1977 USAMO

Part of USAMO

Subcontests

(5)

1

Inequality in Interval [p,q]

a,b,c,d,e

are positive numbers bounded by

p

q

, i.e, if they lie in

[p,q], 0 < p

, prove that (a \plus{} b \plus{} c \plus{} d \plus{} e)\left(\frac {1}{a} \plus{} \frac {1}{b} \plus{} \frac {1}{c} \plus{} \frac {1}{d} \plus{} \frac {1}{e}\right) \le 25 \plus{} 6\left(\sqrt {\frac {p}{q}} \minus{} \sqrt {\frac {q}{p}}\right)^2 and determine when there is equality.

1

Congruent Opposite Sides of Skew Quadrilateral

Prove that if the opposite sides of a skew (non-planar) quadrilateral are congruent, then the line joining the midpoints of the two diagonals is perpendicular to these diagonals, and conversely, if the line joining the midpoints of the two diagonals of a skew quadrilateral is perpendicular to these diagonals, then the opposite sides of the quadrilateral are congruent.

1

Roots of Two Polynomials

a

b

are two of the roots of x^4\plus{}x^3\minus{}1\equal{}0, prove that

ab

is a root of x^6\plus{}x^4\plus{}x^3\minus{}x^2\minus{}1\equal{}0.

1

Weird Area Identity

ABC

A'B'C'

are two triangles in the same plane such that the lines

AA',BB',CC'

are mutually parallel. Let

[ABC]

denotes the area of triangle

ABC

with an appropriate

\pm

sign, etc.; prove that 3([ABC] \plus{} [A'B'C']) \equal{} [AB'C'] \plus{} [BC'A'] \plus{} [CA'B'] \plus{} [A'BC] \plus{} [B'CA] \plus{} [C'AB].

1

Polynomial Divisibility

Determine all pairs of positive integers

(m,n)

such that (1\plus{}x^n\plus{}x^{2n}\plus{}\cdots\plus{}x^{mn}) is divisible by (1\plus{}x\plus{}x^2\plus{}\cdots\plus{}x^{m}).