Problem #2195


A paint brush is swept along both diagonals of a square to produce the symmetric painted area, as shown. Half the area of the square is painted. What is the ratio of the side length of the square to the brush width?

[asy] unitsize(15mm); defaultpen(linewidth(.8pt)); path P=(-sqrt(2)/2,1)--(0,1-sqrt(2)/2)--(sqrt(2)/2,1); path Pc=(-sqrt(2)/2,1)--(0,1-sqrt(2)/2)--(sqrt(2)/2,1)--cycle; path S=(-1,-1)--(-1,1)--(1,1)--(1,-1)--cycle; fill(S,gray); for(int i=0;i<4;++i) { fill(rotate(90*i)*Pc,white); draw(rotate(90*i)*P); } draw(S); [/asy]

$\text{(A)}\ 2\sqrt {2} + 1 \qquad \text{(B)}\ 3\sqrt {2}\qquad \text{(C)}\ 2\sqrt {2} + 2 \qquad \text{(D)}\ 3\sqrt {2} + 1 \qquad \text{(E)}\ 3\sqrt {2} + 2$

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Instructions for entering answers:

  • Reduce fractions to lowest terms and enter in the form 7/9.
  • Numbers involving pi should be written as 7pi or 7pi/3 as appropriate.
  • Square roots should be written as sqrt(3), 5sqrt(5), sqrt(3)/2, or 7sqrt(2)/3 as appropriate.
  • Exponents should be entered in the form 10^10.
  • If the problem is multiple choice, enter the appropriate (capital) letter.
  • Enter points with parentheses, like so: (4,5)
  • Complex numbers should be entered in rectangular form unless otherwise specified, like so: 3+4i. If there is no real component, enter only the imaginary component (i.e. 2i, NOT 0+2i).

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