Difference between revisions of "1993 USAMO Problems"
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== Problem 3== | == Problem 3== | ||
− | Consider functions <math>f : [0, 1] \rightarrow \ | + | Consider functions <math>f : [0, 1] \rightarrow \mathbb{R}</math> which satisfy |
<table><tr> | <table><tr> | ||
<td> </td><td>(i)</td><td><math>f(x)\ge0</math> for all <math>x</math> in <math>[0, 1]</math>,</td></tr> | <td> </td><td>(i)</td><td><math>f(x)\ge0</math> for all <math>x</math> in <math>[0, 1]</math>,</td></tr> |
Latest revision as of 07:25, 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.