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> | ||
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<center><math>\frac{a_0+\cdots+a_n}{n+1}\cdot\frac{a_1+\cdots+a_{n-1}}{n-1}\ge\frac{a_0+\cdots+a_{n-1}}{n}\cdot\frac{a_1+\cdots+a_{n}}{n}</math>.</center> | <center><math>\frac{a_0+\cdots+a_n}{n+1}\cdot\frac{a_1+\cdots+a_{n-1}}{n-1}\ge\frac{a_0+\cdots+a_{n-1}}{n}\cdot\frac{a_1+\cdots+a_{n}}{n}</math>.</center> | ||
+ | |||
+ | [[1993 USAMO Problems/Problem 5 | Solution]] | ||
== See Also == | == See Also == | ||
{{USAMO box|year=1993|before=[[1992 USAMO]]|after=[[1994 USAMO]]}} | {{USAMO box|year=1993|before=[[1992 USAMO]]|after=[[1994 USAMO]]}} | ||
− | + | {{MAA Notice}} |
Latest revision as of 08:25, 19 July 2016
Problem 1
For each integer , determine, with proof, which of the two positive real numbers
and
satisfying
![$a^n = a + 1, \quad b^{2n} = b + 3a$](http://latex.artofproblemsolving.com/5/c/8/5c8cac81e3d9c2751d3306d933e8d4212fa64c08.png)
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
![$f(x) \le cx$](http://latex.artofproblemsolving.com/b/1/e/b1e93ec7efefb0af408a87d3d13e41a9e5bd902e.png)
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
,
![$\frac{a_0+\cdots+a_n}{n+1}\cdot\frac{a_1+\cdots+a_{n-1}}{n-1}\ge\frac{a_0+\cdots+a_{n-1}}{n}\cdot\frac{a_1+\cdots+a_{n}}{n}$](http://latex.artofproblemsolving.com/d/a/2/da21136d8f04ee35b3510f5fb8653b483dc8131b.png)
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.