Difference between revisions of "1980 USAMO Problems"

(Problem 5)
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==Problem 1==
 
==Problem 1==
 
A balance has unequal arms and pans of unequal weight. It is used to weigh three objects. The
 
A balance has unequal arms and pans of unequal weight. It is used to weigh three objects. The
first object balances against a weight <math>A</math>, when placed in the left pan and against a weight a, when
+
first object balances against a weight <math>A</math>, when placed in the left pan and against a weight <math>a</math>, when
 
placed in the right pan. The corresponding weights for the second object are <math>B</math> and <math>b</math>. The third
 
placed in the right pan. The corresponding weights for the second object are <math>B</math> and <math>b</math>. The third
 
object balances against a weight <math>C</math>, when placed in the left pan. What is its true weight?
 
object balances against a weight <math>C</math>, when placed in the left pan. What is its true weight?

Revision as of 20:53, 21 March 2012

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?

Problem 2

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

Problem 3

$A + B + C$ is an integral multiple of $\pi$. $x, y,$ and $z$ 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 $x^n\sin(na)+y^n \sin(nb) +z^n \sin(nc)=0$ for any positive integer $n$.

Problem 4

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

Problem 5

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