[/quote]
,
be inscribed in the circle 
.
intersects 
at points 
be the reflection of 
be the reflection of 
of triangle 
lies on the circle 
and 
coincide.
on the board. For each procces, the stutent deletes the number x and replaces it with either 
or 
or 
.
on the board?
satisfying
Prove
:![\[
1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2n + 1) \mid (4n + 3)(4n + 5) \cdot \ldots \cdot (8n + 3).
\]](http://latex.artofproblemsolving.com/3/1/a/31a0eac3c7ca298c8fa221083dc94e127e705784.png)
and 
.
?
and 
.
.
.
.
the number of positive integer 
satisfying 
is less than or equal to 
where 
is a constant independent of 
be different real numbers. Prove that![\[\sum_{1 \leqslant i \leqslant n} \prod_{j \neq i} \frac{1-x_{i} x_{j}}{x_{i}-x_{j}}=\left\{\begin{array}{ll}
0, & \text { if } n \text { is even; } \\
1, & \text { if } n \text { is odd. }
\end{array}\right.\]](http://latex.artofproblemsolving.com/6/5/2/6525fc6b451ff5a3c83ff2a0541b063a7cf065c3.png)
satisfying
for all real numbers 
and 
.
be its altitudes. Let 
and 
be the circumcenters of triangles 
and 
.
and 
.
,
,
are concurrent.
that satisfy the condition 
for the natural number 
Here 
and 
are natural numbers. Is there a natural number 
such that each of the numbers 
is the last digit of one of the numbers written by the teacher on the board? Justify your opinion.
we have 
.
.
Y by Adventure10, Rounak_iitr
Let 
be a cyclic quadrilateral. Let 
be the intersection of lines parallel to 
and 
passing through points 
and 
, respectively. The lines 
and 
intersect the circumcircle of 
again at 
and 
, respectively. Prove that points 
, 
, , and lie on a circle.
be a cyclic quadrilateral. Let 
be the intersection of lines parallel to 
and 
passing through points 
and 
, respectively. The lines 
and 
intersect the circumcircle of 
again at 
and 
, respectively. Prove that points 
, 
, , and lie on a circle.
.
is a parallelogram and so 
.
which gives the desired result.
cut 
at 
Then 
is cyclic 
is the image of 
under inversion with center 
and 
are pairs of inverse points under this inversion 
is cyclic.
be the tangent to the circumcircle of 
at 
,
,
is a parallelogramm, 
at 
cyclic.
