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Difference between revisions of "1998 AHSME Problems/Problem 29"

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== Solution 1 ==
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== Solution ==
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This is the best square:
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<asy>
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[asy]
 
 
real e = 0.1;
 
real e = 0.1;
 +
 
dot((0,-1));
 
dot((0,-1));
 
dot((1,-1));
 
dot((1,-1));
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dot((1,1));
 
dot((1,1));
 
dot((0,2));
 
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);
 
draw((0.8, -1.4+e)--(1.8-e, 0.6)--(-0.2, 1.6-e)--(-1.2+e, -0.4)--cycle);
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[/asy]
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</asy>
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This square's side length is slightly less than <math>\sqrt 5</math>, yielding an answer of <math>\textbf{(D) }5.0</math>
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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>
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== Solution 2 ==
 
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Apply Pick's Theorem. 4 lattice points on the border edges, 3 points in the interior. <math>A = I + \frac{B}{2} -1</math>, implying that <math>max(A) = 4</math>, <math>\boxed{A}</math> (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 ==
 
== See also ==
 
{{AHSME box|year=1998|num-b=28|num-a=30}}
 
{{AHSME box|year=1998|num-b=28|num-a=30}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 23:54, 29 January 2019

Problem

A point $(x,y)$ in the plane is called a lattice point if both $x$ and $y$ are integers. The area of the largest square that contains exactly three lattice points in its interior is closest to

$\mathrm{(A) \ } 4.0 \qquad \mathrm{(B) \ } 4.2 \qquad \mathrm{(C) \ } 4.5 \qquad \mathrm{(D) \ } 5.0 \qquad \mathrm{(E) \ } 5.6$


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 $\sqrt 5$, yielding an answer of $\boxed{\textbf{(D) }5.0}$

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

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