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.
such that![\[
f(f(x)) + x f(xy) = x + f(y)
\]](http://latex.artofproblemsolving.com/9/1/5/9157eb92f89f6c96333ce105a88ebed02c5307ef.png)
for all positive real numbers 
and 
.(Andrew Brahms, USA)
be positive real numbers such that 
. Prove that:![\[\frac{1}{a^2+155}+\frac{1}{b^2+155}+\frac{1}{c^2+155}\le \frac{11}{780}\]](http://latex.artofproblemsolving.com/4/2/7/427e12cbfdbbba6f26407dbb5c2430aee26c15eb.png)
When does the equality hold ?
be a function satisfying
for all integers 
and 
. Show that there exist positive integers 
and 
such that 
for all integers .
be a function taking in positive integers and outputting nonnegative integers, defined as follows:
is the number of positive integers with 
such that the equation 
has an integer solution 
. such that
and 
(Adit Aggarwal, India.)
such that: 
. Find the maximum: 
. Let 
and 
be the feet of the altitudes from 
to 
and from 
to 
respectively.
and 
as the reflections of across lines 
and , respectively. Let 
be the circumcircle of 
. Denote by 
the second intersection of line 
with , and by 
the intersection of ray 
with .
is the circumcenter of , prove that , , and are collinear if and only if quadrilateral 
can be inscribed within a circle.
have,where and are natural numbers.
be the set of the positive real numbers. Find all functions 
such that 
for all positive real numbers and .
be defined by![\[
a_1 = 1, \quad a_{n+1} = a_n + \frac{1}{\sqrt[2024]{a_n}} \quad \text{for } n \geq 1, \, n \in \mathbb{N}
\]](http://latex.artofproblemsolving.com/8/8/f/88f423d87c08b20dc552703fcf30f2a8e9585902.png)
Prove that![\[
a_n^{2025} >n^{2024}
\]](http://latex.artofproblemsolving.com/a/6/8/a6806c817edb7ce29a399981137c1a3e93f63c63.png)
for all positive integers 
.
with 
and the circle 
of radius centered at 
Let 
be the midpoint of 
The line 
intersects a second time at 
Let be a point on such that 
Let 
be the intersection of 
and Prove that 
such that for any set of consecutive integers taken from the given set of integers, there will always be a colour colouring exactly one of them? That is, for all integers 
such that 
, there will be a colour coloring exactly one integer from the set 
.
be a triangle inscribed circle 
, orthocenter 
. 
lie on such that 
. is midpoint of 
. The line passing though and parallel to 
cuts at 
. Prove that 
.
be a circumconic of 
passing through 
and 
. is the isogonal conjugate of the infinity point on 
WRT , is the isogonal conjugate of WRT which is a rectangle hyperbola , is the forth intersection of 
and 
is the center of .
WRT pass through the center 
of (well-known) ,
.
be midpoint of and 
is diameter of thus is midpoint of 
. 
cuts again at . 
is reflection of through 
then lies on . 
so 
is diameter of hence 
. We see 
by parallel lines. Therefore 
.
is projection of on 
. Note that, 
, we have ![\[\angle EAP=\angle EAC-\angle PAC=\angle BAF-\angle PAC=\angle BCF-\angle PAC=\angle ACF-\angle ACB-\angle PAC=\angle TAP-\angle PAC-\angle ACB=\angle TAC-(\angle GAT-\angle TCA)=180^\circ-\angle GTA-\angle ATC=\angle GTS.\]](http://latex.artofproblemsolving.com/f/3/8/f38233c1d044cf427a0f6b7aa2133cba84ce3a03.png)
deduce 
.
thus 
deduce 
.
suy ra 
.
thus 
. Hence if is midpoint of then ![\[\angle NMD=\angle NMB+\angle BMD=\angle ACB+\angle MCD+\angle MDC=\angle ACD+\angle QTA=\angle ACD+\angle QAC=\angle QCD.\]](http://latex.artofproblemsolving.com/8/8/b/88bdc2551c05000142da6f669b8d7842afb1f934.png)
be altitude, easily seen 
is cyclic deduce 
. Thus, 
is cyclic 
. We are done.
be the nine point center of .
be the midpoint of 
and 
be the reflection of in .
be the reflection of 
in 
, respectively .
are isogonal conjugate of 
.
,
,
. 
is the reflection of 
in 
, respectively , we get 
.
be the projections of 
on 
and let 
be the midpoint of 
cuts at and 
cuts 
at 
Clearly is on 9-point circle 
of 
is cyclic 
Thus since 
with corresponding circumcenters 
Hence 
and 
is cyclic 
, 
, 
, 
as variables and with chord 
being the diameter of parallel to 
, so 
. Then 
Moving points approach Fix and animate on the circumcircle. Then varies on line projectively via the map 
Also, varies projectively on the nine-point circle.![[asy] import graph; size(7cm); pen zzttqq = rgb(0.6,0.2,0.); pen qqwuqq = rgb(0.,0.39215,0.); pen cqcqcq = rgb(0.75294,0.75294,0.75294); pair O = (0.,0.), A = (-0.49774,0.86732), B = (-0.86295,-0.50528), C = (0.86211,-0.50672), H = (-0.49858,-0.14468), F = (0.75998,0.64993), D = (-0.62874,0.25325), G = (-0.69648,0.12037), X = (-0.22262,0.58933), Y = (-0.69700,-0.50542); draw(A--B--C--cycle, linewidth(0.6) + zzttqq); draw((-0.65022,0.21113)--(-0.60809,0.18965)--(-0.58661,0.23178)--D--cycle, linewidth(0.6) + qqwuqq);
draw(circle(O, 1.), linewidth(0.6)); draw(A--B, linewidth(0.6) + zzttqq); draw(B--C, linewidth(0.6) + zzttqq); draw(C--A, linewidth(0.6) + zzttqq); draw(O--G, linewidth(0.6)); draw(A--F, linewidth(0.6)); draw((-0.75890,0.65120)--F, linewidth(0.6)); draw(H--(-0.75890,0.65120), linewidth(0.6)); draw(circle((-0.24929,-0.07234), 0.5), linewidth(0.6)); draw(G--D, linewidth(0.6)); draw(C--D, linewidth(0.6)); dot("$O$", O, dir((0.857, 2.328))); dot("$A$", A, dir((0.923, 2.308))); dot("$B$", B, dir((0.887, 2.257))); dot("$C$", C, dir((0.912, 2.178))); dot("$H$", H, dir((0.785, 1.862))); dot("$E$", (-0.75890,0.65120), dir((0.959, 2.297))); dot("$F$", F, dir((0.871, 2.201))); dot("$D$", D, dir((0.872, 1.746))); dot("$G$", G, dir((0.959, 1.882))); // dot("$X$", X, dir((0.829, 1.797))); // dot("$Y$", Y, dir((0.788, 1.825))); // dot("$E'$", (0.75890,-0.65120), dir((0.979, 1.691))); [/asy]](http://latex.artofproblemsolving.com/4/8/5/485fd79f24a032c98f3759833c36f6101c1ab39e.png)
which has degree 
. On the other hand, 
has degree 
. Hence, it suffices to verify the result for 
points, in order for the rotation of 
by 
to coincide with 
. be the foot of the altitude and the midpoint of each of the three sides. (Proof should be written out in a contest, but here it is omitted.)
. I also had posted a proof but it was quite long. I was also surprised that the solution using complex numbers is quite concise compared to the purely geometric method, which is a good example for complex coordinates. I would like to contribute with a general problem. For my original general method, you can see the topic An extension of Nine-point circle. be a triangle inscribed in circle . Points and lie on such that 
. A circle 
passes through , 
and meets sides 
, at , 
, respectively. 
meets 
at . Let 
be the isogonal of with respect to triangle . is a point on line such that 
. Prove that line 
goes through a intersection of 
and 
.
be a triangle inscribed in circle . Points and lie on such that . A circle passes through , and meets sides , at , , respectively. meets at . Let be the isogonal of with respect to triangle . is a point on line such that . Prove that line goes through a intersection of and .
intersect 
for the second time at and let 
intersect at . it is obvious that 
so 
is cyclic we only need to prove that 
is cyclic.
then it is easy to prove that 
is a line (you can simply use moving point here and check on the lines 
,
and ). is cyclic we have to prove that is the miquel point of 
. So then the requirement is that 
. this is trivial by 
and 
. so we are done.
with 
? I think it must be 
?
?
and 
Prove that
