Difference between revisions of "1972 USAMO Problems"
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− | Problems from the '''1972 [[United States of America Mathematical Olympiad | USAMO]]'''. | + | Problems from the '''1972 [[United States of America Mathematical Olympiad|USAMO]]'''. |
==Problem 1== | ==Problem 1== | ||
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<center><math>\frac{[a,b,c]^2}{[a,b][b,c][c,a]}=\frac{(a,b,c)^2}{(a,b)(b,c)(c,a)}</math>.</center> | <center><math>\frac{[a,b,c]^2}{[a,b][b,c][c,a]}=\frac{(a,b,c)^2}{(a,b)(b,c)(c,a)}</math>.</center> | ||
− | [[1972 USAMO Problems/Problem 1 | Solution]] | + | [[1972 USAMO Problems/Problem 1|Solution]] |
==Problem 2== | ==Problem 2== | ||
A given tetrahedron <math>ABCD</math> is isosceles, that is, <math>AB=CD, AC=BD, AD=BC</math>. Show that the faces of the tetrahedron are acute-angled triangles. | A given tetrahedron <math>ABCD</math> is isosceles, that is, <math>AB=CD, AC=BD, AD=BC</math>. Show that the faces of the tetrahedron are acute-angled triangles. | ||
− | [[1972 USAMO Problems/Problem 2 | Solution]] | + | [[1972 USAMO Problems/Problem 2|Solution]] |
==Problem 3== | ==Problem 3== | ||
A random number selector can only select one of the nine integers 1, 2, ..., 9, and it makes these selections with equal probability. Determine the probability that after <math>n</math> selections (<math>n>1</math>), the product of the <math>n</math> numbers selected will be divisible by 10. | A random number selector can only select one of the nine integers 1, 2, ..., 9, and it makes these selections with equal probability. Determine the probability that after <math>n</math> selections (<math>n>1</math>), the product of the <math>n</math> numbers selected will be divisible by 10. | ||
− | [[1972 USAMO Problems/Problem 3 | Solution]] | + | [[1972 USAMO Problems/Problem 3|Solution]] |
==Problem 4== | ==Problem 4== | ||
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<center><math>\left|\frac{aR^2+bR+c}{dR^2+eR+f}-\sqrt[3]{2}\right|<|R-\sqrt[3]{2}|</math></center> | <center><math>\left|\frac{aR^2+bR+c}{dR^2+eR+f}-\sqrt[3]{2}\right|<|R-\sqrt[3]{2}|</math></center> | ||
− | [[1972 USAMO Problems/Problem 4 | Solution]] | + | [[1972 USAMO Problems/Problem 4|Solution]] |
==Problem 5== | ==Problem 5== | ||
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<asy> | <asy> | ||
− | size( | + | size(120); |
− | defaultpen(fontsize( | + | defaultpen(fontsize(10)); |
− | pair A=( | + | pair A=dir(90), B=dir(90-72), C=dir(90-2*72), D=dir(90-3*72), E=dir(90-4*72); |
− | + | draw(A--B--C--D--E--cycle); | |
− | draw( | + | draw(A--C--E--B--D--cycle); |
− | label("A",A, | + | label("A",A,A);label("B",B,B);label("C",C,C);label("D",D,D);label("E",E,E); |
</asy> | </asy> | ||
− | [[1972 USAMO Problems/Problem 5 | Solution | + | [[1972 USAMO Problems/Problem 5|Solution]] |
− | |||
− | |||
− | |||
+ | == See Also == | ||
{{USAMO box|year=1972|before=First USAMO|after=[[1973 USAMO]]}} | {{USAMO box|year=1972|before=First USAMO|after=[[1973 USAMO]]}} | ||
+ | {{MAA Notice}} |
Latest revision as of 15:28, 2 June 2018
Problems from the 1972 USAMO.
Problem 1
The symbols and
denote the greatest common divisor and least common multiple, respectively, of the positive integers
. For example,
and
. Prove that
![$\frac{[a,b,c]^2}{[a,b][b,c][c,a]}=\frac{(a,b,c)^2}{(a,b)(b,c)(c,a)}$](http://latex.artofproblemsolving.com/a/3/7/a3797d87dfc7855e74fd6a01e4f48819256cecb4.png)
Problem 2
A given tetrahedron is isosceles, that is,
. Show that the faces of the tetrahedron are acute-angled triangles.
Problem 3
A random number selector can only select one of the nine integers 1, 2, ..., 9, and it makes these selections with equal probability. Determine the probability that after selections (
), the product of the
numbers selected will be divisible by 10.
Problem 4
Let denote a non-negative rational number. Determine a fixed set of integers
, such that for every choice of
,
![$\left|\frac{aR^2+bR+c}{dR^2+eR+f}-\sqrt[3]{2}\right|<|R-\sqrt[3]{2}|$](http://latex.artofproblemsolving.com/e/4/8/e48c707907f83429e12a32d297e18dcbe4621168.png)
Problem 5
A given convex pentagon has the property that the area of each of the five triangles
,
,
,
, and
is unity. Show that all pentagons with the above property have the same area, and calculate that area. Show, furthermore, that there are infinitely many non-congruent pentagons having the above area property.
See Also
1972 USAMO (Problems • Resources) | ||
Preceded by First USAMO |
Followed by 1973 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.