Difference between revisions of "1993 USAMO Problems"
(Created page with '== Problem 1== For each integer <math>n\ge2</math>, determine, with proof, which of the two positive real numbers <math>a</math> and <math>b</math> satisfying <center><math>a^n…') |
m (→Problem 3) |
||
Line 27: | Line 27: | ||
Find, with proof, the smallest constant <math>c</math> such that | Find, with proof, the smallest constant <math>c</math> such that | ||
− | <math>f(x) \le cx</math> | + | <center><math>f(x) \le cx</math></center> |
for every function <math>f</math> satisfying (i)-(iii) and every <math>x</math> in <math>[0, 1]</math>. | for every function <math>f</math> satisfying (i)-(iii) and every <math>x</math> in <math>[0, 1]</math>. | ||
[[1993 USAMO Problems/Problem 3 | Solution]] | [[1993 USAMO Problems/Problem 3 | Solution]] | ||
− | |||
== Problem 4== | == Problem 4== |
Revision as of 18:59, 22 April 2010
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 fn 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)