AoPS Academy
be an integer. Consider an 
chessboard consisting of 
unit squares. A configuration of 
rooks on this board is peaceful if every row and every column contains exactly one rook. Find the greatest positive integer 
such that, for each peaceful configuration of rooks, there is a 
square which does not contain a rook on any of its 
unit squares.
students. Let, for 
, the 
student have 
many friends. For 
let us define 
to be the number of students who have strictly more than 
friends. Show that 
are positive real numbers, then
passes through points 
and 
and intersects the sides 
and 
of triangle 
at points 
and 
, respectively. The circumcircles of triangles and 
meet at distinct points 
and 
. Prove that 
.
passes through points and and intersects the sides and of triangle at points and , respectively. The circumcircles of triangles and meet at distinct points and . Prove that .
denote the image of 
. An inversion 
around causes 
to be collinear, and 
to be collinear. Then, also sends the circumcircle of 
to the circumcircle 
of 
. As such, 
lies on the midpoint of the tangents from to .
lies on the polar of , hence 
. Inverting back implies the result. 
![[asy]
import graph; size(10cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -9.627272727272732, xmax = 11.245454545454548, ymin = -6.2381818181818245, ymax = 7.652727272727277; /* image dimensions */
pen ffxfqq = rgb(1,0.4980392156862745,0); pen yqqqqq = rgb(0.5019607843137255,0,0);
/* draw figures */
draw(circle((-1.1727272727272753,0.16181818181818255), 3.4075754207256947), linewidth(1) + ffxfqq);
draw((-3.2454545454545487,5.052727272727277)--(-4.136876511792053,-1.51907132570026), linewidth(1) + red);
draw((-4.136876511792053,-1.51907132570026)--(1.1909090909090891,-2.292727272727274), linewidth(1) + red);
draw((1.1909090909090891,-2.292727272727274)--(-3.2454545454545487,5.052727272727277), linewidth(1) + red);
draw(circle((-0.9928108389069386,1.4008136157456776), 4.290789790551068), linewidth(1) + ffxfqq);
draw(circle((-3.4253709792748843,3.8137318387997823), 1.2519902589284826), linewidth(1) + red);
draw((-4.662867079350029,3.6237765526121537)--(-1.1727272727272753,0.16181818181818255), linewidth(1) + yqqqqq);
draw((-4.662867079350029,3.6237765526121537)--(-3.2454545454545487,5.052727272727277), linewidth(1) + yqqqqq);
/* dots and labels */
dot((-1.1727272727272753,0.16181818181818255),dotstyle);
label("$O$", (-1.1,0.3436363636363624), NE * labelscalefactor);
dot((1.1909090909090891,-2.292727272727274),dotstyle);
label("$A$", (1.3,-2.7290909090909126), NE * labelscalefactor);
dot((-4.136876511792053,-1.51907132570026),dotstyle);
label("$C$", (-4.063636363636367,-1.3290909090909115), NE * labelscalefactor);
dot((-3.2454545454545487,5.052727272727277),dotstyle);
label("$B$", (-3.172727272727276,5.234545454545457), NE * labelscalefactor);
dot((-2.2450261848480517,3.3962803150017655),linewidth(4pt) + dotstyle);
label("$K$", (-2.1727272727272755,3.543636363636365), NE * labelscalefactor);
dot((-3.5820063289067603,2.57157847640949),linewidth(4pt) + dotstyle);
label("$N$", (-3.9909090909090943,2.8709090909090915), NE * labelscalefactor);
dot((-4.662867079350029,3.6237765526121537),linewidth(4pt) + dotstyle);
label("$M$", (-5.318181818181822,3.6890909090909103), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
[/asy]](http://latex.artofproblemsolving.com/2/b/3/2b31342bc35be2d6d6015784950642be638b62ce.png)
such that:
with all 
is real number
