[/quote]
,
rows and 
columns. Each point is coloured with one of the colours red, blue or green. Show that no matter whatever way the colouring is done, there always exist four points
and 
Show that
Let 
Show that
Let 
Show that
,
,
are decimal digits. Find the necessary conditions in 
be a positive integer. Show that there exist a polynomial 
with integer coefficient that satisfy the following
for each 
,
holds for all positive reals 
and for every positive real 
,
,
be a polynomial where 
and 
are coprime integers in the set 
,
such that for every positive integer 
,
.
there exists a positive integer 
.
.![\[\sum ab +\sum ab^2 \le 6\]](http://latex.artofproblemsolving.com/9/f/6/9f632ab1e6df5029bcd4416eb06623123ad2cd8c.png)
be a triangle with 
.
we consider points 
such that 
and 
.
be the circumcenter of triangle 
and
,
be the points of intersection of the lines 
,
and 
,
respectively. Let 
be
,
the circle with center 
,
the circle
.
.
unit in any of the directions, left, right, up or down. Find the number of ways to go from 
to 
in 
steps. (Note: Two paths where identical set of points is traversed are considered different if the order of traversal of each point is different in both paths.)
be positive real numbers. Prove the inequality![\[
\frac{x_1 + 3x_2}{x_2 + x_3}
\;+\;
\frac{x_2 + 3x_3}{x_3 + x_4}
\;+\;
\frac{x_3 + 3x_4}{x_4 + x_1}
\;+\;
\frac{x_4 + 3x_1}{x_1 + x_2}
\;\ge\;8.
\]](http://latex.artofproblemsolving.com/5/c/0/5c062a37f6ff120232ae87eb0093a4ce73ee0eb1.png)
be a function satisfying the following conditions for all 
:![\[
\begin{cases}
f(n+1) > f(n) \\
f(f(n)) = 2n + 2
\end{cases}
\]](http://latex.artofproblemsolving.com/0/4/7/047e28c9d899d7de443ae40ac2c756fda851fafb.png)
Find the value of 
.
is the foot of the altitude from 
of triangle 
and 
points 
and 
are marked such that the circumcircles of triangles 
and 
are tangent, call this circles 
and 
respectively. Tangent lines to circles 
.
.
Y by Kizaruno
Let ABC be an isosceles triangle with AB = AC and M the midpoint of BC. Consider a point E outside the triangle such that BE = BM and CE perpendicular to AB. The point of intersection of the perpendicular bisector of segment EB with the circumcircle of triangle AMB, which is on the same side as A with respect to BE, is point F. Show that angle FME = 90°
This post has been edited 3 times. Last edited by AlexCenteno2007, Apr 17, 2025, 5:45 PM
Reason: Error
Reason: Error
,
.
intersects the circle 
in the point 
;
.
intersects the circumcircle 
.
,
