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 real numbers. Four solutions on the complex plane of the quartic equation in 
form the four vertices of a quadrilateral. Find the condition such that the area of the quadrilateral is less than 1, and also sketch the domain of the set of the points 
satisfying the condition.
be the set of real numbers. Find all functions 
where there exist a real constant 
such that 
holds for all reals 
, 
, 
that satisfy 
.
, denote the sum of digits of in its decimal representation by 
. Find all polynomials 
with integer coefficients such that for any positive integer 
, the integer 
is positive and 
be a triangle with incentre 
and circumcircle 
. Let 
be the midpoints of the arcs 
of not containing 
respectively. Let 
be the point of diametrically opposite to 
. Show that 
and the midpoint 
of 
lie on a line.
such that, for every positive integer 
the integer
is a multiple of 
(Note that 
denotes the greatest integer less than or equal to 
For example, 
and 
)
such that 
for all 
?
be a monic polynomial with complex coefficients such that there exist a polynomial 
with complex coefficients for which ![\[P(x^2-2)=P(x)Q(x).\]](http://latex.artofproblemsolving.com/a/6/2/a62b06890b6bbcc5c9634f6de8c7ae5fe9f37828.png)
Determine all complex numbers that could be the root of .
such that for all 
,![\[xf(x+f(y))=(y-x)f(f(x)).\]](http://latex.artofproblemsolving.com/e/4/a/e4a3bbb8b91d2aa62d699c24df342fa59be71915.png)
. Find all functions 
such that for all 
,![\[
f(xf(x+y)+z) = f(z) + f(x)y + f(xf(x)).
\]](http://latex.artofproblemsolving.com/f/9/3/f93b7526ff4338eea89a1c59bc009aa001ffc15d.png)
be a parallelogram with center 
such that 
and 
. Consider points 
and 
on the rays 
and 
respectively, such that 
is parallel to 
and 
. Prove that 
is perpendicular to 
.
be a parallelogram with center such that and . Consider points and on the rays and respectively, such that is parallel to and . Prove that is perpendicular to .
and 
Prove that
.
and 
Prove that
Prove that
![[asy]
import graph; size(8cm);
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 = -14.9943560079638, xmax = 9.209305301315823, ymin = -11.292660196107217, ymax = 7.3846233374467944; /* image dimensions */
/* draw figures */
draw((2.76,-0.93)--(4.295554760029417,2.6339878219780375), linewidth(1.2));
draw((4.295554760029417,2.6339878219780375)--(-5.004445239970583,2.5139878219780374), linewidth(1.2));
draw((-5.004445239970583,2.5139878219780374)--(-6.54,-1.05), linewidth(1.2));
draw((xmin, 0.012903225806451623*xmin-0.9656129032258066)--(xmax, 0.012903225806451623*xmax-0.9656129032258066), linewidth(1.2)); /* line */
draw((-6.54,-1.05)--(-2.7185433386072324,1.5000559740200694), linewidth(1.2));
draw((1.1054987466558457,1.5493984525395927)--(2.76,-0.93), linewidth(1.2));
draw((xmin, 0.667299461951945*xmin + 5.85345143797838)--(xmax, 0.667299461951945*xmax + 5.85345143797838), linewidth(1.2)); /* line */
draw((xmin, -1.4985775607773748*xmin + 9.071209796448564)--(xmax, -1.4985775607773748*xmax + 9.071209796448564), linewidth(1.2)); /* line */
draw((4.295554760029417,2.6339878219780375)--(xmin, 0.33999069762146966*xmin + 1.1735391624444114), linewidth(1.2)); /* ray */
draw((-5.004445239970583,2.5139878219780374)--(xmax, -0.4435587753583134*xmax + 0.29422221998894477), linewidth(1.2)); /* ray */
draw((-14.29815753668234,-3.687701393153892)--(8.319271370104877,-3.395863600808253), linewidth(1.2));
/* dots and labels */
dot((-6.54,-1.05),linewidth(3pt) + dotstyle);
label("$A$", (-6.751623054339161,-2.0757860752360298), NE * labelscalefactor);
dot((2.76,-0.93),linewidth(3pt) + dotstyle);
label("$B$", (2.3341166786334524,-1.9446516873374562), NE * labelscalefactor);
dot((-5.004445239970583,2.5139878219780374),linewidth(3pt) + dotstyle);
label("$D$", (-5.66508098317955,2.7574527930256902), NE * labelscalefactor);
dot((4.295554760029417,2.6339878219780375),linewidth(3pt) + dotstyle);
label("$C$", (4.675802176822271,2.9073206649097747), NE * labelscalefactor);
dot((-1.1222226199852914,0.7919939109890185),linewidth(3pt) + dotstyle);
label("$O$", (-1.2065117946280401,1.0339722663587203), NE * labelscalefactor);
dot((1.1054987466558457,1.5493984525395927),linewidth(3pt) + dotstyle);
label("$V$", (0.8541714437781196,1.6334437538950577), NE * labelscalefactor);
dot((-2.7185433386072324,1.5000559740200694),linewidth(3pt) + dotstyle);
label("$Z$", (-2.6677235454978625,1.5772433019385261), NE * labelscalefactor);
dot((-10.420390528786735,-1.1000695552101516),linewidth(3pt) + dotstyle);
label("$W$", (-10.910456499122501,-0.8393761321923341), NE * labelscalefactor);
dot((6.640390528786738,-0.8799304447898486),linewidth(3pt) + dotstyle);
label("$U$", (6.71775193124292,-0.7644421962502919), NE * labelscalefactor);
dot((-14.29815753668234,-3.687701393153892),linewidth(3pt) + dotstyle);
label("$A_1$", (-14.919422072021758,-3.0311937584970674), NE * labelscalefactor);
dot((8.319271370104877,-3.395863600808253),linewidth(3pt) + dotstyle);
label("$B_1$", (7.935428390301105,-2.7127245307433885), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
[/asy]](http://latex.artofproblemsolving.com/d/c/a/dca9b5fe26d73cbdfca59cb9a1866106844e1576.png)
be respectively bisectors of angles 
.
are intersections of line 
.
Also we have
By angle bisector theorem
so reversed Tales says 
Because 
we have 
QED
