Difference between revisions of "1993 USAMO Problems"
m (→See Also) |
|||
Line 55: | Line 55: | ||
{{USAMO box|year=1993|before=[[1992 USAMO]]|after=[[1994 USAMO]]}} | {{USAMO box|year=1993|before=[[1992 USAMO]]|after=[[1994 USAMO]]}} | ||
− | |||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 07:54, 19 July 2016
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) | for all in , | |
(ii) | , | |
(iii) | whenever , , and are all in . |
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.