Difference between revisions of "1972 USAMO Problems"
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− | 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> | ||
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.