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

Difference between revisions of "1991 USAMO Problems"
m
 
Line 5: Line 5:
 
In triangle <math>\, ABC, \,</math> angle <math>\,A\,</math> is twice angle <math>\,B,\,</math> angle <math>\,C\,</math> is obtuse, and the three side lengths <math>\,a,b,c\,</math> are integers.  Determine, with proof, the minimum possible perimeter.
 
In triangle <math>\, ABC, \,</math> angle <math>\,A\,</math> is twice angle <math>\,B,\,</math> angle <math>\,C\,</math> is obtuse, and the three side lengths <math>\,a,b,c\,</math> are integers.  Determine, with proof, the minimum possible perimeter.
  
−
* [[1991 USAMO Problems/Problem 1 | Solution]]
+
[[1991 USAMO Problems/Problem 1 | Solution]]
  
 
== Problem 2 ==
 
== Problem 2 ==
Line 13: Line 13:
 
where "<math>\Sigma</math>" denotes a sum involving all nonempty subsets <math>S</math> of <math>\{1,2,3, \ldots,n\}</math>.
 
where "<math>\Sigma</math>" denotes a sum involving all nonempty subsets <math>S</math> of <math>\{1,2,3, \ldots,n\}</math>.
  
−
* [[1991 USAMO Problems/Problem 2 | Solution]]
+
[[1991 USAMO Problems/Problem 2 | Solution]]
  
 
== Problem 3 ==
 
== Problem 3 ==
Line 23: Line 23:
 
[The tower of exponents is defined by <math>a_1 = 2, \; a_{i+1} = 2^{a_i}</math>. Also <math>a_i \pmod{n}</math> means the remainder which results from dividing <math>\,a_i\,</math> by <math>\,n</math>.]
 
[The tower of exponents is defined by <math>a_1 = 2, \; a_{i+1} = 2^{a_i}</math>. Also <math>a_i \pmod{n}</math> means the remainder which results from dividing <math>\,a_i\,</math> by <math>\,n</math>.]
  
−
* [[1991 USAMO Problems/Problem 3 | Solution]]
+
[[1991 USAMO Problems/Problem 3 | Solution]]
  
 
== Problem 4 ==
 
== Problem 4 ==
Line 31: Line 31:
 
[You may wish to analyze the ratio <math>\,(a^N - N^N)/(a-N),</math> for real <math>\, a \geq 0 \,</math> and integer <math>\, N \geq 1</math>.]
 
[You may wish to analyze the ratio <math>\,(a^N - N^N)/(a-N),</math> for real <math>\, a \geq 0 \,</math> and integer <math>\, N \geq 1</math>.]
  
−
* [[1991 USAMO Problems/Problem 4 | Solution]]
+
[[1991 USAMO Problems/Problem 4 | Solution]]
  
 
== Problem 5 ==
 
== Problem 5 ==
Line 72: Line 72:
 
</center>
 
</center>
  
−
* [[1991 USAMO Problems/Problem 5 | Solution]]
+
[[1991 USAMO Problems/Problem 5 | Solution]]
  
−
== Resources ==
+
== See Also ==
  
 
{{USAMO box|year=1991|before=[[1990 USAMO]]|after=[[1992 USAMO]]}}
 
{{USAMO box|year=1991|before=[[1990 USAMO]]|after=[[1992 USAMO]]}}

Latest revision as of 18:17, 18 July 2016

Problems from the 1991 USAMO. There were five questions administered in one three-and-a-half-hour session.

Problem 1

In triangle $\, ABC, \,$ angle $\,A\,$ is twice angle $\,B,\,$ angle $\,C\,$ is obtuse, and the three side lengths $\,a,b,c\,$ are integers. Determine, with proof, the minimum possible perimeter.

Solution

Problem 2

For any nonempty set $\,S\,$ of numbers, let $\,\sigma(S)\,$ and $\,\pi(S)\,$ denote the sum and product, respectively, of the elements of $\,S\,$. Prove that \[\sum \frac{\sigma(S)}{\pi(S)} = (n^2 + 2n) - \left( 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} \right) (n+1),\] where "$\Sigma$" denotes a sum involving all nonempty subsets $S$ of $\{1,2,3, \ldots,n\}$.

Solution

Problem 3

Show that, for any fixed integer $\,n \geq 1,\,$ the sequence \[2, \; 2^2, \; 2^{2^2}, \; 2^{2^{2^2}}, \ldots \pmod{n}\] is eventually constant.

[The tower of exponents is defined by $a_1 = 2, \; a_{i+1} = 2^{a_i}$. Also $a_i \pmod{n}$ means the remainder which results from dividing $\,a_i\,$ by $\,n$.]

Solution

Problem 4

Let $\, a =(m^{m+1} + n^{n+1})/(m^m + n^n), \,$ where $\,m\,$ and $\,n\,$ are positive integers. Prove that $\,a^m + a^n \geq m^m + n^n$.

[You may wish to analyze the ratio $\,(a^N - N^N)/(a-N),$ for real $\, a \geq 0 \,$ and integer $\, N \geq 1$.]

Solution

Problem 5

Let $\, D \,$ be an arbitrary point on side $\, AB \,$ of a given triangle $\, ABC, \,$ and let $\, E \,$ be the interior point where $\, CD \,$ intersects the external common tangent to the incircles of triangles $\, ACD \,$ and $\, BCD$. As $\, D \,$ assumes all positions between $\, A \,$ and $\, B \,$, prove that the point $\, E \,$ traces the arc of a circle.

[asy] size(220); defaultpen(1); pair A=(0,0), B=(220,0), C=(18.7723,118.523); pair D=(72.6,0); pair Ia=incenter(A,D,C), Ib=incenter(B,D,C); pair Ta=(24.9758,52.5775),Tb=(86.6196,67.4129); pair E=IntersectionPoint((Ta--Tb),(C--D)); path Oa=circle(Ia,inradius(A,D,C)); path Ob=circle(Ib,inradius(B,D,C)); pair Da=IP(Oa,A--B), Db=IP(Ob,A--B); draw(D--C--A--B--C); draw(Ta--Tb); draw(Oa); draw(Ob); dot(A,linewidth(4)); dot(B,linewidth(4)); dot(C,linewidth(4)); dot(D,linewidth(4)); dot(E,linewidth(4)); dot(Ta,linewidth(4)); dot(Tb,linewidth(4)); label("\(A\)",A,SW); label("\(B\)",B,SE); label("\(C\)",C,W); label("\(D\)",D,S); label("\(E\)",E,NNE); [/asy]

Solution

See Also

1991 USAMO (Problems • Resources)
Preceded by
1990 USAMO 		Followed by
1992 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. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1991_USAMO_Problems&oldid=79463"