by sealight2107, Apr 23, 2025, 2:40 PM 
be positive reals such that 
. Find the minimum and maximum value of by jaydenkaka, Apr 23, 2025, 2:22 PM by orangesyrup, Apr 23, 2025, 1:51 PM 
by Assassino9931, Apr 8, 2025, 1:59 PM 
be 
given points in the plane, and let 
be a real number. Alice and Bob play the following game. Firstly, Alice constructs a connected graph with vertices at the points , i.e., she connects some of the points with edges so that from any point you can reach any other point by moving along the edges.Then, Alice assigns to each vertex 
a non-negative real number 
, for 
, such that 
. Bob then selects a sequence of distinct vertices 
such that 
and 
are connected by an edge for every 
. (Note that the length 
is not fixed and the first selected vertex always has to be 
.) Bob wins if![\[
\frac{1}{k+1} \sum_{j=0}^{k} r_{i_j} \geq r;
\]](http://latex.artofproblemsolving.com/1/7/9/1795db8e4a509dd465f6ff462093ae75b04de2b4.png)
otherwise, Alice wins. Depending on 
, determine the largest possible value of 
for which Bobby has a winning strategy.by utkarshgupta, Apr 16, 2016, 5:54 AM 
be a quadrilateral inscribed in a circle 
. The lines 
and 
meet at 
, the lines 
and 
meet at 
, and the diagonals 
and 
meet at 
. Let 
be the midpoint of the segment 
, and let 
be the common point of the segment 
and the circle . Prove that the circumcircle of the triangle 
and are tangent to one another.
be the center of 
.
and 
.
be the reflection of in 
is a straight line (radical axis are concurrent).
is the polar od , 
and 
are straight lines..
and 
are cyclic quadrilaterals.
lies on the polar of ,
is a straight line; is the midpoint of 
and 
is the reflection of about , is the reflection of about 
is a cyclic pentagon.
and centre .
and 
maps to .
are tangent to 
;
and 
are tangent to 
.by SpectralS, Jul 10, 2012, 5:24 PM 
the point 
is the centre of the excircle opposite the vertex 
This excircle is tangent to the side at , and to the lines and at and 
, respectively. The lines 
and 
meet at 
, and the lines 
and 
meet at 
Let 
be the point of intersection of the lines 
and , and let 
be the point of intersection of the lines 
and 
Prove that is the midpoint of 
opposite the vertex 
is the circle that is tangent to the line segment , to the ray beyond 
, and to the ray beyond 
.)by Brut3Forc3, Apr 4, 2010, 2:45 AM 
, and 
are all polynomials such that ![\[ P(x^5)+xQ(x^5)+x^2R(x^5)=(x^4+x^3+x^2+x+1)S(x),\]](http://latex.artofproblemsolving.com/1/6/a/16a71abc110ece558d427a07d7be2d1f72148024.png)
prove that 
is a factor of 
.by rohitsingh0812, Jun 3, 2006, 5:39 AM 
, 
, 
, 
, 
, 
be positive integers and let 
.
divides 
and 
. Prove that is composite.Stay insane,Coz it's your will, labour and pain,which takes you to the top of the mountain.
September 2017
February 2017
September 2016
July 2016
April 2016
March 2016
December 2015
November 2015
August 2015
April 2015
March 2015
February 2015
January 2015
October 2014
September 2014
August 2014
July 2014
such that:
and 
: Prove that :