Difference between revisions of "1991 USAMO Problems"
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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]] | |
== Problem 2 == | == Problem 2 == | ||
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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]] | |
== Problem 3 == | == Problem 3 == | ||
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[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]] | |
== Problem 4 == | == Problem 4 == | ||
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[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]] | |
== Problem 5 == | == Problem 5 == | ||
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</center> | </center> | ||
− | + | [[1991 USAMO Problems/Problem 5 | Solution]] | |
− | == | + | == 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 19:17, 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 angle
is twice angle
angle
is obtuse, and the three side lengths
are integers. Determine, with proof, the minimum possible perimeter.
Problem 2
For any nonempty set of numbers, let
and
denote the sum and product, respectively, of the elements of
. Prove that
where "
" denotes a sum involving all nonempty subsets
of
.
Problem 3
Show that, for any fixed integer the sequence
is eventually constant.
[The tower of exponents is defined by . Also
means the remainder which results from dividing
by
.]
Problem 4
Let where
and
are positive integers. Prove that
.
[You may wish to analyze the ratio for real
and integer
.]
Problem 5
Let be an arbitrary point on side
of a given triangle
and let
be the interior point where
intersects the external common tangent to the incircles of triangles
and
. As
assumes all positions between
and
, prove that the point
traces the arc of a circle.
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.