(player) and 
(items), with valuations given by matrix![$
V = \left[\begin{matrix}
v_{1a} & v_{1b} & v_{1c} & v_{1d} \\
v_{2a} & v_{2b} & v_{2c} & v_{2d}
\end{matrix}\right]
$](http://latex.artofproblemsolving.com/d/4/6/d465348852f12739e1fc2a04a9c2e5fdad0dbcb4.png)
![$v_{ij} \in [0, \infty]$](http://latex.artofproblemsolving.com/7/8/8/788bffd362b46a318a2cb0703a9e2ae77089e4e7.png)
.
'
'
is strictly better than another possible bid 
of his if the total valuation derived from 
.![$B_i = [b^i_1, b^i_2, b^i_3, b^i_4]$](http://latex.artofproblemsolving.com/7/b/2/7b233ca3e9056aa374e937b1133cd9c81726f407.png)
such that ![$b^i_j \in [0,1]$](http://latex.artofproblemsolving.com/b/9/6/b96c2a7618767934c8a44d9238c485fed39b8510.png)
and 
.
and 
such that given 
has a right angle at 
.
be the point on line 
and 
.
and 
is the bisector of 
.
and 
.
be the midpoint of 
is a parallelogram. Prove that 
and 
are concurrent.
satisfying 
,
,
)
.
such that![\[
\sum_{i=1}^{n} \frac{k_i}{P'(a_i)} = \sum_{i=1}^{n} k_i.
\]](http://latex.artofproblemsolving.com/4/4/e/44e5eb17b17531e36c362e062471d9bdffc8144c.png)
Y by
Given a triangle 
with circumcircle 
, and two arbitrary points 
on . Let 
, 
, 
be points on lines 
, 
, 
, respectively, such that 
, 
, and 
concur at a point 
. Let 
be a point on line such that 
, 
, , are concyclic. Similarly, let 
be a point on line such that , , , are concyclic, and let 
be a point on line such that , , , are concyclic. Prove that 
, 
, 
concur at a single point.
Proposed by chengbilly
with circumcircle 
, and two arbitrary points 
on . Let 
, 
, 
be points on lines 
, 
, 
, respectively, such that 
, 
, and 
concur at a point 
. Let 
be a point on line such that 
, 
, , are concyclic. Similarly, let 
be a point on line such that , , , are concyclic, and let 
be a point on line such that , , , are concyclic. Prove that 
, 
, 
concur at a single point.Proposed by chengbilly
This post has been edited 1 time. Last edited by chengbilly, Mar 27, 2025, 5:18 AM
.
.
.
,
be the cevian triangle of 
with respect to 
.
,
be the circumcevian triangle of 
,
,
,
and 
,
,
,
are concyclic. Then 
,
,
,
is concyclic. 
be the point on 
,
,
are convolution pairs. Define 
and 
similarly. Then 
is just 
,
,
is the isoconjugation 
and 
is the centroid of 
,
.
is just 
,
,
is the circumcircle of 
and thus 
,
is the symmedian point of 
be points on lines 
respectively, s.t. 
.
,
Similarly, 
Combine these equations, 
Rewrite it then we can get the result.
.
By Ceva's Theorem again, it's clear that 
,
