Difference between revisions of "1980 USAMO Problems"
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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 19:53, 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 
, when placed in the left pan and against a weight 
, 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 
