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.
with real coefficients of degree greater than 
is given. Prove that there are infinitely many positive integers which cannot be represented in the form ![\[f(n+1)+f(n+2)+\cdots+f(n+k)\]](http://latex.artofproblemsolving.com/a/6/4/a648623198555fa749157c99a83eec8e74c98961.png)
where 
and 
are positive integers.
satisfies 
. Find the value of 
.
be a scalene triangle with incenter 
and incircle 
. Let the tangency points of to 
be 
respectively. Let the line 
intersect the circumcircle of at the points 
. Assume that 
lies between the points 
and 
. Let 
be a circle that passes through and 
and that is tangent to at the point 
which lies on different semi-planes with 
with respect to the line . Let intersect 
at points 
and 
and let the second intersection point of the circumcircle of and the circumcircle of 
be 
. Prove that the intersection point of 
and 
lies on the circumcircle of if and only if the intersection point of 
and 
lies on .
and are situated 
units apart on a level surface. Find the set of all points on the surface which are so situated that the angles of elevation of the tops of the poles are equal.
with 
odd, which satisfy the equation:
denote the set of all real numbers. Find all functions 
such that ![\[ f\left( x^{2}+f(y)\right) =y+\left( f(x)\right) ^{2}\hspace{0.2in}\text{for all}\,x,y\in \mathbb{R}. \]](http://latex.artofproblemsolving.com/8/5/c/85c37ee472f918eace12c8cd7bef6766e617d2aa.png)
(where 
are the digits and 
) we define the \textit{permutation sum} as the value 
For example the permutation sum of 20253 is 
Let 
and be two fivedigit integers with the same permutation sum.
.
pairwise distinct positive integer numbers 
, find the maximum possible number of equal numbers among the fractions of the form![\[
\frac{a_i^2 + a_j^2}{a_i + a_j}
\]](http://latex.artofproblemsolving.com/3/5/3/3537ddec6d793d3831ba82a1e13b98d43b1c29bc.png)
be the incircle of triangle . Line 
is parallel to side 
and tangent to . Line 
is parallel to side 
and tangent to . It turned out that the intersection point of and lies on circumcircle of 
, we have:
satisfy all conditions below:
for all 
for all 
for all
that are solutions of the equation 
.
satisfies the given constraint then so does 
, and we can apply the same argument to that sequence.
if and only if 
have different sign, so for any fixed value of 
, 
should have fixed sign (positive, negative, or zero). Clearly if 
then 
for all 
as well, else take 
and 
, and then the desired claim is true. Thus assume that no terms in the sequence are . Then if 
then 
is forced by comparing 
with 
, and by induction eventually either every term is at most or exactly other term is at most . We only care about the latter scenario, in which case we care only about the subsequence 
of terms greater than , which "behaves" the same way as a complete sequence in itself would. Thus assume that every term is greater than .
for each 
, so each 
is a positive, and it suffices to show is bounded below by a positive real. The condition (after reciprocating both sides) then becomes 
. This is then a linear recurrence whose solutions (over 
) are 
. But if 
then we eventually have 
which is forbidden, so 
and the are bounded below by 
. 
be a sequence of positive numbers satisfying, for any positive integers 
such that 
,
Show that there exist positive numbers 
so that 
for any positive integer 