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.
with circumcenter 
. Let 
be a point on 
such that 
. Denote 
the point on satisfying 
. If 
is the projection of onto 
, show that 
passes through the orthocenter of 
.
+2 w 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 circumscribed about circle 
, and let be the orthocenter of . The circle touches line at 
. The tangent to the circle 
at meets at 
. Let 
be the midpoint of 
, and let the line 
meet again at 
. The tangent to parallel to meets the line 
at 
. Prove that 
is tangent to .
be an acute-angled triangle, with circumcenter 
, circumradius 
and orthocenter . Let 
be a point on such that 
. Define 
and 
similarly.
, prove that is equilateral.
be an acute angled scalene triangle with circumcircle 
. Let and be the circumcenter and orthocenter of 
respectively. Let 
and be points on segments 
and , respectively, such that
Prove that 
and 
intersect on the circumcircle of 
.
we consider the last digit different from zero in the decimal expansion of 
. The infinite sequence of these digits starts with 
. Determine all digits which occur at least once in this sequence, and show that each of those digits occurs in fact infinitely often.
be a quadrilateral such that 
and 
are acute and 
. Suppose that 
, prove that 
.
, 
, and 
. Points and of the sides and respectively, are such that 
and 
. Calculate the lenght of 
.
, then jumps in some order on each of the numbers 
exactly once and finally returns with his last jump to . Can the total distance he travelled with these 
jumps be a) 
, b) 
?
be any point on side of .Let 
and 
be points on 
and such that 
and 
respectively. Prove that the locus of circumcenter of 
is a line.
. Put 
. Then 
, so 
. Note that if 
, then 
, which is impossible. By this reason we have 
and after putting this equality in 
we get 
. Since the condition is cyclic onto 
, we similarly get 
and 
. Note that if 
, then 
, so it is possible situation that are distinct (i.e. there is no equals roots of 
). By Viet's theorem we get 
.
. In this case we have 
and 
. Put 
, then 
, which is impossible (
by condition).
be distinct real numbers such that 
and 
Evaluate all the possible values of 
. be distinct real numbers such that 
and 
. Prove that
. First note that
Multiplying the above equations we get
hence 
. The next step is to note that
hence
. First note thatMultiplying the above equations we gethence . The next step is to note thathence
. First note thatMultiplying the above equations we gethence . The next step is to note thathence
. First note thatMultiplying the above equations we gethence . The next step is to note thathence
. Note that 
, so 
.
, such that 
Prove that :
equality cases : ?
such that ![$$\left(a^{3}+b^{3}+c^{3}-3abc\right)^{2}-\left[a^{3}+b^{3}+c^{3}+3abc-ab(a+b)-bc(b+c)-ca(c+a)\right]^{2}\ge 2k\cdot(a-b)^{2}(b-c)^{2}(c-a)^{2}$$](http://latex.artofproblemsolving.com/d/2/4/d24b46e65a587fbd977467b6e4e31838a0eb57aa.png)
forall 
