[/quote]
,
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)
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 
.
Equality holds when 
or 
and 
,
cookies from one of the piles, of which he eats 
and puts 
for which Marica has a winning strategy.
be non-negative real numbers such that 
Prove that
be the circumcenter, symmedian point of 
.
at 
,
at 
,
at 
are concurrent.
be distinct natural numbers each of which has exactly 
positive integer factors. For instance, 
has exactly 
.
divides any of the 
'
be an integer. Prove that 
can be written as a polynomial with integer coefficients in the variables 
,
and 
.
,
are real numbers such that 
is an integral multiple of 
.
,
for all positive integers 
be the orthocenter of an acute-angled triangle 
.
centered at the midpoint of 
and passing through 
and 
.
,
,
and 
.
with real coefficient. It is known that 
and 
.
can't have 
distinct real roots.
be an acute triangle. Let 
be a point on side 
be a point on side 
are parallel. Let 
.
and 
meet side 
and 
,
and 
intersect at a point 
.
,
Y by
Points 
are midpoints of sides 
, 
and 
respectively of 
. Let 
and 
be two semicircles built using the sides and , respectively, as the diameters, to the outside of . Let 
and 
intersect the semicircles 
and at the points 
and 
respectively. The point 
is the point of intersection of the tangents and at the points 
and 
respectively. Prove that 
.
are midpoints of sides 
, 
and 
respectively of 
. Let 
and 
be two semicircles built using the sides and , respectively, as the diameters, to the outside of . Let 
and 
intersect the semicircles 
and at the points 
and 
respectively. The point 
is the point of intersection of the tangents and at the points 
and 
respectively. Prove that 
.This post has been edited 1 time. Last edited by parmenides51, Jul 14, 2021, 6:13 PM
