Difference between revisions of "1998 AHSME Problems/Problem 29"
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<math> \mathrm{(A) \ } 4.0 \qquad \mathrm{(B) \ } 4.2 \qquad \mathrm{(C) \ } 4.5 \qquad \mathrm{(D) \ } 5.0 \qquad \mathrm{(E) \ } 5.6</math> | <math> \mathrm{(A) \ } 4.0 \qquad \mathrm{(B) \ } 4.2 \qquad \mathrm{(C) \ } 4.5 \qquad \mathrm{(D) \ } 5.0 \qquad \mathrm{(E) \ } 5.6</math> | ||
| + | |||
| + | |||
| + | == Solution == | ||
| + | <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)); | ||
| + | dot((-1,-1)); | ||
| + | dot((2,2)); | ||
| + | dot((1,2)); | ||
| + | dot((2,1)); | ||
| + | dot((2,-1)); | ||
| + | dot((-1,2)); | ||
| + | draw((0.8, -1.4+e)--(1.8-e, 0.6)--(-0.2, 1.6-e)--(-1.2+e, -0.4)--cycle); | ||
| + | </asy> | ||
| + | The best square's side length is slightly less than <math>\sqrt 5</math>, yielding an answer of <math>\boxed{\textbf{(D) }5.0}</math> | ||
== See also == | == See also == | ||
| − | {{AHSME box|year=1998|num-b= | + | {{AHSME box|year=1998|num-b=28|num-a=30}} |
| + | {{MAA Notice}} | ||
Latest revision as of 00:54, 30 January 2019
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
![[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)); dot((-1,-1)); dot((2,2)); dot((1,2)); dot((2,1)); dot((2,-1)); dot((-1,2)); draw((0.8, -1.4+e)--(1.8-e, 0.6)--(-0.2, 1.6-e)--(-1.2+e, -0.4)--cycle); [/asy]](http://latex.artofproblemsolving.com/b/5/9/b59f1ddd792a8c20dc70c287f59d5cdb62c1aa48.png)
The best square's side length is slightly less than 
, yielding an answer of 
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. 
