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 
.
and 
by 
,
,
for 
.![\[(a_n - a_{n-1})(b_n-b_{n-1})<0.\]](http://latex.artofproblemsolving.com/a/5/c/a5c8844d595bb1186d4804beadd69a3bc1d40ea3.png)
is marked on side 
of parallelogram 
,
- point 
so that the circumscribed circle of 
is tangent to line segment 
.
is tangent to line 
.
be a scalene acute triangle with incentre 
and circumcentre 
.
cross 
.
and 
be the mid-arc points of 
,
cross 
at 
cross 
at 
.
and 
are concurrent on the external bisector of 
.
Y by
Let 
be a positive integer. In an 
board, two opposite sides have been joined, forming a cylinder. Determine whether it is possible to place queens on the board such that no two threaten each other when:
.
.
be a positive integer. In an 
board, two opposite sides have been joined, forming a cylinder. Determine whether it is possible to place queens on the board such that no two threaten each other when:
.
.This post has been edited 2 times. Last edited by EmersonSoriano, Apr 5, 2025, 5:25 PM
Reason: change
Reason: change
