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 
and at most one of them being zero, show that![\[\frac{1}{\max(a^2,b)}+\frac{1}{\max(b^2,c)}+\frac{1}{\max(c^2,a)}\geq\frac{27}{4}\]](http://latex.artofproblemsolving.com/9/e/1/9e14a9ef2d1a0a98006bcf716fa33c3d9ea29bd5.png)
+1 w
be an integer. Find all sequences 
satisfying the following conditions:![\[ \text{ (a) } a_i \in \left\{0,1\right\} \text{ for all } 1 \leq i \leq n^2 + n;
\]](http://latex.artofproblemsolving.com/3/c/5/3c509ec2e9013e8d3be492c8eb44a7c33841b74e.png)
![\[ \text{ (b) } a_{i + 1} + a_{i + 2} + \ldots + a_{i + n} < a_{i + n + 1} + a_{i + n + 2} + \ldots + a_{i + 2n} \text{ for all } 0 \leq i \leq n^2 - n.
\]](http://latex.artofproblemsolving.com/9/7/d/97d2a467d1c0dc8594ec024c3bb9b8c87ee85b19.png)
and 
. Prove that
Equality holds when 
or 
and 
, Ivica and Marica play the following game: Given two piles of cookies, on each turn a player takes 
cookies from one of the piles, of which he eats 
and puts of is arbitrary in every move. Players take turns alternatively, with Ivica going
for which Marica has a winning strategy.
be non-negative real numbers such that 
Prove that
be the circumcenter, symmedian point of 
. Show that the tangent of 
at 
,the tangent of 
at 
, the tangent of 
at 
are concurrent.
be distinct natural numbers each of which has exactly 
positive integer factors. For instance, 
has exactly positive integer factors 
. Assume that no prime larger than 
divides any of the 
's. Show that there must be some perfect cube among the 's.
be an integer. Prove that 
can be written as a polynomial with integer coefficients in the variables 
, 
and 
.
, where 
are real numbers such that 
is an integral multiple of 
. Using (a) or otherwise show that if 
, then 
for all positive integers .
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.
with real coefficient. It is known that 
and 
. Prove that 
can't have 
distinct real roots.
is given. Prove that 
, where 
is the radius of the circumscribed circle, 
is the radius of the inscribed circle, is the length of the longest side, 
is the length of the shortest altitude.
is given. Prove that , where is the radius of the circumscribed circle, is the radius of the inscribed circle, is the length of the longest side, is the length of the shortest altitude.
is semiperimeter and is area.)
is the length of the longest side is the length of the shortest altitude.?????
?? if 
?? then 
; with 
, but 
isn't the length of the shortest altitude, too. 
, having the longest side implies 
is the shortest altitude. 
& 
.
and 
,![\[2R\ge a\]](http://latex.artofproblemsolving.com/f/2/6/f26e2a8e7e54939383edd5c96daafe79594880be.png)
,
.![\[Rs>\frac{a^2}{2}\]](http://latex.artofproblemsolving.com/1/7/5/1753963d4741a08594c72172972a70112d32b369.png)
,![\[\frac{R}{r}>\frac{a}{h}\]](http://latex.artofproblemsolving.com/7/8/a/78aedffa31c8e82f6aa8f0f81993541689f3f727.png)
.
.
which implies 
.
,
.
.
)
. So we need to prove
.