Difference between revisions of "1998 AHSME Problems/Problem 29"
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== Solution 1 == | == Solution 1 == | ||
| − | + | This is the best square: | |
| + | [asy] | ||
| + | real e = 0.1; | ||
| + | dot((0,-1)); | ||
| + | dot((1,-1)); | ||
| + | dot((-1,0)); | ||
| + | dot((0,0)); | ||
| + | dot((1,0)); | ||
| + | dot((2,0)); | ||
| + | dot((-1,1)); | ||
| + | dot((0,1)); | ||
| + | dot((1,1)); | ||
| + | dot((0,2)); | ||
| + | draw((0.8, -1.4+e)--(1.8-e, 0.6)--(-0.2, 1.6-e)--(-1.2+e, -0.4)--cycle); | ||
| + | [/asy] | ||
| + | This square's side length is slightly less than <math>\sqrt 5</math>, yielding an answer of <math>\textbf{(D) }5.0</math> | ||
== Solution 2 == | == Solution 2 == | ||
Revision as of 23:46, 6 February 2017
Contents
Problem
A point 
in the plane is called a lattice point if both 
and 
are integers. The area of the largest square that contains exactly three lattice points in its interior is closest to
Solution 1
This is the best square:
[asy]
real e = 0.1;
dot((0,-1));
dot((1,-1));
dot((-1,0));
dot((0,0));
dot((1,0));
dot((2,0));
dot((-1,1));
dot((0,1));
dot((1,1));
dot((0,2));
draw((0.8, -1.4+e)--(1.8-e, 0.6)--(-0.2, 1.6-e)--(-1.2+e, -0.4)--cycle);
[/asy]
This square's side length is slightly less than 
, yielding an answer of 
Solution 2
Apply Pick's Theorem. 4 lattice points on the border edges, 3 points in the interior. 
, implying that 
, 
(This is incorrect, because the vertices are not necessarily lattice points. The key idea is in fact to consider the lines on which the sides lie, and in fact not the vertices.
See also
| 1998 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 28 |
Followed by Problem 30 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
