1993 USAMO Problems
Problem 1
For each integer , determine, with proof, which of the two positive real numbers
and
satisfying

is larger.
Problem 2
Let be a convex quadrilateral such that diagonals
and
intersect at right angles, and let
be their intersection. Prove that the reflections of
across
,
,
,
are concyclic.
Problem 3
Consider functions which satisfy
(i) | ![]() ![]() ![]() | |
(ii) | ![]() | |
(iii) | ![]() ![]() ![]() ![]() ![]() |
Find, with proof, the smallest constant such that

for every function satisfying (i)-(iii) and every
in
.
Problem 4
Let ,
be odd positive integers. Define the sequence
by putting
,
, and by letting
for
be the greatest odd divisor of
.
Show that
is constant for
sufficiently large and determine the eventual
value as a function of
and
.
Problem 5
Let be a sequence of positive real numbers satisfying
for
. (Such a sequence is said to be log concave.) Show that for
each
,

See Also
1993 USAMO (Problems • Resources) | ||
Preceded by 1992 USAMO |
Followed by 1994 USAMO | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.