1988 USAMO

Part of USAMO

Subcontests

(5)

1

1988 USAMO Problem 5

A polynomial product of the form

(1-z)^{b_1}(1-z^2)^{b_2}(1-z^3)^{b_3}(1-z^4)^{b_4}(1-z^5)^{b_5}\cdots(1-z^{32})^{b_{32}},

b_k

are positive integers, has the surprising property that if we multiply it out and discard all terms involving

z

to a power larger than

32

, what is left is just

1-2z

. Determine, with proof,

b_{32}

1

1988 USAMO Problem 4

I

be the incenter of triangle

ABC

A'

B'

C'

be the circumcenters of triangles

IBC

ICA

IAB

, respectively. Prove that the circumcircles of triangles

ABC

A'B'C'

are concentric.

1

1988 USAMO Problem 3

f(S)

assigns to each nine-element subset of

S

\{1,2,\ldots, 20\}

a whole number from

1

20

. Prove that regardless of how the function

f

is chosen, there will be a ten-element subset

T\subset\{1,2,\ldots, 20\}

f(T - \{k\})\neq k

k\in T

1

1988 USAMO Problem 2

The cubic equation

x^3 + ax^2 + bx + c = 0

has three real roots. Show that

a^2-3b\geq 0

\sqrt{a^2-3b}

is less than or equal to the difference between the largest and smallest roots.

1

1988 USAMO Problem 1

By a pure repeating decimal (in base

10

), we mean a decimal

0.\overline{a_1\cdots a_k}

which repeats in blocks of

k

digits beginning at the decimal point. An example is

.243243243\cdots = \tfrac{9}{37}

. By a mixed repeating decimal we mean a decimal

0.b_1\cdots b_m\overline{a_1\cdots a_k}

which eventually repeats, but which cannot be reduced to a pure repeating decimal. An example is

.011363636\cdots = \tfrac{1}{88}

.
Prove that if a mixed repeating decimal is written as a fraction

\tfrac pq

in lowest terms, then the denominator

q

is divisible by

2

5