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.
be a positive integer. A Japanese triangle consists of 
circles arranged in an equilateral triangular shape such that for each 
, 
, 
, , the 
row contains exactly 
circles, exactly one of which is coloured red. A ninja path in a Japanese triangle is a sequence of circles obtained by starting in the top row, then repeatedly going from a circle to one of the two circles immediately below it and finishing in the bottom row. Here is an example of a Japanese triangle with 
, along with a ninja path in that triangle containing two red circles., find the greatest 
such that in each Japanese triangle there is a ninja path containing at least red circles.
, such that 
and 
.
such that 
is prime, 
with 
and ?
be a function such that for every regular 
-gon 
we have 
. Prove that 
for all reals 
.
be a triangle (
). 
are the altitudes of the triangle. The bisector of 
intersects 
at 
. Let 
be a point such that 
and 
. Prove that 
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 
. 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 midpoints of 
. Let 
be 
. By DIT on 
with line 
, we see that 
, 
, 
are involutions ( is parallel to ). Projecting this through 
, 
, 
, and 
are pairs of an involution.
and are corresponding midpoints so 
and 
are isogonal in 
. Clearly, 
and 
are also isogonal, hence this characterizes our involution, and 
, 
are isogonal as desired.
& 
.
parallel to 
of 
,
intersects 
.
,
.
,
.
and 
,
.
.
,
.![[asy]
import olympiad;import geometry;
size(6cm);defaultpen(fontsize(10pt));pen org=magenta;pen med=mediummagenta;pen light=pink;pen deep=deepmagenta;
pair O,A,B,C,D,mb,mc,P,Q,M,T;
O=(0,0);A=dir(120);B=dir(200);C=dir(340);path w=circumcircle(A,B,C);mc=midpoint(A--B);mb=midpoint(A--C);
P=intersectionpoints(w,mb--100mc-99mb)[0];Q=intersectionpoints(w,mb--100mb-99mc)[0];
T=intersectionpoint(line(B,C),perpendicular(A,line(A,O)));
M=extension(A,T,P,Q);
draw(A--B--C--cycle,deep);draw(w,deep);draw(A--T--B,deep);draw(M--Q,deep);
dot("$A$",A,dir(A));
dot("$B$",B,dir(B));
dot("$C$",C,dir(C));
dot("$P$",P,dir(P));
dot("$Q$",Q,dir(Q));
dot(mb);
dot(mc);
dot("$M$",M,dir(M));
dot("$T$",T,dir(T));
[/asy]](http://latex.artofproblemsolving.com/9/f/6/9f6f29c08c005290e335360abfeca9c13972c930.png)
