[/quote]
,
such that:
Prove that
be a triangle with circumcenter 
.
be the midpoints of 
and 
respectively and let 
be the projection of 
.
be the projection of 
on 
.
intersect 
at points 
and 
.
intersct 
.
intersect 
at 
and 
intersect 
.
are concurrent.
and 
respectively be the incentre, circumcentre and circumcircle of triangle 
.
are chosen on 
,
is the foot of the altitude from 
to 
.
are similarly defined, prove that lines 
and 
concurr on 
.
and let 
and 
be tangency points of the incircle of 
,
and 
respectively. Let the circle with centre 
and 
.
,
and 
,
.
and 
be feet of perpendiculars from 
and 
respectively. Prove that 
,
and 
are concurrent.
be a sequence that consists of an initial block of 
positive distinct integers that then repeat periodically. This means that 
are 
for every positive integer 
.
prime numbers, find for which values of 
terms of the sequence.
such that, for all real numbers 
with 
,
s
of positive integers for which there exist positive integers 
and 
such that
holds for all integers 
(Note that 
denotes the greatest common divisor of integers 
and 
)
number. I will allow you to ask me questions. A question consists of you saying a three-digit number, to which I respond 'yes' if at least two of its digits are correct, or 'no' otherwise. (A digit is correct if it matches the corresponding digit in my number.)
on the diagonal 
of a given rectangle 
,
.
and the parallel line to 
that contains 
.
is equilateral if and only if 
![$P \in \mathbb R[x,y]$](http://latex.artofproblemsolving.com/8/a/0/8a0576299e976c51607959fb68c7c742b2a49c6a.png)
with ![$P \not\in \mathbb R[x] \cup \mathbb R[y]$](http://latex.artofproblemsolving.com/a/f/7/af71ba8c643d5bd8a2928fa6714ccd91713bd641.png)
and 
homeomorphismes of 
,
is an homoemorphisme too.
checkered board is cut into several squares with integer side length. The size of a square is it's side length.
Y by sami1618
In a complete graph with 
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 
.
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 
.![$38=\lceil 3\sqrt[3]{2025} \rceil$](http://latex.artofproblemsolving.com/a/d/5/ad50c8bbf464e861797791b2efd16b95d515bdac.png)
.
.
choices for the first partition, 
for the second, and 
for the third. Thus, there are a total of 
ways to assign a vertex across all three partitions.
,
and 
that are in the same group for all three partitions. However, this means the edge 
is not colored with ![\[
a_1 + a_2 + a_3 \geq \lceil 3\sqrt[3]{a_1 a_2 a_3} \rceil \geq \lceil 3\sqrt[3]{2025} \rceil \geq \lceil 30\sqrt[3]{2} \rceil \geq \lceil 30 \cdot \frac{5}{4} \rceil = 38.
\]](http://latex.artofproblemsolving.com/a/7/c/a7c395290f869ce557d583e198d864946788519d.png)
lattice points 
with 
and 
.
.
,
,
,
,
