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![\[
4k^3 + 4k + 1
\]](http://latex.artofproblemsolving.com/f/3/5/f355e456cd47ecfdb13597d52fec3d848b67c594.png)
. Two subsets 
and 
of 
are said to be friends if the following conditions are true: equals the product of all elements from . that are friends such that each one of them contains at least 
elements.[/quote]
be a circle on the plane. Let 
and 
be circles which are internally tangent to at points and respectively. Let the centers of and be 
and 
respectively and let the intersection points of and be 
and 
. Assume that lies on the line 
. Let the common external tangent of and that is closer to point be tangent to the circles and at 
and 
respectively. Let the second intersection point of the line 
and be 
and let the second intersection point of the circumcircle of 
and be . Let the circumcenter of 
be 
and let the intersection points of 
and 
be 
. Prove that
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 .
people at a party. Prove that there are two people such that, of the remaining 
people, there are at least 
of them, each of whom either knows both or else knows neither of the two. Assume that knowing is a symmetric relation, and that 
denotes the greatest integer less than or equal to 
.
be a polygon formed by the edges of an infinite chessboard, which does not intersect itself. Let the numbers 
represent the number of unit squares that have exactly 
edges on the boundary of respectively. Find the largest real number 
such that the inequality 
holds for each polygon constructed with these conditions.
such that, for all 
,
, the function 
is defined as, 
and for all 
, if the prime factorization of is 
then 
. We have an arithmetic sequence 
. If for a positive integer 
, the sequence 
is also an arithmetic sequence, show that the sequence has to be constant.
such that![\[f(f(f(\frac{x+y}{2}))+x+y)=f(x)+f(y)+f(\frac{x+y}{2})\]](http://latex.artofproblemsolving.com/6/2/4/624afa78b68f810aaaecf0e1e2251047beeddc36.png)
for all rational numbers 
.
vertices, each edge has one of the colors 
, 
, or 
. For each 
, if the vertices can be divided into 
groups such that any two vertices connected by an edge of color 
are in different groups, find the minimum possible value of 
.
be given pairwise coprime positive integers where 
. Let 
be positive integers. We call 
to be a grandson of if and only if, for all possible piles of stones whose total mass adds up to and consist of stones with masses , it's possible to take some of the stones out from this pile in a way that in the end, we can obtain a new pile of stones with total mass of . Find the greatest possible number that doesn't have any grandsons.
, for which 
for all functions 
, who satisfy the equation 
for all non-negative real numbers and 
.
in term of a function 
such that 
Find the value of 
Find the value of 
Find the value of 
Prove that 
,
,
be defined as![\[
x=\frac{a}{b-c},\quad y=\frac{b}{c-a},\quad z=\frac{c}{a-b}.
\]](http://latex.artofproblemsolving.com/c/6/b/c6b2629dfafd73409a6d911d1dd984ff4c70a94c.png)
Then, the system of equations![\[
a - xb + xc = 0,
\]](http://latex.artofproblemsolving.com/d/9/b/d9b088945fa707c2bfd4cad278308b35642dbc49.png)
![\[
ya + b - yc = 0,
\]](http://latex.artofproblemsolving.com/1/6/e/16eace37b2c4c09aa59968e0d299e1f982e0213d.png)
![\[
-za + zb + c = 0,
\]](http://latex.artofproblemsolving.com/8/6/f/86f4f7da9c109d2621815fef961b7f16c5d164fc.png)
admits a nonzero solution 
.![\[
\begin{vmatrix}
1 & -x & x \\
y & 1 & -y \\
-z & z & 1
\end{vmatrix}
= xy + yz + zx + 1 = 0.
\]](http://latex.artofproblemsolving.com/3/a/1/3a10ec43739f905365053fb18558037bda2cce8e.png)
It follows that![\[
(x+y+z)^2=x^2+y^2+z^2+2(xy+yz+zx)=2-2=0.
\]](http://latex.artofproblemsolving.com/4/6/0/4605588d4b8c63e3de8fdbeec43959607c06c5c3.png)
