Difference between revisions of "1980 USAMO Problems"
(Created page with "==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 <math>A</math>, when placed in ...") |
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==Problem 5== | ==Problem 5== | ||
| − | If <math>x, y, z</math> are reals such that <math>0\le x, y, z \le 1, show that < | + | If <math>x, y, z</math> are reals such that <math>0\le x, y, z \le 1</math>, show that <math>\frac{x}{y + z + 1} + \frac{y}{z + x + 1} + \frac{z}{x + y + |
| − | 1} \le 1 - (1 - x)(1 - y)(1 - z) | + | 1} \le 1 - (1 - x)(1 - y)(1 - z)</math> |
Revision as of 20:59, 19 August 2011
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 a, 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 
