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.
. We call a distribution the act of distributing the integers from 
represented by tokens to players 
to 
so that they all have the same number of tokens in their urns.
beats 
when, when and each draw a token from their urn, has a strictly greater chance of drawing a larger number than . We then denote 
. A distribution is said to be chicken-fox-viper when 
What is 
satisfy 
. The circumcircle 
and the incircle 
of are tangent to the sides 
at 
, respectively. The line 
meets 
at 
and intersects 
at 
, while the line 
meets 
at 
. Construct the common external tangent 
(different from 
) to the incircles of the triangles 
and 
. Show that is parallel to the line 
.
is situated inside the parallelogram 
such that 
. Prove that 
.
inscribed in with being the midpoint of . The tangents at 
of intersect at 
. Let 
be the projection of onto 
. On the perpendicular bisector of , take a point 
that is not on and different from M. Circle 
intersects 
at 
. Lines 
and 
intersect at 
. Prove that 
is an isosceles triangle.
be a triangle with circumcircle 
. Arbitrary points 
on 
respectively. Circumcircle 
of triangle intersects at 
. 
intersects 
at 
. 
cuts and at 
respectively. Construct parallelogram 
. Prove that 
are collinear.
a grid of 9*9, we choose a little square in of this grid three times, we can choose three times the same.
is a strictly increasing sequence of positive integers such that the sub-sequences ![\[s_{s_1},\, s_{s_2},\, s_{s_3},\, \ldots\qquad\text{and}\qquad s_{s_1+1},\, s_{s_2+1},\, s_{s_3+1},\, \ldots\]](http://latex.artofproblemsolving.com/2/6/8/2686c6e4c1110bce322dd876aba37d5273bffe05.png)
are both arithmetic progressions. Prove that the sequence 
is itself an arithmetic progression.
: 
such :
be an interior point of a triangle and 
meet the sides 
in 
respectively. Show that ![\[ \frac{AP}{PD} = \frac{AF}{FB} + \frac{AE}{EC}. \]](http://latex.artofproblemsolving.com/7/5/9/759482bace109eb619aaea692625f7854faa9fec.png)
is tangent to the sides , 
and 
on the points , and , respectively. intersect the circumference on the point . Show that the line 
meet the segment 
at its midpoint if and only if 
.
Prove that
Let 
Prove that
Let 
Prove that
-gon. A move consists in choosing an edge of the -gon and swapping the two tokens placed at the endpoints of that edge. After a finite number of moves have been performed, it turns out that every two tokens have been swapped exactly once. Prove that some edge has never been chosen.
, 
, 
be positive real numbers such that 
. Prove that ![$$\sqrt[3]{(a^2+1)(b^2+1)(c^2+1)} \le \left(\frac{a+b+c}{3}\right)^2 + 1.$$](http://latex.artofproblemsolving.com/2/0/7/207934bca53047858a3820f37705171ff0abfd45.png)
given points 
and 
which are symmetric to its circumcenter 
with respect to 
and 
and the straight line 
which contains its altitude to 
meets the line 
at the orthocenter H. The circle through H centered meets the circle through H centered 
at C. The circle 
meets the perp line from C to at B.
