[/quote]
,
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 
are positive integers.
satisfies 
.
.
be a scalene triangle with incenter 
and incircle 
.
be 
respectively. Let the line 
intersect the circumcircle of 
.
lies between the points 
and 
.
be a circle that passes through 
and that is tangent to 
which lies on different semi-planes with 
with respect to the line 
at points 
and 
and let the second intersection point of the circumcircle of 
be 
.
and 
lies on the circumcircle of 
and 
lies on 
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 
)
For example the permutation sum of 20253 is 
Let 
and 
.
pairwise distinct positive integer numbers 
,![\[
\frac{a_i^2 + a_j^2}{a_i + a_j}
\]](http://latex.artofproblemsolving.com/3/5/3/3537ddec6d793d3831ba82a1e13b98d43b1c29bc.png)
is parallel to side 
and tangent to 
is parallel to side 
and tangent to 
,
satisfy all conditions below:
for all 
for all 
for all 
that are solutions of the equation 
.
![\[
2^a + 3^b + 1 = 6^c
\]](http://latex.artofproblemsolving.com/1/9/e/19eb5efa2ccdf54cd277339f8d3b99f75168779d.png)
Y by
Given a chess board 
with 
with all the unit squares are initially white coloured. Every move, we can turn the color (from white to black or otherwise) from the 5 unit squares that form this T-pentomino which can be rotated or reflexed (see the image below). Determine all natural numbers 
such that all unit squares on the board can be made into all black after a finite number of moves.
with 
with all the unit squares are initially white coloured. Every move, we can turn the color (from white to black or otherwise) from the 5 unit squares that form this T-pentomino which can be rotated or reflexed (see the image below). Determine all natural numbers 
such that all unit squares on the board can be made into all black after a finite number of moves.Attachments:
This post has been edited 1 time. Last edited by iStud, Yesterday at 4:05 AM
