AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
be a finite sequence of positive integers. Prove that there exist nonnegative integers 
and 
such that 
holds for all integers 
be a positive integer. Dominoes are placed on a 
board in such a way that every cell of the board is adjacent to exactly one cell covered by a domino. For each , determine the largest number of dominoes that can be placed in this way.
or 
. Dominoes are placed on the board in such a way that each domino covers exactly two cells of the board, and dominoes do not overlap. Two cells are said to be adjacent if they are different and share a common side.)
+1 w
of positive real numbers that satisfy the system:![\[
\begin{aligned}
11bc - 36b - 15c &= abc \\
12ca - 10c - 28a &= abc \\
13ab - 21a - 6b &= abc.
\end{aligned}
\]](http://latex.artofproblemsolving.com/1/7/0/170e1cfaee90f5c37428fa75192ac889261a8f83.png)
be positive reals. Prove that there is a cyclic permutation 
of 
such that the inequality:![\[
\frac{a}{xa + yb + zc} + \frac{b}{xb + yc + za} + \frac{c}{xc + ya + zb} \geq \frac{3}{x + y + z}
\]](http://latex.artofproblemsolving.com/5/5/3/553527fc40a1b7d29dbdfb4b5e63635308f46c85.png)
and 
.
be a scalene triangle with orthocenter 
and circumcenter 
. Denote by 
, the midpoints of 
, 
. Suppose the circle 
with diameter meets the circumcircle of at 
, and meets line 
at a point 
. The tangent to at 
meets line 
at 
. Show that the circumcircles of 
and 
intersect at a point 
on 
.
Prove that: 
is the sum of 
square numbers
Let 
and 
is the sum of square numbers. Prove that: 
is the sum of square numbers
+1 w be a triangle with circumcircle 
and incenter 
and let be the midpoint of . The points 
, 
, 
are selected on sides , 
, 
such that 
, 
, and 
. Suppose that the circumcircle of 
intersects at a point 
other than 
. Prove that lines 
and 
meet on .
, 
, 
, 
, 
, 
initially contains one coin. The following operations are allowed
, 
, remove one coin from and add two coins to 
;
, 
, remove one coin from and swap the contents (maybe empty) of the boxes 
and 
., , , , become empty, while box contains exactly 
coins. be a triangle, its orthocenter, its circumcenter, and 
its circumradius. Let be the reflection of the point across the line 
, let be the reflection of the point 
across the line 
, and let be the reflection of the point 
across the line 
. Prove that the points , and are collinear if and only if 
.
be a real number and let 
be arbitrary real numbers in 
. The sequences 
and 
are then defined recursively by
. Prove that 
for all sufficiently large .
is a prime number
is cyclic.
are isogonal and 
, we know 
implying the result. 
, 
, and 
finishes. 
. But it's trivial that 
. Hence 
cyclic.
Prove that
and 
be altitudes of triangle 
and 
intersect 
again at 
and 
,
,
and 
intersect at a point.
is a cyclic quadrilateral.
,
we have that 
,
,![[asy]
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */
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 = -22.772584895993297, xmax = 16.656389049878932, ymin = -4.260015997360821, ymax = 12.887480484307561; /* image dimensions */
pen ffxfqq = rgb(1,0.4980392156862745,0); pen zzttqq = rgb(0.6,0.2,0); pen qqwuqq = rgb(0,0.39215686274509803,0);
draw((-8,10)--(-12,2)--(1,2)--cycle, linewidth(0.8) + zzttqq);
/* draw figures */
draw(circle((-5.5,3.75), 6.731456008918131), linewidth(0.8) + ffxfqq);
draw((-8,10)--(-12,2), linewidth(0.8) + zzttqq);
draw((-12,2)--(1,2), linewidth(0.8) + zzttqq);
draw((1,2)--(-8,10), linewidth(0.8) + zzttqq);
draw((-3,-2.5)--(-6.826143756350537,10.349533524234307), linewidth(0.8) + qqwuqq);
draw((-3,-2.5)--(-10.262495372084413,8.507219548315437), linewidth(0.8) + qqwuqq);
draw((-12,2)--(-3,-2.5), linewidth(0.8));
draw((-3,-2.5)--(1,2), linewidth(0.8));
draw((-6.826143756350537,10.349533524234307)--(-10.262495372084413,8.507219548315437), linewidth(0.8) + qqwuqq);
draw((-9.4,7.2)--(-6.2620689655172415,8.455172413793104), linewidth(0.8) + qqwuqq);
draw(circle((-8.256933115823818,8.892332789559545), 2.0422027745350433), linewidth(0.8) + red);
/* dots and labels */
dot((1,2),linewidth(4pt) + dotstyle);
label("$C$", (1.1260965777092078,1.4643727981661987), NE * labelscalefactor);
dot((-12,2),linewidth(4pt) + dotstyle);
label("$B$", (-12.53029283583508,1.515712607991553), NE * labelscalefactor);
dot((-8,10),linewidth(4pt) + dotstyle);
label("$A$", (-8.62846728910814,10.089460848825745), NE * labelscalefactor);
dot((-3,-2.5),linewidth(4pt) + dotstyle);
label("$D$", (-2.6473794444543453,-2.950850846814283), NE * labelscalefactor);
dot((-6.2620689655172415,8.455172413793104),linewidth(4pt) + dotstyle);
label("$E$", (-6.164156417491126,8.65194617371582), NE * labelscalefactor);
dot((-9.4,7.2),linewidth(4pt) + dotstyle);
label("$F$", (-10.065981964218066,7.009072259304478), NE * labelscalefactor);
dot((-6.826143756350537,10.349533524234307),linewidth(4pt) + dotstyle);
label("$Y$", (-6.7288943255700255,10.551519137253933), NE * labelscalefactor);
dot((-10.262495372084413,8.507219548315437),linewidth(4pt) + dotstyle);
label("$Z$", (-10.810409206685705,8.549266554065111), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */
[/asy]](http://latex.artofproblemsolving.com/3/4/7/34751d96817490123912aec97d8ecf66e7e7d8be.png)
is cyclic.
.
and 
are cyclic. Thus, by radical axis we conclude that ![[asy]
import olympiad;
size(6cm);defaultpen(fontsize(10pt));pen org=magenta;pen med=mediummagenta;pen light=pink;pen deep=deepmagenta;
pair O,A,B,C,D,E,F,Y,Z,R;
O=(0,0);A=dir(110);B=dir(200);C=dir(340);path w=circumcircle(A,B,C);D=-A;
E=foot(B,A,C);F=foot(C,A,B);Y=intersectionpoints(w,D--100F-99D)[0];Z=intersectionpoints(w,D--100E-99D)[0];
R=extension(A,O,E,F);
draw(A--B--C--cycle,deep);draw(w,deep);
draw(E--F,med);draw(A--D,med);draw(Y--D--Z,deep);draw(circumcircle(Y,Z,E),deep);
draw(foot(A,B,C)--A,deep);
dot("$A$",A,dir(A));
dot("$B$",B,dir(B));
dot("$C$",C,dir(C));
dot("$D$",D,dir(D));
dot("$O$",O,dir(90));
dot("$E$",E,dir(E));
dot("$F$",F,dir(F));
dot("$Y$",Y,dir(Y));
dot("$Z$",Z,dir(Z));
dot("$R$",R,dir(R));
[/asy]](http://latex.artofproblemsolving.com/2/1/2/212bd4c8f3708fee4620ff371ec96808c67575f3.png)