[/quote]
,
be a positive integer. A Japanese triangle consists of 
circles arranged in an equilateral triangular shape such that for each 
,
,
,
row contains exactly 
circles, exactly one of which is coloured red. A ninja path in a Japanese triangle is a sequence of 
,
such that in each Japanese triangle there is a ninja path containing at least 
,
and 
.
is prime, 
with 
and 
be a function such that for every regular 
-
we have 
.
for all reals 
.
be a triangle 
)
are the altitudes of the triangle. The bisector of 
intersects 
at 
.
be a point such that 
and 
.
passes through the midpoint of 
.
& 
.
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 
for which Marica has a winning strategy.
be non-negative real numbers such that 
Prove that
Y by Adventure10
Acute-angled triangle 
with 
and let be 
it’s circumcircle. Diameters 
and 
are drawn. Circle 
interescts 
at 
. Circle 
intersects 
at 
.(
lies between 
and ). Prove that lines 
and 
intersect at circle 
.
by Evangelos Psychas (Greece)
with 
and let be 
it’s circumcircle. Diameters 
and 
are drawn. Circle 
interescts 
at 
. Circle 
intersects 
at 
.(
lies between 
and ). Prove that lines 
and 
intersect at circle 
.by Evangelos Psychas (Greece)
This post has been edited 2 times. Last edited by parmenides51, Dec 11, 2022, 6:32 PM
and 
are diameters of the circumference, we have 
such as 
.
and 
,
and 
are both rectangle and isosceles 
There’s a spiral similarity that takes 
and 
is the angle of the spiral similarity. Now if 
,
+ 
is 180
on 
.
and 
are collinear, which comes from easy angle chase.
)
