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 the orthocentre of an acuteangled triangle 
. A point 
is taken on the line segment 
such that 
is a rightangled triangle. Prove that the area of the triangle is the geometric mean of the areas of triangles and 
.
is the orthocentre of an acuteangled triangle . A point is taken on the line segment such that is a rightangled triangle. Prove that the area of the triangle is the geometric mean of the areas of triangles and .
meet 
at 
.Now the given question is equivalent to proving 
=
×
is also the geometric mean of 
and 
.( because it is right angled and is altitude).So, =××=×, which is true because triangle 
is similar to triangle 
.
we 
similarly 
, WE know ![$[ABC]= 1/2 ab sin C$](http://latex.artofproblemsolving.com/1/b/4/1b4caab0705c9386d07f95188c25cd2866701a58.png)
and ![$[ABH]=2R^2cos A cos B sin C $](http://latex.artofproblemsolving.com/9/8/5/9858949f3a1ce89dd85587bc81cdcb6a96366e7b.png)
,![$\sqrt([ABH]*[ABC])= R sin C \sqrt (ab cos A cos B)$](http://latex.artofproblemsolving.com/0/8/5/085c4bc30b5530b7e3aec453c025bb81a2a99cb4.png)
= c /2* \sqrt(AD*DB)= c/2* DE = [ABE]
& 
.
