[/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 
.
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.
Y by Adventure10
Two circles 
and 
, both with radius 
, touch each other externally.
Four circles
and 
, all four with the same radius 
, lie such that
externally touches on 
and ,
externally touches on 
and 
,
externally touches on and ,
externally touches on 
and .
Calculate the length of
![[asy]
unitsize(0.3 cm);
pair A, B, P, Q, R, S;
real r = (3 + sqrt(17))/2;
A = (-1,0);
B = (1,0);
P = intersectionpoint(arc(A,r + 1,0,180), arc(B,r + 1,0,180));
R = -P;
Q = (r + 2,0);
S = (-r - 2,0);
draw(Circle(A,1));
draw(Circle(B,1));
draw(Circle(P,r));
draw(Circle(Q,r));
draw(Circle(R,r));
draw(Circle(S,r));
label("$A$", A);
label("$B$", B);
label("$P$", P);
label("$Q$", Q);
label("$R$", R);
label("$S$", S);
[/asy]](//latex.artofproblemsolving.com/4/c/7/4c719a0b71a3d884107a1362a46ac582aab3c484.png)
and 
, both with radius 
, touch each other externally.Four circles
and 
, all four with the same radius 
, lie such that
externally touches on 
and ,
externally touches on 
and 
, externally touches on and , externally touches on 
and .Calculate the length of
![[asy]
unitsize(0.3 cm);
pair A, B, P, Q, R, S;
real r = (3 + sqrt(17))/2;
A = (-1,0);
B = (1,0);
P = intersectionpoint(arc(A,r + 1,0,180), arc(B,r + 1,0,180));
R = -P;
Q = (r + 2,0);
S = (-r - 2,0);
draw(Circle(A,1));
draw(Circle(B,1));
draw(Circle(P,r));
draw(Circle(Q,r));
draw(Circle(R,r));
draw(Circle(S,r));
label("$A$", A);
label("$B$", B);
label("$P$", P);
label("$Q$", Q);
label("$R$", R);
label("$S$", S);
[/asy]](http://latex.artofproblemsolving.com/4/c/7/4c719a0b71a3d884107a1362a46ac582aab3c484.png)
This post has been edited 2 times. Last edited by nsato, Feb 14, 2023, 10:36 PM
denote the centres of circles 
and 
and 
is isosceles with apex 
is isosceles with apex 
so the angle 
is also 
it is equal to the sum of angles 
.
and 
.
gives 
and 
.
.
.
.
