1978 USAMO

Part of USAMO

Subcontests

(5)

1

1978 USAMO #5

Nine mathematicians meet at an international conference and discover that among any three of them, at least two speak a common language. If each of the mathematicians speak at most three languages, prove that there are at least three of the mathematicians who can speak the same language.

1

1978 USAMO #4

(a) Prove that if the six dihedral (i.e. angles between pairs of faces) of a given tetrahedron are congruent, then the tetrahedron is regular.
(b) Is a tetrahedron necessarily regular if five dihedral angles are congruent?

1

1978 USAMO #3

n

will be called good if we can write

n=a_1+a_2+\cdots+a_k,

a_1,a_2, \ldots, a_k

are positive integers (not necessarily distinct) satisfying

\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=1.

Given the information that the integers 33 through 73 are good, prove that every integer

\ge 33

1

1978 USAMO #2

ABCD

A'B'C'D'

are square maps of the same region, drawn to different scales and superimposed as shown in the figure. Prove that there is only one point

O

on the small map that lies directly over point

O'

of the large map such that

O

O'

each represent the same place of the country. Also, give a Euclidean construction (straight edge and compass) for

O

.
[asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); real theta = -100, r = 0.3; pair D2 = (0.3,0.76); string[] lbl = {'A', 'B', 'C', 'D'}; draw(unitsquare); draw(shift(D2)*rotate(theta)*scale(r)*unitsquare); for(int i = 0; i < lbl.length; ++i) { pair Q = dir(135-90*i), P = (.5,.5)+Q/2^.5; label("

"+lbl+"'

", P, Q); label("

"+lbl+"

",D2+rotate(theta)*(r*P), rotate(theta)*Q); }[/asy]

1

1978 USAMO #1

a,b,c,d,e

are real numbers such that

a+b+c+d+e=8

a^2+b^2+c^2+d^2+e^2=16

.
Determine the maximum value of

e