Difference between revisions of "1972 USAMO Problems"
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[[1972 USAMO Problems/Problem 5 | Solution]] | [[1972 USAMO Problems/Problem 5 | Solution]] | ||
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Revision as of 13:02, 16 April 2012
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 |