AoPS Academy
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, 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 non-negative real numbers such that 
Prove that
satisfying![\[1=\frac{b_1}{1^2} > \frac{b_2}{2^2} > \frac{b_3}{3^2} > \frac{b_4}{4^2} > \dotsb\]](http://latex.artofproblemsolving.com/a/e/6/ae65e3d53931fe9d60d0de76dc79540ccf44c26e.png)
and let 
denote the largest real number satisfying 
for all positive integers 
. What are the possible values of across all possible choices of the sequence 
?
such that for all any 
be positive real numbers such that 
. Prove
.
be a scalene triangle with circumcenter 
, incenter 
, and incircle 
. Let touch the sides 
, 
, and 
at points 
, 
, and 
respectively. Let 
be the projection of to 
. The line 
intersects the circumcircle of again at point 
. The circumcircles of 
and 
intersect again at points 
and 
respectively. Prove that the lines 
, 
, and 
are concurrent.
be positive real numbers such that : 
. Prove that : 
and 
such that divides 
. Show that there exists integers 
, 
, 
satisfying 
.
be an acute triangle with 
. Let 
be the centroid of triangle 
and let be the foot of the perpendicular from 
to side 
. The median 
intersects the circumcircle 
of triangle at a second point 
. Let 
be the point where intersects . The line 
intersects the circle at a point 
, such that lies between and . Prove that the points 
and lie on a circle. be an acute triangle with orthocenter 
and circumcenter . Let and be points on the line segments 
and respectively such that 
is a parallelogram. Prove that 
.
be the set of positive real numbers. Determine all functions 
such that ![\[x \big(f(x) + f(y)\big) \geqslant \big(f(f(x)) + y\big) f(y)\]](http://latex.artofproblemsolving.com/e/7/1/e71b6e0d0b858eb46b81149f1e6be8c41e13d301.png)
for every 
.
, and be pairwise distinct nonnegative real numbers. Prove that![\[
(a + b + c) \left( \frac{a}{(b - c)^2} + \frac{b}{(c - a)^2} + \frac{c}{(a - b)^2} \right) > 4.
\]](http://latex.artofproblemsolving.com/2/0/9/209f9807481e7a1e97f023bf65a603cb462c62cb.png)
(Karl Czakler)
that can be written in the form![\[
n = \frac{a^2 - b^2}{b},
\]](http://latex.artofproblemsolving.com/f/9/0/f90a3ce2e1b51d6e1cb53a15cad6f36c2b30492a.png)
where and are positive integers.
without further conclusions.
is possible for a function - this does not mean that 
.
or the "cancellation" of is called injectivity. You have to prove the function is injective first before cancellation.
from the equation 
in any way - so 
is wrong for that function as well. (thus is not injective)
without further conclusions. is possible for a function - this does not mean that . or the "cancellation" of is called injectivity. You have to prove the function is injective first before cancellation.
,we can divide by 
on both sides to get 
(or simply avoid pointwise trap).
.Let 
for 
.
gives 
.Take some 
such that 
.Then we have 
or 
.But 
gives that is even which means 
or 
, contradiction. Thus we are done.
be the assertion of the problem.
value we get that 
or 
(Just 
works).
and 
Prove that
such that the equality 
holds for all pairs of real numbers 
.
is an obvious solution. Now, let 
.
or 
.
Adding these two together we get:
