[/quote]
,
,
are parallel and the angles 
and 
are acute. Show that it is possible to divide the triangle 
into 4 disjoint triangle 
and the triangle 
into 4 disjoint triangles 
such that the triangles 
and 
are congruent for all 
.
is given in the plane, let 
be the set of points which lie strictly inside it. Also let 
be the set of circles in the plane. For a point 
,
be the reflection of 
in sides 
respectively. Define a function 
such that 
is the circumcircle of 
.
injective?
,
board, some squares are painted red. Let 
be the number of rows that have an odd number of red squares and let 
be the number of columns that have an odd number of red squares. Find all possible values of 
.
sides are marked on the blackboard. Ana and Beto play alternately, Ana begins. Each player, in turn, must do the following:
join two vertices with a segment, without cutting another already marked segment; or
be the set of positive real numbers. Determine all functions 
such that ![\[x \big(f(x) + f(y)\big) \geqslant \big(f(f(x)) + y\big) f(y)\]](http://latex.artofproblemsolving.com/e/7/1/e71b6e0d0b858eb46b81149f1e6be8c41e13d301.png)
for every 
.
friends, 
people are seated around a circular table in a fixed order. Ali places 
such that: 
.
be the foot of altitudes from 
onto the opposite sides respectively and 
to obtain 
.
such that 
is the incentre of 
.
and 
and 
respectively, then show that 
are collinear.
square is divided into unit squares. Is it possible to fill each unit square with a number 
in such a way that, whenever one places the tile so that it fully covers nine unit squares, the tile will cover nine different numbers?
be a positive integer. There're 
angles formed by intersection points with 
in any order.
if he goes through 
,
(or vice versa). Katappa colors two numbers 
with 
with the same color iff there's a path traced by Baahubali which touches the angles numbered 
colors will be used by Katappa to color all the numbers?
: In the picture attached below, total 
colors will be used to color the numbers in each set with same color 
.
will be colored with the same color because the red path traced by Baahubali touches the angles 
.
such that for any 
:
,
denote the number of solutions to 
,
such that 
for all 
infintiely many often.
.
such that 
Y by Adventure10, Mango247
Let 
and 
be two circles which intersect at points 
and 
. Consider 
another circle which cuts in 
, and it is tangent to in the point 
, and also tangent to 
in 
. Consider 
, and 
the symetric point of w.r.t 
. Find 
.
and 
be two circles which intersect at points 
and 
. Consider 
another circle which cuts in 
, and it is tangent to in the point 
, and also tangent to 
in 
. Consider 
, and 
the symetric point of w.r.t 
. Find 
.
is the radical axis of 
.
.
.
is right at 
