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Difference between revisions of "1980 USAMO Problems"

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==Problem 3==
 
==Problem 3==
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<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>.
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<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)+y\sin(B)+z\sin(C)=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]]
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== See Also ==
 
== See Also ==
 
{{USAMO box|year=1980|before=[[1979 USAMO]]|after=[[1981 USAMO]]}}
 
{{USAMO box|year=1980|before=[[1979 USAMO]]|after=[[1981 USAMO]]}}
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{{MAA Notice}}

Latest revision as of 12:41, 26 December 2015

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 $A$, when placed in the left pan and against a weight $a$, when placed in the right pan. The corresponding weights for the second object are $B$ and $b$. The third object balances against a weight $C$, when placed in the left pan. What is its true weight?

Solution

Problem 2

Find the maximum possible number of three term arithmetic progressions in a monotone sequence of $n$ distinct reals.

Solution

Problem 3

$A + B + C$ is an integral multiple of $\pi$. $x, y,$ and $z$ are real numbers. If $x\sin(A)+y\sin(B)+z\sin(C)=x^2\sin(2A)+y^2\sin(2B)+z^2\sin(2C)=0$, show that $x^n\sin(nA)+y^n \sin(nB) +z^n \sin(nC)=0$ for any positive integer $n$.

Solution

Problem 4

The insphere of a tetrahedron touches each face at its centroid. Show that the tetrahedron is regular.

Solution

Problem 5

If $x, y, z$ are reals such that $0\le x, y, z \le 1$, show that $\frac{x}{y + z + 1} + \frac{y}{z + x + 1} + \frac{z}{x + y + 1} \le 1 - (1 - x)(1 - y)(1 - z)$

Solution

See Also

1980 USAMO (Problems • Resources)
Preceded by
1979 USAMO 		Followed by
1981 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. AMC logo.png

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