Interesting problem
by deraxenrovalo, Mar 23, 2025, 4:53 PM
Given 
with circumcenter 
Let 
be an arbitrary point on 
such that is outside 
Let 
be an arbitrary point on 
cuts 
again at 
and 
cuts 
again at 
The intersection of 
and 
is 
Let 
and 
be the intersection of 
with 
and 
respectively such that , , are pairwise distinct.
Show that :
, 
, 
are coaxial circles
with circumcenter 
Let 
be an arbitrary point on 
such that is outside 
Let 
be an arbitrary point on 
cuts 
again at 
and 
cuts 
again at 
The intersection of 
and 
is 
Let 
and 
be the intersection of 
with 
and 
respectively such that , , are pairwise distinct.Show that :
, 
, 
are coaxial circlesThis post has been edited 2 times. Last edited by deraxenrovalo, an hour ago
Reason: Error displayed
Reason: Error displayed
,
,
,
be 
integers. If 
is an integer, prove that the only solution to
is is 
.
with four distinct elements is special if there exists a rearrangement 
of 
such that 
.
cannot be partitioned into 
and ![$P_1(x),P_2(x), \ldots P_n(x) \in \mathbb{Z}[x]$](http://latex.artofproblemsolving.com/c/6/4/c64176e52d94032e571b4579e35f9e6adfde675a.png)
such that 
.
such that every equation: 
has no real roots
such that 
,
such that 
,
and 
e
are tangent to 
at 
and 
intersects line 
.
intersects 
intersects 
.
intersects 
.
.
such that 
is a perfect square but none of the integers 
are perfect squares.
be a cyclic quadrilateral an let 
The diagonals 
at 
The line through 
mmets the extension of the side 
beyond 
at 
The line through 
meets the extension of the side 
Prove that the circumcircles of the triangles 
and 
are tangent .
(first player) and 
,
chips from the remaining pile of chips, where 
. No one can skip a turn. The player who at some point is unable to make a move (cannot remove chips from the pile) loses the game. Who among the two players can force a win on this game?
July 2014
June 2014