[/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, Mango247
Circles ω1 and ω2 have no common point. Where is outerior tangents a and b, interior tangent c. Lines a, b and c touches circle
ω1 respectively on points A1, B1 and C1, and circle ω2 – respectively
on points A2, B2 and C2. Prove that triangles A1B1C1 and A2B2C2
area ratio is the same as ratio of ω1 and ω2 radii.
ω1 respectively on points A1, B1 and C1, and circle ω2 – respectively
on points A2, B2 and C2. Prove that triangles A1B1C1 and A2B2C2
area ratio is the same as ratio of ω1 and ω2 radii.
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This post has been edited 2 times. Last edited by TautasJ, Mar 21, 2017, 4:21 PM
.
that carries 
.
is the antipode of 
and so 
but 
but as the ratio of the homothety is equal to the ratio of the radii we're done
and 
have no common point, have externals tangents 
.
and 
,
and 
,
,
and 
and 
is the same as the ratio of radii of 
