Difference between revisions of "1972 USAMO Problems"
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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> | ||
| − | [[1972 USAMO Problems/Problem 5 | Solution]] | + | [[1972 USAMO Problems/Problem 5|Solution]] |
== See Also == | == See Also == | ||
{{USAMO box|year=1972|before=First USAMO|after=[[1973 USAMO]]}} | {{USAMO box|year=1972|before=First USAMO|after=[[1973 USAMO]]}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 20:55, 31 March 2014
Problems from the 1972 USAMO.
Problem 1
The symbols 
and ![$[a,b,\ldots, g]$](http://latex.artofproblemsolving.com/a/4/3/a435cfa873528fda8011ab1fee0f179f1233b4d5.png)
denote the greatest common divisor and least common multiple, respectively, of the positive integers 
. For example, 
and ![$[6,15]=30$](http://latex.artofproblemsolving.com/c/e/6/ce61f21da055164e6c4aa95b4ac1c682bb8075a7.png)
. 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.
![[asy] size(80); defaultpen(fontsize(7)); pair A=(0,7), B=(5,4), C=(3,0), D=(-3,0), E=(-5,4), P; P=extension(B,D,C,E); draw(D--E--A--B--C--D--B--E--C); label("A",A,(0,1));label("B",B,(1,0));label("C",C,(1,-1));label("D",D,(-1,-1));label("E",E,(-1,0));label("P",P,(0,1)); [/asy]](http://latex.artofproblemsolving.com/3/2/3/323bdb5c3f41250e3db1af79ed1bd258fad55799.png)
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. 
