AoPS Academy
and 
be two sequences of integers such that 
and 
for all 
. Suppose that 
divides 
for an infinity of integers . Prove that there exist an integer 
such that 
for all
are coprime. Consider the sequence 
given by 
. Suppose that there exist positive integers 
such that 
is even and 
is an integer. Prove that the numerator in its simplest form of 
is an odd integer greater than 
.
be an integer and let 
be 
distinct subsets of 
. Show that there exists 
such that the n subsets 
are also disjoint.
![[asy]
import graph;import olympiad;import geometry;size(7cm);
pen fonted=fontsize(8.5);
defaultpen(fonted);
pair B=(0,0),A=(2,8),C=(10,0),D,E,S,T,M,N,P,Q;
D=A+(0.6)*(B-A);E=A+(0.6)*(C-A);
path c=circumcircle(A,B,C);pair O=circumcenter(A,B,C);
pair Bt=rotate(90,B)*O;pair Ct=rotate(90,C)*O;
T=extension(C,D,B,Bt);S=extension(B,E,C,Ct);
pair x[]=intersectionpoints(T--(T+10*(A-T)),c);
pair y[]=intersectionpoints(S--A,c);
M=x[0];N=y[0];
P=extension(B,E,C,D);
Q=extension(B,Bt,C,Ct);
draw(A--B--C--cycle,magenta);draw(T--Q--S--A--cycle,heavycyan);draw(C--T^^B--S^^D--E^^M--N^^A--P^^A--Q,green);draw(c,orange);
dot(A);dot(B);dot(C);dot(D);dot(E);dot(T);dot(S);dot(x[0]);dot(y[0]);dot(P);dot(Q);
label("$A$",A,NNE);label("$B$",B,SE);label("$C$",C,SE);label("$D$",D,NE);label("$E$",E,NW);label("$S$",S,NNE);label("$T$",T,NNE);label("$M$",M,NW);label("$N$",N,NE);label("$P$",P,SE);label("$Q$",Q,SE);[/asy]](http://latex.artofproblemsolving.com/f/6/c/f6cc2edacad258df2cc9700a37b0d886d03cbbf6.png)
and 
Then 
is the 
median and 
is the symmedian. So 
are isogonal w.r.t. 
also, 
are trivially isogonal w.r.t. 
So by Isogonal Line Lemma, 
are isogonal w.r.t. So 
is an isosceles trapezium 
, as desired. 
be arbritary points on the tangents of 
respectively, such that 
are in the same side with respect to 
. Let also 
, 
be the intersection points of 
and 
with 
and 
respectively. Finally, let 
and 
the intersections of the circumcircle of 
with 
and 
.
, and 
are concurrent.
such that ![\[f(2x+y) + f(x+f(2y)) = f(x)f(y) - xy\]](http://latex.artofproblemsolving.com/0/6/2/0623b7c328af6d088cfc5dd288afe3e17a94ff29.png)
for all reals 
and 
.
s
.
be points on the segments 
.
of 
and 
.
and 
with 
.
.
![[asy]import graph; size(9cm);
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 = -4.319332900674542, xmax = 12.46238917406057, ymin = -3.2834042311679315, ymax = 6.211517469011155; /* image dimensions */
pen aqaqaq = rgb(0.6274509803921569,0.6274509803921569,0.6274509803921569); pen uququq = rgb(0.25098039215686274,0.25098039215686274,0.25098039215686274); pen qqwuqq = rgb(0.,0.39215686274509803,0.);
draw((0.3989981582330802,2.5552050251389775)--(0.,0.)--(4.,0.)--cycle, aqaqaq);
/* draw figures */
draw((0.3989981582330802,2.5552050251389775)--(0.,0.), uququq);
draw((0.,0.)--(4.,0.), uququq);
draw((4.,0.)--(0.3989981582330802,2.5552050251389775), uququq);
draw(circle((2.,0.9964522548556711), 2.234483630776236));
draw((0.,0.)--(-0.8529264345008228,1.7119263473880404));
draw((-0.8529264345008228,1.7119263473880404)--(4.,0.));
draw((0.,0.)--(5.839458241941738,3.692014811503779));
draw((5.839458241941738,3.692014811503779)--(4.,0.));
draw((-0.8529264345008228,1.7119263473880404)--(1.1571205626701906,3.0658655413534492));
draw((0.3989981582330802,2.5552050251389775)--(5.839458241941738,3.692014811503779));
draw((1.1571205626701906,3.0658655413534492)--(2.8428794373298087,3.0658655413534492), linewidth(1.6) + red);
draw((0.20883313408761875,1.3373782876573805)--(2.115258216224203,1.3373782876573805), linewidth(1.6) + qqwuqq);
/* dots and labels */
dot((0.3989981582330802,2.5552050251389775),linewidth(3.pt) + dotstyle);
label("$A$", (0.15,2.710067938380329), NE * labelscalefactor);
dot((0.,0.),linewidth(3.pt) + dotstyle);
label("$B$", (-0.4,-0.03431142346545344), NE * labelscalefactor);
dot((4.,0.),linewidth(3.pt) + dotstyle);
label("$C$", (4.134617317425102,-0.0027668330994099592), NE * labelscalefactor);
dot((0.20883313408761875,1.3373782876573805),linewidth(3.pt) + dotstyle);
label("$D$", (-0.1,1.5271457996536988), NE * labelscalefactor);
dot((2.115258216224203,1.3373782876573805),linewidth(3.pt) + dotstyle);
label("$E$", (2.084218943632278,1.4956012092876552), NE * labelscalefactor);
dot((-0.8529264345008228,1.7119263473880404),linewidth(3.pt) + dotstyle);
label("$T$", (-1.0860123881550887,1.7952748177650684), NE * labelscalefactor);
dot((5.839458241941738,3.692014811503779),linewidth(3.pt) + dotstyle);
label("$S$", (5.901114377923536,3.7825840108258073), NE * labelscalefactor);
dot((1.1571205626701906,3.0658655413534492),linewidth(3.pt) + dotstyle);
label("$M$", (1.106336642284931,3.2147813842370248), NE * labelscalefactor);
dot((2.8428794373298087,3.0658655413534492),linewidth(3.pt) + dotstyle);
label("$N$", (2.9043782931494078,3.1674644986879597), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */[/asy]](http://latex.artofproblemsolving.com/8/b/c/8bc5a4677758a485d1b7a7a60546fb1b3e379fe0.png)