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.
black and 
red cards. You pull out 
cards, one after another, and check their colour. If both cards are the same colour, then a black card is added to the deck. However, if the cards are of different colours, then a red card is used to replace them. Once the cards are taken out of the deck, they are not returned to the deck, and thus the number of cards keeps reducing. What is the likelihood the last card left in the deck is black?
be positive real numbers such that 
Prove that we have![\[(a+b)(b+c)(c+a)\geq 8.\]](http://latex.artofproblemsolving.com/2/a/6/2a63f19de634fce70ff4ebc02ad13d442a20c378.png)
be a scalene triangle with orthocenter 
and circumcenter 
. Denote by 
, 
the midpoints of 
, 
. Suppose the circle 
with diameter meets the circumcircle of at 
, and meets line 
at a point 
. The tangent to at 
meets line 
at 
. Show that the circumcircles of 
and 
intersect at a point 
on 
. with circumcenter . 
and 
intersect the altitude from 
to 
at points and 
respectively. 
is the circumcenter of triangle 
and 
is the reflection of over . 
is the second intersection of circumcircles of triangles 
and 
. Show that 
are collinear.
be a cyclic quadrilateral with 
. Let be the midpoint of the arc 
not containing . Suppose there is a point inside such that 
and 
.
, and are concurrent.
be the circumcenter of triangle . The altitudes from 
of triangle intersects the circumcircle of the triangle at 
respectively. 
meets 
at 
respectively. Prove that the circumcircles of triangles 
share two common points. with the midpoint of . Draw the perpendicular 
from the orthocenter of to 
.
.
be the orthocenter of an acute-angled triangle 
. The circle 
centered at the midpoint of 
and passing through intersects the sideline at points 
and 
. Similarly, define the points 
, 
, 
and 
., , , , and are concyclic.
be a diameter of a circle 
, and let 
be the tangent at 
. Furthermore, let 
be a fixed, positive real, and consider all pairs of points 
and 
lying on , on opposite sides of , such that 
. The lines 
and 
intersect at points 
and 
, respectively. Show that all the lines 
pass through a common point.
is the midpoint of the arc 
, containing 
, in the circumcircle of 
, and let 
be the -excircle's center. Assume that the external angle bisector of 
intersects at 
. Prove that 
is perpendicular to 
, where 
is the incenter of .
be 
. The tangents to at 
meet at point 
. For a point 
on line which is not on the segment , let the midpoint of 
be . Lines 
meet again at points 
respectively. Let be the midpoint of . Prove that the points 
lie on a circle.
be the Fermat point of a 
. Prove that the Euler line of the triangles 
, 
, 
are concurrent and the point of concurrence is , the centroid of .
until such that 
is equilateral now, taking as homothetic center the midpoint of (3:1) sending X to circumcenter of , P to centroid of and A to centroid of that's enough
, 
and 
are the bisectors of 
, 
cutting 
, 
at 
, respectively. If 
and 
, find the angles of 
, and are the bisectors of , cutting , at , respectively. If and , find the angles of
, be the midpoint of . Prove that
.
and 
are points on the circumcircle of such that 
is parallel to . Prove that 
is perpendicular to the Simson line of .
be the Fermat point of a . Prove that the Euler line of the triangles , , are concurrent and the point of concurrence is , the centroid of . then the problem is trivial by Liang-Zelich , consider the midpoint of 
, 
, as 
,let the centroid of be 
then we know 
so 
is parallel to 
which is also parallel to 
. now meets the point outside of , as 
such that 
is an equilateral triangle, and now by 
, where 
is the midpoint of we have 
is parallel to so 
lies on therefore it's done
be an acute triangle whose incircle touches sides and at and , respectively. Let the angle bisectors of 
and 
meet at and , respectively, and let the midpoint of be 
. Show that 
is equilateral if and only if 
.
are drawn perpendicular to the sides respectively of then prove that
Not been able to solve this even though I know I can do this.
is parallel to , show that tan 
tan 
.
but nvm be the orthocenter of and let be the circumcenter
from and be and 
then just use 
, is the circumcentre and is the orthocentre. Then, prove that 
.
be the antipode and then 
is a rhombus
is parallel to , show that tan tan . but nvm be the orthocenter of and let be the circumcenter from and be and then just use 
and 
Prove that
now they are the circumcenters of the equilateral triangle formed outside of 
,
,
is concurrent at 
.
,