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.
and 
such that
be reals such that 
and 
Find the value of 
players take part. Every two play exactly one match. The schedule is such that no two matches are played at the same time, and each player, after taking part in a match, is free in at least 
next (consecutive) matches. Prove that one of the players who play in the opening match will also play in the closing match.
and 
Find the value of 
be a triangle and 
, 
, 
be the midpoints of arcs 
, 
, 
on the circumcircle. Line 
passes through the feet of the perpendiculars from 
to 
and 
. Line 
passes through the feet of the perpendiculars from to and 
. Let 
denote the intersection of lines and . Define points 
and 
similarly. Prove that triangle 
and 
are similar to each other.
such that : 
and 
table filled with natural numbers, we say it is a divisor table if:
-th row are exactly all the divisors of some natural number 
,
-th column are exactly all the divisors of some natural number 
,
for every 
.
is given. Determine the smallest natural number 
, divisible by , such that there exists an divisor table, or prove that such does not exist.
. Determine the smallest Norwegian number.
.)
, prove that![\[\frac{a^3}{(b-c)^2+bc}+\frac{b^3}{(c-a)^2+ca}+\frac{c^3}{(a-b)^2+ab}\ge a+b+c.\]](http://latex.artofproblemsolving.com/1/c/8/1c80f369f5e5daf2155c9620dff0463ed8d284a0.png)
be a triangle with circumcenter 
and internal bisector 
. Let cuts 
again at 
and 
cuts again at 
. Point 
lie on such that 
. The line pass through and perpendicular to intersects 
at 
respectively. Let circumcircle of 
cuts at 
. Prove that 
.
eggs, each with a hardness value 
, where 
is a permutation of the set 
. He is throwing the eggs from an 
-floor building.-th egg is dropped from the -th floor, its new hardness becomes![\[
\left\lfloor \frac{k_i}{j+1} \right\rfloor.
\]](http://latex.artofproblemsolving.com/7/5/8/7583af9655a43eb3f6167f955148984a468d4e70.png)
If 
, then the egg breaks and cannot be used again. and can Arthur always determine the correct ordering of the eggs according to their initial hardness values?
be a right-angled triangle with 
and let 
be the altitude from 
on to . Draw 
and 
. Let 
and 
be respectively the incentres of triangle 
and 
. Suppose 
are collinear. Prove that 
lie on a circle.
?
.
centered at taking triangle 
to 
.
and thus the corresponding incenters will also be mapped to each other that is
maps line 
to 
.
.
are collinear we get 
are concyclic
.
so it suffices to show that 
.
and 
.
,
as desired.