Difference between revisions of "1991 USAMO Problems"

Revision as of 19:50, 3 July 2013

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

Resources

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

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 * [http://www.unl.edu/amc/a-activities/a7-problems/USAMO-IMO/q-usamo/-tex/usamo1991.tex 1991 USAMO Problems (TEX)]
 * [http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=27&year=1991 1991 USAMO Problems on the resources page]
+{{MAA Notice}}

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