[/quote]
,
,
is the angle bisector of 
.
in 
respectively and the line through 
in 
respectively. Prove that point 
lie on a circle.
kids in a class. Is it true that for all positive integers 
it is possible to create several clubs each with 3 kids such that any pair of kids are both present in exactly one club?
be an integer. In a configuration of an 
board, each of the 
cells contains an arrow, either pointing up, down, left, or right. Given a starting configuration, Turbo the snail starts in one of the cells of the board and travels from cell to cell. In each move, Turbo moves one square unit in the direction indicated by the arrow in her cell (possibly leaving the board). After each move, the arrows in all of the cells rotate 
counterclockwise. We call a cell good if, starting from that cell, Turbo visits each cell of the board exactly once, without leaving the board, and returns to her initial cell at the end. Determine, in terms of 
,
lamps 
in a row, each of them being either on or off. Every second we simultaneously modify the state of each lamp as follows: if the lamp 
and its neighbours (only one neighbour for 
or 
,
)
Prove that there are infinitely many integers 
for which all the lamps will eventually be off.
Prove that there are infinitely many integers 
integers 
,
,
be positive integers such that no one of them is divisible by any other member of the sequence. Then![\[
a_1 \geq 2^k,
\]](http://latex.artofproblemsolving.com/1/f/e/1fecb6d6811bfc4aff7ccefa20412f0a5b860066.png)
where 
is defined by the inequalities![\[
3^k < 2n < 3^{k+1}.
\]](http://latex.artofproblemsolving.com/2/8/f/28f12a76029727a828167f1cb8e118b84d710d62.png)
This estimate for 
is the best possible.
![\[
\max \left( (a_i, a_j) \right) > \frac{38n}{147} - c,
\]](http://latex.artofproblemsolving.com/6/5/d/65df2fa2c65d90d0548cb3c11dd36ea42c4060e6.png)
where 
is a constant independent of 
,
denotes the greatest common divisor of 
and 
.
satisfying 
and ![$ (x + 1)[f(x)]^2 - 1$](http://latex.artofproblemsolving.com/9/c/1/9c1e80d035c923dd13711bff624d8a1a3e6a3b15.png)
is an odd function.
be the inscribed circle of a triangle 
.
and 
be the tangency points of 
and 
and 
intersect 
and 
,
and 
intersects 
at 
,
intersects 
at 
, and the tangent line of 
at 
.
and 
is a prime and 
,![\[
n! \sum_{pi+j=n} \frac{1}{p^i i! j!} \equiv 0 \pmod{p}.
\]](http://latex.artofproblemsolving.com/1/6/6/166f07aa9e69c0eb35833971acb9b04932fea087.png)
Prove that
Let 
Prove that![$$ (a+b)(a+1)(b+1) \leq \frac{3}{2}+\sqrt[3]{6}+\sqrt[3]{36}$$](http://latex.artofproblemsolving.com/f/5/e/f5e68204507aa9c3e71c28844bbee3a804546b94.png)
,
,
,
.
points on the segment 
.
is a point on segment 
intersects 
and 
at 
and 
,
and 
intersect at 
and 
,
is cyclic.
Y by Adventure10, Mango247
Let 
be an acute triangle with orthocenter 
. Segments 
and 
intersect segments 
and 
in points 
and 
respectively. The segments 
and 
meet at point 
. Let 
be the midpoint of the segment . Let 
be the reflection of the point in 
. Prove that quadrilateral 
is cyclic.
Proposed by Matko Ljulj.
be an acute triangle with orthocenter 
. Segments 
and 
intersect segments 
and 
in points 
and 
respectively. The segments 
and 
meet at point 
. Let 
be the midpoint of the segment . Let 
be the reflection of the point in 
. Prove that quadrilateral 
is cyclic.Proposed by Matko Ljulj.
and let 
be the midpoints of 
,
is a kite, hence 
is a diameter of nine-point circle) ![\[MN\cdot MP=MA_1^2 = MA\cdot MC.\]](http://latex.artofproblemsolving.com/a/e/1/ae1c4758f64d531f537bb994d2b7d2610a7c43e9.png)
Denoting by 
the reflection of 
is cyclic (power of 
is cyclic.
is 
-
and let 
-
is cyclic 
)
(nine-point circle) and ![\[ XD\cdot XN = XA_1\cdot XM = XC_1\cdot XA_1 = XA\cdot XC.\]](http://latex.artofproblemsolving.com/7/2/2/7226c365fe482775c02f74a7ad03500ab74a5f01.png)
.
and 
is cyclic (and all those points lie onto the 9-point circle of 
)
.
and 
is symmetrical of 
w.r.t. 
intersect circle 
at 
and 
,
becomes 
,
we have 
;
,
.
are the polars of 
with respect to the circumcircle of 
.
and 
are similar thus
.
so 
.
.
,
is the orthocenter of 
,
.
be Cyclic.
as acute angles with perpendicular sides and since 
due to the isosceles triangle 
, 
and then its obvious.
are autopolar by Brocard's and 
We can observe that 
