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.
board, 
pieces are placed (vertically or horizontally) so that they occupy entirely 
boxes on the board and do not have a common point.
, let 
be the sum of the digits of . Determines the smallest positive integer 
such that there are infinite positive integers for which you have 
.
, 
, 
be fixed positive integers. There are 
ducks sitting in a ducks ducks picking paper, and ducks picking scissors., , and , the maximum number of moves which could take
be the circumcenter of triangle 
. Line 
intersects the altitude from 
at point 
. Let 
be the midpoints of 
, 
respectively. If 
intersects 
at 
, and the circumcircle of triangle 
meets 
at 
, prove that 
is cyclic. be a triangle with incentre 
and circumcentre . Let 
be the touchpoints of the incircle with , 
, respectively. Prove that 
is the Euler line of 
.
with its incircle 
. Let 
and 
be the points of tangency of with and respectively. On the edges 
and 
, take points 
and 
such that 
is tangent to at 
. Suppose 
is the intersection point of the lines 
and 
. Prove that two lines 
and 
are parallel or coincide.
be a triangle with orthocenter 
and let 
be the midpoint of 
Suppose that and are distinct points on the circle with diameter 
different from 
such that lies on line 
Prove that the orthocenter of 
lies on the circumcircle of 
. Find all functions 
that satisfy
for all integers 
and rational numbers 
.
such that 
( is sum of digit of in base 10)
be a scalene oblique triangle with circumcenter and orthocenter , and (
, 
) a point in the plane.
, 
be the circumcevian triangles of , , respectively.
be the pedal triangle of with respect to .
be the reflection in 
of 
. Define 
, 
cyclically.
be the reflection in of 
. Define 
, 
cyclically.
, 
be the circumcenters of 
, 
, respectively., , are collinear if and only if lies on the Jerabek hyperbola of ., , are collinear if and only if lies on the Lemoine cubic (= 
) of .
is a triangle with incenter . The foot of the perpendicular from to is 
, and the foot of the perpendicular from to 
is . Prove that 
.
and 
, do not lie in one plane and intersect at a point . Let 
be center of sphere that passes through the points 
and . Prove that 
and
intersect at one point. (P.Kozhevnikov)
, then the circumcenter is at:![\[\frac{(\mathbf{a\cdot a})(\mathbf{b\times c})+(\mathbf{b\cdot b})(\mathbf{c\times a})+(\mathbf{c\cdot c})(\mathbf{a\times b})}{12V}\]](http://latex.artofproblemsolving.com/5/8/6/5867d30918b2da1262f68b5b88ea601b424e13e7.png)
is the volume.
, 
, and 
, where 
so that is at the origin. Then we have:
in space such that 
for three points 
is the line passing through the circumcenter of 
perpendicular to the plane containing .
passing through their midpoints, which is exactly the line described.
, 
lie on a plane 
. Then by the lemma 
for all 
, so 
for 
. This is true for planes 
and 
so in fact the solid created by all the circumcenters is just a rectangular prism. But we're done, since the space diagonals of a rectangular prism obviously intersect the center.
. Let 
be images of points 
respectively. Let spheres 
intersect by circles 
. To prove that lines 
are concurrent it's enough to prove that circles lie on the same sphere.
of planes 
and 
. Line is image of circle , so it's enough to prove that all lines lie on the same plane.
easy to see that lines , 
intersect at point 
. So any two line , lie on the same plane. So all lines lie on the same plane. 
![\[
\text{Find all functions } f : \mathbb{R} \to \mathbb{R} \text{ such that:} \\
f(a^4 + a^2b^2 + b^4) = f\left((a^2 - f(ab) + b^2)(a^2 + f(ab) + b^2)\right)
\]](http://latex.artofproblemsolving.com/2/a/7/2a73a5e58eab4a994fbd83160fb79a0b00152951.png)
are one of such 
