Difference between revisions of "1980 USAMO Problems"

Revision as of 11:55, 16 May 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 $\frac{x}{y + z + 1} + \frac{y}{z + x + 1} + \frac{z}{x + y + 1} \le 1 - (1 - x)(1 - y)(1 - z)$

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