Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus
During AMC testing, the AoPS Wiki is in read-only mode and no edits can be made.

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)
 
(6 intermediate revisions by 5 users not shown)
Line 17: Line 17:
 
== Problem 3==
 
== Problem 3==
  
−
Consider functions <math>f : [0, 1] \rightarrow \Re</math> which satisfy
+
Consider functions <math>f : [0, 1] \rightarrow \mathbb{R}</math> which satisfy
 
<table><tr>
 
<table><tr>
 
<td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td><td>(i)</td><td><math>f(x)\ge0</math> for all <math>x</math> in <math>[0, 1]</math>,</td></tr>
 
<td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td><td>(i)</td><td><math>f(x)\ge0</math> for all <math>x</math> in <math>[0, 1]</math>,</td></tr>
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==
  
 
Let <math>a</math>, <math>b</math> be odd positive integers. Define the sequence <math>(f_n)</math> by putting <math>f_1 = a</math>,
 
Let <math>a</math>, <math>b</math> be odd positive integers. Define the sequence <math>(f_n)</math> by putting <math>f_1 = a</math>,
−
<math>f_2 = b</math>, and by letting fn for <math>n\ge3</math> be the greatest odd divisor of <math>f_{n-1} + f_{n-2}</math>.
+
<math>f_2 = b</math>, and by letting <math>f_n</math> for <math>n\ge3</math> be the greatest odd divisor of <math>f_{n-1} + f_{n-2}</math>.
 
Show that <math>f_n</math> is constant for <math>n</math> sufficiently large and determine the eventual
 
Show that <math>f_n</math> is constant for <math>n</math> sufficiently large and determine the eventual
 
value as a function of <math>a</math> and <math>b</math>.
 
value as a function of <math>a</math> and <math>b</math>.
Line 51: Line 50:
 
<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 ==
 +
{{USAMO box|year=1993|before=[[1992 USAMO]]|after=[[1994 USAMO]]}}
  
−
[[1993 USAMO Problems/Problem 5 | Solution]]
+
{{MAA Notice}}

Latest revision as of 07:25, 19 July 2016

Problem 1

For each integer $n\ge2$, determine, with proof, which of the two positive real numbers $a$ and $b$ satisfying

$a^n = a + 1, \quad b^{2n} = b + 3a$

is larger.

Solution

Problem 2

Let $ABCD$ be a convex quadrilateral such that diagonals $AC$ and $BD$ intersect at right angles, and let $E$ be their intersection. Prove that the reflections of $E$ across $AB$, $BC$, $CD$, $DA$ are concyclic.

Solution

Problem 3

Consider functions $f : [0, 1] \rightarrow \mathbb{R}$ which satisfy

     (i)$f(x)\ge0$ for all $x$ in $[0, 1]$,
     (ii)$f(1) = 1$,
     (iii)     $f(x) + f(y) \le f(x + y)$ whenever $x$, $y$, and $x + y$ are all in $[0, 1]$.

Find, with proof, the smallest constant $c$ such that

$f(x) \le cx$

for every function $f$ satisfying (i)-(iii) and every $x$ in $[0, 1]$.

Solution

Problem 4

Let $a$, $b$ be odd positive integers. Define the sequence $(f_n)$ by putting $f_1 = a$, $f_2 = b$, and by letting $f_n$ for $n\ge3$ be the greatest odd divisor of $f_{n-1} + f_{n-2}$. Show that $f_n$ is constant for $n$ sufficiently large and determine the eventual value as a function of $a$ and $b$.

Solution

Problem 5

Let $a_0, a_1, a_2,\cdots$ be a sequence of positive real numbers satisfying $a_{i-1}a_{i+1}\le a^2_i$ for $i = 1, 2, 3,\cdots$ . (Such a sequence is said to be log concave.) Show that for each $n > 1$,

$\frac{a_0+\cdots+a_n}{n+1}\cdot\frac{a_1+\cdots+a_{n-1}}{n-1}\ge\frac{a_0+\cdots+a_{n-1}}{n}\cdot\frac{a_1+\cdots+a_{n}}{n}$.

Solution

See Also

1993 USAMO (Problems • Resources)
Preceded by
1992 USAMO 		Followed by
1994 USAMO
1 • 2 • 3 • 4 • 5
All USAMO Problems and Solutions

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. AMC Logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1993_USAMO_Problems&oldid=79559"