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is described on a circle 
, and is not a trapezius inscribed in a circle. Let the tangency points of and 
be 
respectively. A circle with a center 
, different from is tangent to the segement 
and lines 
. A circle with center 
, different from is tangent to segment 
and lines 
. A circle with center 
, different from is tangent to segment 
and lines . A circle with center 
, different from is tangent to segment 
and lines . Prove that the lines 
are concurrent.
define 
be the product of first 
primes. Determine if there exists positive integers 
with the following property :![\[ m?=n(n+1)(n+2)(n+3) \]](http://latex.artofproblemsolving.com/4/5/c/45cb4a00f9395b8c6edb08a343c3bbd664916a4d.png)
be reals such that 
and 
Prove that
Let be reals such that and 
Prove that
Let be reals such that and 
Prove that
, 
. A circle passing through 
and 
intersects at 
. Angle bisector from intersects the circle at 
. (different from ) Prove that orthocenter of 
is on this circle.
, . A circle passing through and intersects at . Angle bisector from intersects the circle at . (different from ) Prove that orthocenter of is on this circle.
at 
.We claim that is the orthocenter of 
.Let 
meet at 
and 
meet at 
.Since,
,it is enough to prove 
cyclic.Notice that 
,so is cyclic and is the desired orthocenter.
satisfying![\[\begin{aligned}
\begin{cases}
a+b+c+d=0,\\
a^2+b^2+c^2+d^2=12,\\
abcd=-3.\\
\end{cases}
\end{aligned}\]](http://latex.artofproblemsolving.com/8/b/1/8b1923e0303c0aa79332ee4540ee1885a9cf3f08.png)
