1980 USAMO Problems

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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?


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 $x\sin(A)+y\sin(B)+z\sin(C)=x^2\sin(2A)+y^2\sin(2B)+z^2\sin(2C)=0$, show that $x^n\sin(nA)+y^n \sin(nB) +z^n \sin(nC)=0$ for any positive integer $n$.


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 $\frac{x}{y + z + 1} + \frac{y}{z + x + 1} + \frac{z}{x + y +  1} \le 1 - (1 - x)(1 - y)(1 - z)$


See Also

1980 USAMO (ProblemsResources)
Preceded by
1979 USAMO
Followed by
1981 USAMO
1 2 3 4 5
All USAMO Problems and Solutions

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