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.
and 
, of equal radius intersect at different points 
and 
. Consider a circle 
externally tangent to at 
and internally tangent to at point 
. Prove that lines 
and 
intersect at a point lying on .
be inscribed in the circle 
. A line through point 
intersects 
and 
at points 
and 
, respectively. Let 
be the reflection of across the midpoint of , and 
be the reflection of across the midpoint of . Prove that:
of triangle across line 
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 
. Is this possible to make the number 
on the board?
and 
.
?
lies inside triangle such that 
and 
. Point is the midpoint of segment 
. Point lies on segment with 
. Prove that 
.
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 a triangle, and let 
be its altitudes. Let be its orthocenter, and let 
and 
be the circumcenters of triangles 
and 
. Let 
be the second intersection of the circumcircles of triangles 
and 
. Prove that the lines 
, 
, and 
-median of 
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.
with center 
touch and at and respectively. 
and 
intersect at 
and 
respectively. Prove, that circumcircle of 
touch incircle of iff 
.
touch at 
Since 
it follows that 
passes through the circumcenter of 
Therefore, if 
is tangent to through 
then 
is the antipode of 
WRT 
cut again at 
Then we have 
Hence, 
is the antipode of WRT the incircle 
is the Nagel point of 
Now, it is well-known that lies on the small arc of if and only if 
implies that the antipode of WRT coincides with the Nagel point of Then 
coincides with (orthogonal projection of C onto BI) is the antipode of WRT the circle is tangent to through 
where 
are the incenter and centroid of Further, it's the antipode of 
of lies on the A-sideline of its incentral triangle. is equidistant from and the midpoint of 
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)
, Can someone help me with finding some pages about it ?