We regard

k

as fixed. Now consider all straight line segments

XY

which have one endpoint

X

on

C_2

, one endpoint

Y

on

C_3

, and contain the point

B

. For what ratio

XB/BY

will the segment

XY

have minimal length?

3

1

If r(x),s(x) are distinct elements of $x$ then 0<x<1

Construct a set

S

of polynomials inductively by the rules:
(i)

x\in S

; (ii) if

f(x)\in S

, then

xf(x)\in S

and

x+(1-x)f(x)\in S

.
Prove that there are no two distinct polynomials in

S

whose graphs intersect within the region

\{0 < x < 1\}

.

2

1

Find all possible values of angle A

AD

,

BE

, and

CF

are the bisectors of the interior angles of triangle

ABC

, with

D

,

E

, and

F

lying on the perimeter. If angle

EDF

is

90

degrees, determine all possible values of angle

BAC

.

1

(m^2+n)(m+n^2)=(m-n)^3

Determine all solutions in non-zero integers

a

and

b

of the equation

(a^2+b)(a+b^2) = (a-b)^3.

1987 USAMO

Subcontests

T=sum_{i=1}^n f(i)* (fi-1)/2 and n is odd, find min(T)

Value of |PY|/|QY| such that |PQ| is maximized

If r(x),s(x) are distinct elements of $x$ then 0<x<1

Find all possible values of angle A

(m^2+n)(m+n^2)=(m-n)^3

1987 USAMO

Subcontests

T=sum_{i=1}^n f(i)* (fi-1)/2 and n is odd, find min(T)

Value of |PY|/|QY| such that |PQ| is maximized

If r(x),s(x) are distinct elements of $x$ then 0&lt;x&lt;1

Find all possible values of angle A

(m^2+n)(m+n^2)=(m-n)^3

If r(x),s(x) are distinct elements of $x$ then 0<x<1