[/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.
be non-negative real numbers such that 
Prove that
Y by
Two circles inscribed in an angle with a vertex 
meet at points 
and 
. A line is drawn through A that intersects the smaller circle at point 
, and the larger one at point 
. It turned out that 
.
1. Prove that the tangents to the circles at point are perpendicular.
2. Let and coincide with the points of tangency of the circles and the angle. Prove that the 
is right.
3. Let and coincide with the points of tangency of the circles and the angle. Find the angle 
.
4. Prove that if is right, then and coincide with the tangency points of the circles and the angle.
5. Let
. The perpendicular from A to the nearest side of the angle intersects the smaller circle at point 
, the perpendicular from to the second side intersects 
at point 
. Finally, let 
and 
be the centers of the original circles, 
be the center of the circle circumscribed around 
. Prove that 
is the bisector of the angle 
.
6. What values can the angle
take, where is the center of the smaller circle?
Grade 9: 1-4 , Grade 10: 2-5 Grade 11: 2-4,6
meet at points 
and 
. A line is drawn through A that intersects the smaller circle at point 
, and the larger one at point 
. It turned out that 
.1. Prove that the tangents to the circles at point
are perpendicular.2. Let
and coincide with the points of tangency of the circles and the angle. Prove that the 
is right.3. Let
and coincide with the points of tangency of the circles and the angle. Find the angle 
.4. Prove that if
is right, then and coincide with the tangency points of the circles and the angle.5. Let
. The perpendicular from A to the nearest side of the angle intersects the smaller circle at point 
, the perpendicular from to the second side intersects 
at point 
. Finally, let 
and 
be the centers of the original circles, 
be the center of the circle circumscribed around 
. Prove that 
is the bisector of the angle 
.6. What values can the angle
take, where is the center of the smaller circle?Grade 9: 1-4 , Grade 10: 2-5 Grade 11: 2-4,6
This post has been edited 1 time. Last edited by parmenides51, Mar 5, 2021, 4:52 PM
,
are the reflections of 
,
,
yields that 
,
,
,
