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

(Solution 1)
(Solution 1)
 
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== Solution 1 ==
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== Solution ==
The answer is actually (D)
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<asy>
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real e = 0.1;
  
== Solution 2 ==
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dot((0,-1));
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>
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dot((1,-1));
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dot((-1,0));
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dot((0,0));
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dot((1,0));
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dot((2,0));
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dot((-1,1));
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dot((0,1));
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dot((1,1));
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dot((0,2));
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dot((-1,-1));
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dot((2,2));
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dot((1,2));
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dot((2,1));
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dot((2,-1));
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dot((-1,2));
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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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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=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 (ProblemsAnswer KeyResources)
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
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