Difference between revisions of "1982 USAMO Problems"
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+ | Problems from the '''1982 [[United States of America Mathematical Olympiad | USAMO]]'''. | ||
+ | |||
==Problem 1== | ==Problem 1== | ||
− | + | In a party with <math>1982</math> persons, among any group of four there is at least one person who knows each of the other three. What is the minimum number of people in the party who know everyone else? | |
[[1982 USAMO Problems/Problem 1 | Solution]] | [[1982 USAMO Problems/Problem 1 | Solution]] | ||
==Problem 2== | ==Problem 2== | ||
− | + | Let <math>S_r=x^r+y^r+z^r</math> with <math>x,y,z</math> real. It is known that if <math>S_1=0</math>, | |
+ | |||
+ | <math>(*) </math> <math>\frac{S_{m+n}}{m+n}=\frac{S_m}{m}\frac{S_n}{n}</math> | ||
+ | |||
+ | for <math>(m,n)=(2,3),(3,2),(2,5)</math>, or <math>(5,2)</math>. Determine ''all'' other pairs of integers <math>(m,n)</math> if any, so that <math>(*)</math> holds for all real numbers <math>x,y,z</math> such that <math>S_1=0</math>. | ||
[[1982 USAMO Problems/Problem 2 | Solution]] | [[1982 USAMO Problems/Problem 2 | Solution]] | ||
==Problem 3== | ==Problem 3== | ||
− | <math> | + | If a point <math>A_1</math> is in the interior of an equilateral triangle <math>ABC</math> and point <math>A_2</math> is in the interior of <math>\triangle{A_1BC}</math>, prove that |
+ | |||
+ | <math>I.Q. (A_1BC) > I.Q.(A_2BC)</math>, | ||
+ | |||
+ | where the ''isoperimetric quotient'' of a figure <math>F</math> is defined by | ||
+ | |||
+ | <math>I.Q.(F) = \frac{\text{Area (F)}}{\text{[Perimeter (F)]}^2}</math> | ||
[[1982 USAMO Problems/Problem 3 | Solution]] | [[1982 USAMO Problems/Problem 3 | Solution]] | ||
==Problem 4== | ==Problem 4== | ||
− | + | Prove that there exists a positive integer <math>k</math> such that <math>k\cdot2^n+1</math> is composite for every positive integer <math>n</math>. | |
[[1982 USAMO Problems/Problem 4 | Solution]] | [[1982 USAMO Problems/Problem 4 | Solution]] | ||
==Problem 5== | ==Problem 5== | ||
− | <math> | + | <math>A,B</math>, and <math>C</math> are three interior points of a sphere <math>S</math> such that <math>AB</math> and <math>AC</math> are perpendicular to the diameter of <math>S</math> through <math>A</math>, and so that two spheres can be constructed through <math>A</math>, <math>B</math>, and <math>C</math> which are both tangent to <math>S</math>. Prove that the sum of their radii is equal to the radius of <math>S</math>. |
[[1982 USAMO Problems/Problem 5 | Solution]] | [[1982 USAMO Problems/Problem 5 | Solution]] | ||
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== See Also == | == See Also == | ||
{{USAMO box|year=1982|before=[[1981 USAMO]]|after=[[1983 USAMO]]}} | {{USAMO box|year=1982|before=[[1981 USAMO]]|after=[[1983 USAMO]]}} | ||
+ | {{MAA Notice}} |
Latest revision as of 22:36, 19 March 2020
Problems from the 1982 USAMO.
Problem 1
In a party with persons, among any group of four there is at least one person who knows each of the other three. What is the minimum number of people in the party who know everyone else?
Problem 2
Let with
real. It is known that if
,
for , or
. Determine all other pairs of integers
if any, so that
holds for all real numbers
such that
.
Problem 3
If a point is in the interior of an equilateral triangle
and point
is in the interior of
, prove that
,
where the isoperimetric quotient of a figure is defined by
Problem 4
Prove that there exists a positive integer such that
is composite for every positive integer
.
Problem 5
, and
are three interior points of a sphere
such that
and
are perpendicular to the diameter of
through
, and so that two spheres can be constructed through
,
, and
which are both tangent to
. Prove that the sum of their radii is equal to the radius of
.
See Also
1982 USAMO (Problems • Resources) | ||
Preceded by 1981 USAMO |
Followed by 1983 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.