1984 USAMO

Part of USAMO

Subcontests

(5)

1

USAMO 1984 Problem 5 - Polynomial of degree 3n

P(x)

is a polynomial of degree

3n

such that
\begin{eqnarray*} P(0) = P(3) = \cdots &=& P(3n) = 2, \\ P(1) = P(4) = \cdots &=& P(3n-2) = 1, \\ P(2) = P(5) = \cdots &=& P(3n-1) = 0, \text{ and }\\ && P(3n+1) = 730.\end{eqnarray*}
Determine

n

1

USAMO 1984 Problem 4 - Tests taken per person

A difficult mathematical competition consisted of a Part I and a Part II with a combined total of

28

problems. Each contestant solved

7

problems altogether. For each pair of problems, there were exactly two contestants who solved both of them. Prove that there was a contestant who, in Part I, solved either no problems or at least four problems.

1

USAMO 1984 Problem 3 - Optimize <APC+<BPD

P, A, B, C,

D

are five distinct points in space such that

\angle APB = \angle BPC = \angle CPD = \angle DPA = \theta

\theta

is a given acute angle. Determine the greatest and least values of

\angle APC + \angle BPD

1

USAMO 1984 Problem 2 - Geometric Mean

The geometric mean of any set of

m

non-negative numbers is the

m

-th root of their product.
(\text{i}) For which positive integers

n

is there a finite set

S_n

n

distinct positive integers such that the geometric mean of any subset of

S_n

is an integer? (\text{ii}) Is there an infinite set

S

of distinct positive integers such that the geometric mean of any finite subset of

S

1

Polynomial Roots

The product of two of the four roots of the quartic equation

x^4 - 18x^3 + kx^2+200x-1984=0

-32

. Determine the value of

k