AoPS Academy
people be arranged in a circle, numbered clockwise from 
to 
. These people are eliminated one by one in order, until only one person remains. The elimination follows this rule: among the remaining people, start counting clockwise from the person with the smallest number, and eliminate the -th person in that count. Then, among the remaining people, start counting again from the person with the smallest number and eliminate the -th person. Repeat this process until only one person remains. Let 
denote the number of the last remaining person.
, people are eliminated in the following order: 
. Thus, 
. It is known that 
under certain conditions. Prove that the necessary and sufficient condition for this is that both 
and 
are prime numbers.
be the unit circle in the complex plane. Let 
be the map given by 
. We define 
and 
for 
. The smallest positive integer such that 
is called the period of 
. Determine the total number of points in 
of period 
.
)
and 
and 
consecutive even numbers, such that the sum of their squares divides the square of their product.
for some real number 
and ![\[ a_n=na_{n-1}+(n-1)(n!(n-1)!-1) \]](http://latex.artofproblemsolving.com/4/1/f/41f0cc7b041a8bba056882b5c794503195c374fd.png)
for integers 
. Given that 
, and given that 
, where 
and 
are positive integers whose greatest common divisor is , compute 
and 
are in the coordinate plane such that 
. Let 
denote the locus of all points 
in the coordinate plane satisfying 
, and let 
be the midpoint of 
. Points 
and 
are on such that 
and 
. The value of 
can be expressed in the form 
, where 
and are relatively prime positive integers. Find 
.
, then
. (d) None of these.
and 
Then 
in terms of 
and 
Prove that
Let 
Prove that
satisfying:
points on the plane with diameter 
.
and two not necessarily distinct points 
not equal to ![\[|AX-XY|+|BY-XY|\leq \frac{2D}{N-2}.\]](http://latex.artofproblemsolving.com/f/3/6/f3649872c33e1db8880b4aedace38d9439f2b4ae.png)
.
as a closer intersection of circle centered at 
with line 
to 
.
denote the distance between 
.
be 
point with maximum distance.
,
.
,
Assume that 
is closer to 
.
,
,
Assume that 
is closer to 
We also have, 
Hence, 
as desired. 
![[asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */
import graph; size(5cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.3) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -9.32877658344286, xmax = 4.842239436750148, ymin = -5.75205506184172, ymax = 11.315227006273846; /* image dimensions */
pen ccqqqq = rgb(0.8,0,0);
/* draw figures */
draw(circle((-5.619344145657033,3.3112938265867515), 5.9426734764007), linewidth(0.8) + linetype("4 4"));
draw(circle((0.3232,3.3505), 5.9426734764007), linewidth(0.5) + linetype("4 4"));
draw(shift((-5.619344145657033,3.3112938265867515))*xscale(2.136442552966922)*yscale(2.136442552966922)*arc((0,0),1,0.37800571953037904,20.9360584959399), linewidth(0.5));
draw(shift((-5.619344145657033,3.3112938265867515))*xscale(3.2117023387111594)*yscale(3.2117023387111594)*arc((0,0),1,0.3780057195303871,48.997923906553446), linewidth(0.5) + blue);
draw(shift((-5.619344145657033,3.3112938265867515))*xscale(3.8248226592259553)*yscale(3.8248226592259553)*arc((0,0),1,-66.56545505654675,0.3780057195303883), linewidth(0.5) + blue);
draw(shift((-5.619344145657033,3.3112938265867515))*xscale(4.9957336781311055)*yscale(4.9957336781311055)*arc((0,0),1,0.3780057195303922,28.859027968338925), linewidth(0.5) + blue);
draw((-5.619344145657033,3.3112938265867515)--(-2.4077117034080557,3.3324827004737796), linewidth(0.8));
draw((-1.7946047262659561,3.3365276983530587)--(0.3232,3.3505), linewidth(0.5));
draw((-2.4077117034080557,3.3324827004737796)--(-1.7946047262659561,3.3365276983530587), linewidth(0.5) + ccqqqq);
draw((-6.712144852938944,5.8324357481909646)--(-6.352895216349681,5.0036301726720405), linewidth(0.5));
draw((-5.807916353516233,3.746338657099152)--(-4.0982077610349315,-0.19803837112652733), linewidth(0.5));
draw(shift((-5.619344145657033,3.3112938265867515))*xscale(3.2117023387111594)*yscale(3.2117023387111594)*arc((0,0),1,-66.56545505654675,0.3780057195303871), linewidth(0.5) + blue);
draw(shift((-4.0982077610349315,-0.19803837112652733))*xscale(4.298978205573835)*yscale(4.298978205573835)*arc((0,0),1,83.6663114594277,113.43454494345325), linewidth(0.5) + ccqqqq);
draw(shift((-4.0982077610349315,-0.19803837112652733))*xscale(5.6693007647061044)*yscale(5.6693007647061044)*arc((0,0),1,38.74986978582823,113.43454494345325), linewidth(0.5) + linetype("4 4") + ccqqqq);
draw(shift((-4.0982077610349315,-0.19803837112652733))*xscale(6.572616314998897)*yscale(6.572616314998897)*arc((0,0),1,64.26224620718133,113.43454494345323), linewidth(0.5) + ccqqqq);
draw((-6.352895216349681,5.0036301726720405)--(-5.807916353516233,3.746338657099152), linewidth(0.5));
draw(shift((-4.0982077610349315,-0.19803837112652733))*xscale(5.962028603849296)*yscale(5.962028603849296)*arc((0,0),1,84.35919416051627,113.43454494345325), linewidth(0.5) + ccqqqq);
/* dots and labels */
dot((-5.619344145657033,3.3112938265867515),dotstyle);
label("$P$", (-5.527427395544371,3.557371520766769), NE * labelscalefactor);
dot((0.3232,3.3505),dotstyle);
label("$Q$", (0.4202618100110891,3.6090905573368164), NE * labelscalefactor);
dot((-3.51219,5.73512),dotstyle);
label("$W$", (-3.4069468961724243,5.9881662395589865), NE * labelscalefactor);
dot((-3.62395,4.0747),dotstyle);
label("$U$", (-3.5103849693125193,4.333157069317477), NE * labelscalefactor);
dot((-1.2440311483629778,5.7225157488749225),dotstyle);
dot((-4.0982077610349315,-0.19803837112652733),dotstyle);
label("$Y$", (-4.0017158167279705,0.06633655228858504), NE * labelscalefactor);
dot((-3.4829480882119115,3.325388785287022),linewidth(4pt) + dotstyle);
dot((-2.4077117034080557,3.3324827004737796),linewidth(4pt) + dotstyle);
dot((-1.7946047262659561,3.3365276983530587),linewidth(4pt) + dotstyle);
dot((-0.6237191899644594,3.3442526626542244),linewidth(4pt) + dotstyle);
dot((-4.3420464199166915,0.36450872544664914),linewidth(4pt) + dotstyle);
dot((-6.352895216349681,5.0036301726720405),linewidth(4pt) + dotstyle);
dot((-5.807916353516233,3.746338657099152),linewidth(4pt) + dotstyle);
dot((-6.712144852938944,5.8324357481909646),linewidth(4pt) + dotstyle);
dot((-6.469313415723834,5.272212358723019),linewidth(4pt) + dotstyle);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */[/asy]](http://latex.artofproblemsolving.com/c/d/2/cd2a7f81fc999d57c1e7ca839aa2dffc8be3e4e2.png)