1957 AHSME Problems/Problem 34
Problem
The points that satisfy the system 
, constitute the following set:
Solution
![[asy] import geometry; // Axes draw((0,-8)--(0,8),arrow=Arrows); label("$y$",(0,9)); draw((-8,0)--(8,0),arrow=Arrows); label("$x$",(9,0)); // Circle and Line draw(circle((0,0),5),red+dashed); draw(line((0,1),(1,0)),blue); [/asy]](http://latex.artofproblemsolving.com/b/8/d/b8d0c917ef633c50269b9ae3fab786f05a474f55.png)
The line 
is in standard form. Thus, it has a slope of 
and a 
-intercept at 
. It is graphed in blue in the diagram. All the points that satisfy this system must lie on this line.
The inequality 
is true for all points strictly less than distance 
from the origin. This area is a disk with radius 
, whose open boundary is represented in red in the diagram.
Because the -intercept of the line, , is inside the disk, it cannot be tangent to the disk. Thus, there must be infiniely many points along the line which are within the disk and thereby satisfy the system. However, because the inequality is strict, the segment of solutions does not include the endpoints, so our answer is 
.
See Also
| 1957 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 33 |
Followed by Problem 35 | |
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