Difference between revisions of "1980 USAMO Problems"
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==Problem 3== | ==Problem 3== | ||
| − | <math>A + B + C</math> is an integral multiple of <math>\pi</math>. <math>x, y, </math> and <math>z</math> are real numbers. If <math>x\sin(A)\plus{}y\sin(B)\plus{}z\sin(C)\equal{}x^2\sin(2A)+y^2\sin(2B)+z^2\sin(2C)=0</math>, show that <math>x^n\sin( | + | <math>A + B + C</math> is an integral multiple of <math>\pi</math>. <math>x, y, </math> and <math>z</math> are real numbers. If <math>x\sin(A)\plus{}y\sin(B)\plus{}z\sin(C)\equal{}x^2\sin(2A)+y^2\sin(2B)+z^2\sin(2C)=0</math>, show that <math>x^n\sin(nA)+y^n \sin(nB) +z^n \sin(nC)=0</math> for any positive integer <math>n</math>. |
[[1980 USAMO Problems/Problem 3 | Solution]] | [[1980 USAMO Problems/Problem 3 | Solution]] | ||
Revision as of 19:22, 26 April 2013
Problems from the 1980 USAMO.
Problem 1
A balance has unequal arms and pans of unequal weight. It is used to weigh three objects. The first object balances against a weight 
, when placed in the left pan and against a weight 
, when placed in the right pan. The corresponding weights for the second object are 
and 
. The third object balances against a weight 
, when placed in the left pan. What is its true weight?
Problem 2
Find the maximum possible number of three term arithmetic progressions in a monotone sequence of 
distinct reals.
Problem 3
is an integral multiple of 
. 
and 
are real numbers. If $x\sin(A)\plus{}y\sin(B)\plus{}z\sin(C)\equal{}x^2\sin(2A)+y^2\sin(2B)+z^2\sin(2C)=0$ (Error compiling LaTeX. Unknown error_msg), show that 
for any positive integer .
Problem 4
The insphere of a tetrahedron touches each face at its centroid. Show that the tetrahedron is regular.
Problem 5
If 
are reals such that 
, show that 
See Also
| 1980 USAMO (Problems • Resources) | ||
| Preceded by 1979 USAMO |
Followed by 1981 USAMO | |
| 1 • 2 • 3 • 4 • 5 | ||
| All USAMO Problems and Solutions | ||
