Difference between revisions of "1973 USAMO Problems"
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[[1973 USAMO Problems/Problem 5 | Solution]] | [[1973 USAMO Problems/Problem 5 | Solution]] | ||
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*[[USAMO Problems and Solutions]] | *[[USAMO Problems and Solutions]] | ||
{{USAMO box|year=1973|before=[[1972 USAMO]]|after=[[1974 USAMO]]}} | {{USAMO box|year=1973|before=[[1972 USAMO]]|after=[[1974 USAMO]]}} |
Revision as of 08:26, 16 September 2012
Problems from the 1973 USAMO.
Problem 1
Two points and
lie in the interior of a regular tetrahedron
. Prove that angle
.
Problem 2
Let and
denote two sequences of integers defined as follows:
![$X_0=1$](http://latex.artofproblemsolving.com/c/e/b/cebce4cef53ec8891f6b19b9e15609f8f67e3ce4.png)
![$X_1=1$](http://latex.artofproblemsolving.com/4/c/9/4c98f37726f1b237003a06d40752524a28e99f47.png)
![$X_{n+1}=X_n+2X_{n-1}$](http://latex.artofproblemsolving.com/7/9/f/79f9ae578208dd5fd22233a74eb9958b7fa123db.png)
![$(n=1,2,3,\dots),$](http://latex.artofproblemsolving.com/a/2/7/a27887235fc6fe223896299960f50abedf4113f4.png)
![$Y_0=1$](http://latex.artofproblemsolving.com/7/9/7/79793a5127f0c02da1cd1496fc8a96c10a26232a.png)
![$Y_1=7$](http://latex.artofproblemsolving.com/e/0/e/e0e33be65971d1a5f4e92726ac4f5928d8c6ab59.png)
![$Y_{n+1}=2Y_n+3Y_{n-1}$](http://latex.artofproblemsolving.com/0/e/7/0e713f61302bc1d06796c423e76fcc2e456b6df5.png)
![$(n=1,2,3,\dots)$](http://latex.artofproblemsolving.com/0/3/8/0382bde36fe67f5296e632643ceb5d774545d3b6.png)
Thus, the first few terms of the sequences are:
![$X:1, 1, 3, 5, 11, 21, \dots$](http://latex.artofproblemsolving.com/4/2/3/423b49eb7fcf22e80c66be8b01f2cbdd2d9c34fe.png)
![$Y:1, 7, 17, 55, 161, 487, \dots$](http://latex.artofproblemsolving.com/8/9/3/89343c5c6c83f02fb1b692cc7d7d845c7d56e3fe.png)
Prove that, except for the "1", there is no term which occurs in both sequences.
Problem 3
Three distinct vertices are chosen at random from the vertices of a given regular polygon of sides. If all such choices are equally likely, what is the probability that the center of the given polygon lies in the interior of the triangle determined by the three chosen random points?
Problem 4
Determine all the roots, real or complex, of the system of simultaneous equations
![$x+y+z=3$](http://latex.artofproblemsolving.com/a/d/5/ad5ff13ebc979a03c144ba96c88e6281c5556894.png)
,
![$x^3+y^3+z^3=3$](http://latex.artofproblemsolving.com/e/4/7/e47fe3685e8d2812abd631eb34e54bcffd0d2643.png)
Problem 5
Show that the cube roots of three distinct prime numbers cannot be three terms (not necessarily consecutive) of an arithmetic progression.
See Also
1973 USAMO (Problems • Resources) | ||
Preceded by 1972 USAMO |
Followed by 1974 USAMO | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |