Difference between revisions of "1980 USAMO Problems"
m (→Problem 1) |
(→Problem 3) |
||
| (5 intermediate revisions by 4 users not shown) | |||
| Line 1: | Line 1: | ||
| + | Problems from the '''1980 [[United States of America Mathematical Olympiad | USAMO]]'''. | ||
| + | |||
==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 <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 object balances against a weight <math>C</math>, when placed in the left pan. What is its true weight? |
| − | 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 | + | [[1980 USAMO Problems/Problem 1 | Solution]] |
| − | object balances against a weight <math>C</math>, when placed in the left pan. What is its true weight? | ||
==Problem 2== | ==Problem 2== | ||
Find the maximum possible number of three term arithmetic progressions in a monotone sequence of <math>n</math> distinct reals. | Find the maximum possible number of three term arithmetic progressions in a monotone sequence of <math>n</math> distinct reals. | ||
| + | |||
| + | [[1980 USAMO Problems/Problem 2 | Solution]] | ||
==Problem 3== | ==Problem 3== | ||
| − | <math>A + B + C</math> is an integral multiple of <math>\pi</math>. <math>x, y, </math> and <math>z</math> are real numbers. If <math>x\sin(A) | + | <math>A + B + C</math> is an integral multiple of <math>\pi</math>. <math>x, y, </math> and <math>z</math> are real numbers. If <math>x\sin(A)+y\sin(B)+z\sin(C)=x^2\sin(2A)+y^2\sin(2B)+z^2\sin(2C)=0</math>, show that <math>x^n\sin(nA)+y^n \sin(nB) +z^n \sin(nC)=0</math> for any positive integer <math>n</math>. |
| + | |||
| + | [[1980 USAMO Problems/Problem 3 | Solution]] | ||
==Problem 4== | ==Problem 4== | ||
The insphere of a tetrahedron touches each face at its centroid. Show that the tetrahedron is regular. | The insphere of a tetrahedron touches each face at its centroid. Show that the tetrahedron is regular. | ||
| + | |||
| + | [[1980 USAMO Problems/Problem 4 | Solution]] | ||
==Problem 5== | ==Problem 5== | ||
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 + | 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)</math> | 1} \le 1 - (1 - x)(1 - y)(1 - z)</math> | ||
| + | |||
| + | [[1980 USAMO Problems/Problem 5 | Solution]] | ||
| + | |||
| + | == See Also == | ||
| + | {{USAMO box|year=1980|before=[[1979 USAMO]]|after=[[1981 USAMO]]}} | ||
| + | {{MAA Notice}} | ||
Latest revision as of 13:41, 26 December 2015
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 
, 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 
, 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 
See Also
| 1980 USAMO (Problems • Resources) | ||
| Preceded by 1979 USAMO |
Followed by 1981 USAMO | |
| 1 • 2 • 3 • 4 • 5 | ||
| All USAMO Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
