<p>The set <img alt="$G$" height="12" src="https://latex.artofproblemsolving.com/6/e/2/6e28ce12d49d39f160d5a0ef54077fc98e4b9d2b.png" width="14" /> is defined by the points <img alt="$(x,y)$" height="18" src="https://latex.artofproblemsolving.com/e/5/3/e53b1fe25be1c679117fb44a6a886fe1247d189a.png" width="40" /> with integer coordinates, <img alt="$3\le|x|\le7$" height="18" src="https://latex.artofproblemsolving.com/e/6/e/e6e10c9dd38193fb4267ee34f633b1bb4f3898ae.png" width="87" />, <img alt="$3\le|y|\le7$" height="20" src="https://latex.artofproblemsolving.com/a/2/3/a23d0b01c529be8e0459e17665bf33b652a5225e.png" width="88" />. How many squares of side at least <img alt="$6$" height="12" src="https://latex.artofproblemsolving.com/6/0/1/601a7806cbfad68196c43a4665871f8c3186802e.png" width="8" /> have their four vertices in <img alt="$G$" height="12" src="https://latex.artofproblemsolving.com/6/e/2/6e28ce12d49d39f160d5a0ef54077fc98e4b9d2b.png" width="14" />?</p> <p><img alt="[asy] defaultpen(black+0.75bp+fontsize(8pt)); size(5cm); path p = scale(.15)*unitcircle; draw((-8,0)--(8.5,0),Arrow(HookHead,1mm)); draw((0,-8)--(0,8.5),Arrow(HookHead,1mm)); int i,j; for (i=-7;i<8;++i) { for (j=-7;j<8;++j) { if (((-7 <= i) && (i <= -3)) || ((3 <= i) && (i<= 7))) { if (((-7 <= j) && (j <= -3)) || ((3 <= j) && (j<= 7))) { fill(shift(i,j)*p,black); }}}} draw((-7,-.2)--(-7,.2),black+0.5bp); draw((-3,-.2)--(-3,.2),black+0.5bp); draw((3,-.2)--(3,.2),black+0.5bp); draw((7,-.2)--(7,.2),black+0.5bp); draw((-.2,-7)--(.2,-7),black+0.5bp); draw((-.2,-3)--(.2,-3),black+0.5bp); draw((-.2,3)--(.2,3),black+0.5bp); draw((-.2,7)--(.2,7),black+0.5bp); label("$-7$",(-7,0),S); label("$-3$",(-3,0),S); label("$3$",(3,0),S); label("$7$",(7,0),S); label("$-7$",(0,-7),W); label("$-3$",(0,-3),W); label("$3$",(0,3),W); label("$7$",(0,7),W); [/asy]" height="238" src="https://latex.artofproblemsolving.com/6/9/9/6992511f7ecca62a842cd810c131a3730f9032a8.png" width="238" /></p>

Question No: 284

The set $G$ is defined by the points $(x,y)$ with integer coordinates, $3\le|x|\le7$ , $3\le|y|\le7$ . How many squares of side at least $6$ have their four vertices in $G$ ?

$[asy] defaultpen(black+0.75bp+fontsize(8pt)); size(5cm); path p = scale(.15)*unitcircle; draw((-8,0)--(8.5,0),Arrow(HookHead,1mm)); draw((0,-8)--(0,8.5),Arrow(HookHead,1mm)); int i,j; for (i=-7;i<8;++i) { for (j=-7;j<8;++j) { if (((-7 <= i) && (i <= -3)) || ((3 <= i) && (i<= 7))) { if (((-7 <= j) && (j <= -3)) || ((3 <= j) && (j<= 7))) { fill(shift(i,j)*p,black); }}}} draw((-7,-.2)--(-7,.2),black+0.5bp); draw((-3,-.2)--(-3,.2),black+0.5bp); draw((3,-.2)--(3,.2),black+0.5bp); draw((7,-.2)--(7,.2),black+0.5bp); draw((-.2,-7)--(.2,-7),black+0.5bp); draw((-.2,-3)--(.2,-3),black+0.5bp); draw((-.2,3)--(.2,3),black+0.5bp); draw((-.2,7)--(.2,7),black+0.5bp); label("$-7$",(-7,0),S); label("$-3$",(-3,0),S); label("$3$",(3,0),S); label("$7$",(7,0),S); label("$-7$",(0,-7),W); label("$-3$",(0,-3),W); label("$3$",(0,3),W); label("$7$",(0,7),W); [/asy]$

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Tags: Algebra Combinatorics Geometry

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