[/quote]
,
and 
.
lies on segment 
.
.
:![\[
1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2n + 1) \mid (4n + 3)(4n + 5) \cdot \ldots \cdot (8n + 3).
\]](http://latex.artofproblemsolving.com/3/1/a/31a0eac3c7ca298c8fa221083dc94e127e705784.png)
the number of positive integer 
satisfying 
is less than or equal to 
where 
is a constant independent of 
on the board. For each procces, the stutent deletes the number x and replaces it with either 
or 
or 
.
on the board?
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 
be its orthocenter, and let 
and 
be the circumcenters of triangles 
and 
.
be the second intersection of the circumcircles of triangles 
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 
.
.
such that
meets 
,
meets 
.
also moves on a fixed circle.
all of whose digits are odd.
Y by Adventure10, ImSh95, Mango247, and 1 other user
Let 
be an acute-angled triangle whose inscribed circle touches 
and 
at 
and 
respectively. Let 
and 
be the points of intersection of the bisectors of the angles 
and 
with the line 
and let 
be the midpoint of 
. Prove that the triangle 
is equilateral if and only if 
.
be an acute-angled triangle whose inscribed circle touches 
and 
at 
and 
respectively. Let 
and 
be the points of intersection of the bisectors of the angles 
and 
with the line 
and let 
be the midpoint of 
. Prove that the triangle 
is equilateral if and only if 
.
be the incenter. An angle chase shows that 
are concyclic, so, since 
,
.
is a right triangle, so 
.
,
,
,
.
.
,
.
so 
and 
are the midpoints of 
and 
and 
.
are concyclic.
are concyclic.
is equilateral 
be point of tangency of incircle with 
and let 
intersect 
is cyclic quad.
is cyclic.
as 
also
using this with 
intersection point of line 
and angle bisector of 
,
.
and 
denote the circles 
and 
respectively.
other than 
,
Therefore 
,
.
,
.
.
.
.
.
,
.
is isoceles, then
We also have,
which implies that
![[asy]
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dot(A,linewidth(4.pt)+ds); label("$A$",(6.071617213404682,6.134821814771412),NE*lsf); dot(B,linewidth(4.pt)+ds); label("$B$",(4.7034546399703165,0.7273221197689224),NE*lsf); dot(C,linewidth(4.pt)+ds); label("$C$",(11.18593921409981,0.8413356675551195),NE*lsf); dot(D,linewidth(4.pt)+ds); label("$D$",(5.110645882063878,3.0401683748603485),NE*lsf); dot((8.181455294172352,3.818544705827648),linewidth(4.pt)+ds); label("$E$",(7.879546328300093,3.528797865372622),NE*lsf); dot(X,linewidth(4.pt)+ds); label("$X$",(5.811014818464803,3.2844831201164855),NE*lsf); dot(Y,linewidth(4.pt)+ds); label("$Y$",(8.612490564068503,4.164016203038577),NE*lsf); dot(Z,linewidth(4.pt)+ds); label("$Z$",(7.960984576718805,0.6621715210339526),NE*lsf);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
[/asy]](http://latex.artofproblemsolving.com/2/f/9/2f914e1ab971259616b49f4c38961d8c7b2ab8ea.png)
