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by ![\[\displaystyle f(x) = \left\{ \begin{array}{lr} x+\frac 12 & \text{if}\ \ x < \frac 12\\ x^2 & \text{if}\ \ x \ge \frac 12 \end{array} \right.\]](http://latex.artofproblemsolving.com/c/b/d/cbd9b412893774283a438296cdeb05b80cc878bf.png)
Let 
and 
be two real numbers such that 
. We define the sequences 
and 
by 
, and 
, 
for 
. Show that there exists a positive integer 
such that ![\[(a_n - a_{n-1})(b_n-b_{n-1})<0.\]](http://latex.artofproblemsolving.com/a/5/c/a5c8844d595bb1186d4804beadd69a3bc1d40ea3.png)
is marked on side 
of parallelogram 
, and on the side 
- point 
so that the circumscribed circle of 
is tangent to line segment 
. Prove that the circumcircle of triangle 
is tangent to line 
.
be a scalene acute triangle with incentre 
and circumcentre 
. Let 
cross at 
. On circle , let 
and 
be the mid-arc points of and 
, respectively. Let 
cross 
at and let 
cross 
at 
. Prove that the lines 
and 
are concurrent on the external bisector of 
.
+1 w be a triangle with a right angle at 
and let 
be the midpoint of the side 
. Point 
on side 
is such that 
. Prove that 
.
be a triangle with a right angle at and let be the midpoint of the side . Point on side is such that . Prove that .
over 
and consider triangle 
.