be an integer, let 
be a set of 
elements, and let 
,
such that ![\[ A_i \cap A_j \not= \emptyset, A_i \cap A_k \not= \emptyset, A_j \cap A_k \not= \emptyset, \;\textrm{imply}\ \;A_i \cap A_j \cap A_k \not= \emptyset \ .\]](http://latex.artofproblemsolving.com/c/1/1/c113f5cdce00dfec6714c382a8e1bc74f188f025.png)
Show that 
.
such that 
or 
,![\[
1 \leq a_1 < a_2 < \cdots < a_{(p-1)/2} < p
\]](http://latex.artofproblemsolving.com/6/8/e/68e3d73edec43b2cc480832adf23bbbb5f4c7bee.png)
such that![\[
\prod_{\substack{1 \leq i < j \leq (p-1)/2}} (a_i + a_j)^2 \equiv 1 \pmod{p}.
\]](http://latex.artofproblemsolving.com/2/9/e/29edf6ab8cee2a526e7753a140c5a291e987856a.png)
denote the maximum number of 
-
board such that each 
such that![\[
f(N) \leq cN^2
\]](http://latex.artofproblemsolving.com/7/4/8/748109c3d63203f9a67a167434e10a909f38e83f.png)
for all positive integers 
.
be a graph colored using 
colors. We say that a vertex is forced if it has neighbors in all the other 
colors.
-
colors such that at least 
of the colors have a forced vertex of that color.
vertices of the same color may be adjacent.
.
Let 
.
Let 
.
Let 
such that
Proposed by Ngo Van Trang, Vietnam
s.t. 
and 
be positiv integer numbers.
,
and 
,
and 
.
.
be a positive integer.Prove that there is a polynomial 
with integer coefficients so that 
,![$$a^{2n+1}+b^{2n+1}+c^{2n+1}=abc[P(a,b)+P(b,c)+P(a,c)]$$](http://latex.artofproblemsolving.com/5/2/0/520813bebb2f297928208f7b7b2eea76d24043ac.png)
.
.
Y by Adventure10, Mango247
Let 
be an interior point of the convex quadrilateral 
. Construct triangles 
and 
on the outside of the quadrilateral such that the similarities 
and 
hold. Let 
and 
be the projections of on the lines 
and 
, respectively. Prove that if the quadrilateral 
is cyclic, then
![\[EF\cdot CD=EG\cdot DA=EH\cdot AB=EI\cdot BC.\]](//latex.artofproblemsolving.com/0/8/e/08e13ecbf07fb1e8950fa34e40cb7f08a8c5a7e1.png)
be an interior point of the convex quadrilateral 
. Construct triangles 
and 
on the outside of the quadrilateral such that the similarities 
and 
hold. Let 
and 
be the projections of on the lines 
and 
, respectively. Prove that if the quadrilateral 
is cyclic, then![\[EF\cdot CD=EG\cdot DA=EH\cdot AB=EI\cdot BC.\]](http://latex.artofproblemsolving.com/0/8/e/08e13ecbf07fb1e8950fa34e40cb7f08a8c5a7e1.png)
and 
are similar.
= 
= 
and 
are similar, we get:
= 
= 
= 
(from similarities given)
= 
= 
.
= 
.
= 
+ 
= 
= 
(from similarity of quadrilaterals 
+ 
= 
and 
,
,
,
and 
are cyclic.
we do the same for others, the claim follows.
,
and 
,
Similarly, 
,
and 
.
is clear since there is similar quadrilaterals. So let's prove that 
.
.
.
to 
.
to 
,
.
,
.
