[/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 
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 
and 
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 
)
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
Let 
be a sequence that consists of an initial block of 
positive distinct integers that then repeat periodically. This means that 
are distinct positive integers and 
for every positive integer 
. The terms of the sequence are not known and the goal is to find the period . To do this, at each move it possible to reveal the value of a term of the sequence at your choice.
If is one of the first 
prime numbers, find for which values of there exist a strategy that allows to find revealing at most 
terms of the sequence.
be a sequence that consists of an initial block of 
positive distinct integers that then repeat periodically. This means that 
are distinct positive integers and 
for every positive integer 
. The terms of the sequence are not known and the goal is to find the period . To do this, at each move it possible to reveal the value of a term of the sequence at your choice.If
is one of the first 
prime numbers, find for which values of there exist a strategy that allows to find revealing at most 
terms of the sequence.This post has been edited 1 time. Last edited by Anto0110, Yesterday at 7:37 PM
Reason: 9
Reason: 9
into sets which can be computed as 
by A000110, and this 
work.
