[/quote]
,
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 
.
?
lies inside triangle 
such that 
and 
.
is the midpoint of segment 
.
with 
.
.
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 
.
![$ a,b \in [0 ,1] . $](http://latex.artofproblemsolving.com/9/1/9/91980d5ca5d97fe4b61797e6fbb378211674d9cd.png)
Prove that
.
.
Y by Adventure10, Mango247
Let 
be an isosceles triangle with 
. Let 
and 
be points on line segments 
and 
, respectively, such that 
and such that 
is the internal angle bisector of 
. Prove that 
if and only if 
.
be an isosceles triangle with 
. Let 
and 
be points on line segments 
and 
, respectively, such that 
and such that 
is the internal angle bisector of 
. Prove that 
if and only if 
.This post has been edited 4 times. Last edited by parmenides51, Mar 20, 2022, 9:41 AM
and because of 
triangles 
are similar and 
cyclic.
then 
Hence, 
then 
...etc
