AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
, inscribed in the circle 
, satisfies 
and 
, and 
is the intersection point of the diagonals 
and 
. The circle with center 
and radius 
intersects in two points 
and 
. Prove that the line 
is tangent to the circles with diameters 
and 
.
be a triangle inscribed in a circle, where the tangents to the circle at points 
and 
intersect at the point 
. Let 
be a point on the arc (not containing ) such that 
and 
. Let the lines 
and 
intersect at point 
. Let 
be the reflection of with respect to the line . The lines 
and 
intersect at point 
, and intersects the circumcircle of again at point 
. lies on the circumcircle of 
.
be non-negative real numbers such that 
Prove that
such that 
and 
for any real numbers 
and 
.
is written on the blackboard. Every time, Carmela can erase the number 
on the black board and replace it with a new number 
, if and only if 
is a perfect square. Prove or disprove that all positive integers 
can be written exactly once on the blackboard.
and 
as ![\[\sqrt{2}=1.a_1a_2a_3\dots\quad\quad\sqrt{3}=1.b_1b_2b_3\dots\]](http://latex.artofproblemsolving.com/4/d/5/4d5049d5c3b26757a25c5dd34462c5462228b83d.png)
where each 
or 
is a digit between 0 and 9. Prove that there exist at least 1000 values of 
between 
and 
such that 
.
of positive integers with 
is called good if we can colour each integer red, blue, green or purple, in such a way that of each 
consecutive integers at least one is coloured red;
of each 
consecutive integers at least one is coloured blue;
of each 
consecutive integers at least one is coloured green;
of each 
consecutive integers at least one is coloured purple.
. From the fourth term of the sequence onwards, each term of the sequence is the average of the previous three. Given that the first six terms of the sequence are natural numbers and that the sixth number written was 
, what is the fifth term of the sequence?
points in the plane, no three of them belonging to a same line. Each of these points is colored using one of four given colors. Prove that there is a color and at least 
scalene triangles with vertices of that color.
such that :
for all prime numbers 
.
for all positive integers 
such that 
, draw exterior equilateral triangles on sides 
and to obtain 
and 
, respectively. Let 
be the intersection of the altitude through and the median through . Let 
be the intersection of the altitude through and line 
. Let 
be the intersection of the median through and the line 
. Prove that , , and lie on a common line. be an even positive integer. A sequence of real numbers is called complete if for every integer with 
either the sum of the first terms of the sum or the sum of the last terms is integral. Determine numbers.
are positive real numbers satisfied 
. Prove that:
are cyclic.
and 
are collinear.
and 
are collinear. and are collinear, we can say that 
are collinear. 
be orthocentre of triangle . Note that 
and 
are cyclic so by Simson's theorem on 
and 
: 
and 
are collinear.
with radius , and denote inverses with a 
.
and 
, so that 
. Moreover, from 
collinear, we get that 
, hence 
is a rectangle. From 
, we get the desired cyclicity, so inverting back yields collinear
and 
are tangent at point . Similarly 
and are tangent at point . Now using homothety wrt points and we find that 
It's obvious that and are tangent at and have common tangent line in 
. Now using this we have: 
. Hence 
are concyclic. Now again from the power of point and Thales' Theorem we have that 
. Now from 
, are collinear.
be collinear points, such that is between and . Let 
be circles with diametar 
respectively. Let 
be points on 
, both on the same side of . Let 
, 
, 
, 
. Then prove that are collinear 
are concyclic 
, where 
and are tangent at point ?
and are tangent at point ? with radius 
. Then and are each other's inverses, whiles also and are inverse pair. Now let 
be the inverse of 
. Since lies on the polar of , going through , must lines on the polar of , which is the perpendicular to at . So by inverting through the mentioned circle. goes to 
, while goes to . But they are parallel and hence they have only one intesecting point (at infinity), hence they are tangent at . Otherwise the lines would have to intersect at another point. by proving that they have a common tangent line at . Draw the tangent line to at C. Then the angle between the chord 
and the tangent line is 
. Hence by the Tangent and Chord Theorem they share a common tangent line, hence they are tangent to each other. There are lots of other way to prove this. It's easy to prove that is the center of homothety of the two circles, hence they have only one touching point.
and are tangent at point ?
and 
....So by the 
we have and are collinear.(My 100th post!!!!!!! 
)
satisfying the equation![\[
(f(x+y))^2= f(x^2) + f(2xf(y) + y^2), \quad \forall x, y \in \mathbb{R}.
\]](http://latex.artofproblemsolving.com/2/5/e/25eed911f303ed7cc4e03765a2f943ad4641c451.png)
and 
and 
and 
and 
and 
respectively. Prove that 
.