[/quote]
,
be reals such that 
Show that
Let 
and 
Prove that
and 
Find the value range of 
and 
with real coefficients satisfying
such that![\[
4k^3 + 4k + 1
\]](http://latex.artofproblemsolving.com/f/3/5/f355e456cd47ecfdb13597d52fec3d848b67c594.png)
.
and 
of 
are said to be friends if the following conditions are true:
elements.[/quote]
be a circle on the plane. Let 
and 
be circles which are internally tangent to 
and 
respectively and let the intersection points of 
.
and 
respectively. Let the second intersection point of the line 
and 
and let the second intersection point of the circumcircle of 
and 
be 
and let the intersection points of 
and 
be 
.
be 
.
lies between the points 
and 
.
be a circle that passes through 
and that is tangent to 
which lies on different semi-planes with 
with respect to the line 
at points 
.
and 
lies on the circumcircle 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 
.
represent the number of unit squares that have exactly 
edges on the boundary of 
such that the inequality 
holds for each polygon constructed with these conditions.
such that, for all 
,
is defined as, 
and for all 
,
then 
.
.
,
is also an arithmetic sequence, show that the sequence 
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 
.
in term of a function 
such that 
Y by FreakLM10, MatBoy-123, Adventure10, Rounak_iitr
Let 
be an acute-angled triangle with 
. Let 
be the incentre of triangle , and let 
be the points where the incircle touches the sides 
respectively. Let 
meet the line 
at 
respectively. Further assume that both 
and 
are outside the triangle . Prove that
are concyclic.
is also the incentre of triangle 
.
be an acute-angled triangle with 
. Let 
be the incentre of triangle , and let 
be the points where the incircle touches the sides 
respectively. Let 
meet the line 
at 
respectively. Further assume that both 
and 
are outside the triangle . Prove that
are concyclic.
is also the incentre of triangle 
.
being the orthic triangle of 
and more angle chase in right-angled cyclic quadrilaterals
This was stupidly way too easy for RMO.
and 
.
.
to 
.
and 
,
which means that 
is cyclic. Similarly, we get that 
is also cyclic. Thus, 
This means that 
,
,
in 
,
bisects 
,
.
bisects 
,
.
.
,
is its orthic triangle, and so by a well known property, we get that 
,
.Done 
and 
are concylic.
.
thus 
.
.
is cycli
.
is cycli
is cyclic, as desired.
.
.
is cyclic 
.
is cycli
.
bisects 
bisects 
.
,
is cyclic. Thus, since 
,
as well. Now let 
.
is right, we have 
,
,
are collinear.
and so, 
is cyclic. Also, as 
are cyclic with 
by fact 5 we get 
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as 
has its orthocenter as 
and 
.
a homothety sending 
,
and we complete our first part.
(
is cyclic). Hence 
bisects 
and by Iran Lemma we have 
.
externally bisects 
i.e. internally bisects 
also bisects 
internally bisects 
in projective terms.
And since 
we get our desired conclusion
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, so we get 
, also we have 
, hence 
are concyclic points, hence 
are concyclic points , hence 
hence we have points 
concyclic 
and we also quadrilateral 
to be cyclic quadrilateral hence we have 
and similarly 
, by using the fact that quadrilateral 
is also cyclic , it is not hard to observe that 
hence we have 
.
are concyclic, So 
and 
.
,
.
,
(say). 
(say).
,
,
.
,
.
and 
are tangents to the incircle of 
.
(vertically opposite). Thus, by angle chasing in 
,
is cyclic. Part (i) proved.
,
is cyclic.
is cyclic 
.
is cyclic 
.
is cyclic and 
.
