1993 USAMO Problems
For each integer , determine, with proof, which of the two positive real numbers and satisfying
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.
Consider functions which satisfy
|(i)||for all in ,|
|(iii)||whenever , , and are all in .|
Find, with proof, the smallest constant such that
for every function satisfying (i)-(iii) and every in .
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 .
Let be a sequence of positive real numbers satisfying for . (Such a sequence is said to be log concave.) Show that for each ,
|1993 USAMO (Problems • Resources)|
|1 • 2 • 3 • 4 • 5|
|All USAMO Problems and Solutions|