Best bid given the other's bid
by fancyfairy, Mar 30, 2025, 10:04 PM
Consider sets 
(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]
$](//latex.artofproblemsolving.com/d/4/6/d465348852f12739e1fc2a04a9c2e5fdad0dbcb4.png)
where![$v_{ij} \in [0, \infty]$](//latex.artofproblemsolving.com/7/8/8/788bffd362b46a318a2cb0703a9e2ae77089e4e7.png)
.
A player’s total valuation is the sum of item values won, with half-value for half-won items.
Given player
's bid, player 
's bid 
is strictly better than another possible bid 
of his if the total valuation derived from is strictly more than that derived from 
.
Each player simultaneously bids on items with a bid vector![$B_i = [b^i_1, b^i_2, b^i_3, b^i_4]$](//latex.artofproblemsolving.com/7/b/2/7b233ca3e9056aa374e937b1133cd9c81726f407.png)
such that ![$b^i_j \in [0,1]$](//latex.artofproblemsolving.com/b/9/6/b96c2a7618767934c8a44d9238c485fed39b8510.png)
and 
. The highest bidder wins an item; ties split ownership equally.
Given valuation matrix
, can we always find bid vectors 
and 
such that given , the second player can not submit a bid strictly better than and vice-versa?
(player) and 
(items), with valuations given by matrix![$
V = \left[\begin{matrix}
v_{1a} & v_{1b} & v_{1c} & v_{1d} \\
v_{2a} & v_{2b} & v_{2c} & v_{2d}
\end{matrix}\right]
$](http://latex.artofproblemsolving.com/d/4/6/d465348852f12739e1fc2a04a9c2e5fdad0dbcb4.png)
where
![$v_{ij} \in [0, \infty]$](http://latex.artofproblemsolving.com/7/8/8/788bffd362b46a318a2cb0703a9e2ae77089e4e7.png)
.A player’s total valuation is the sum of item values won, with half-value for half-won items.
Given player
's bid, player 
's bid 
is strictly better than another possible bid 
of his if the total valuation derived from is strictly more than that derived from 
.Each player simultaneously bids on items with a bid vector
![$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 
. The highest bidder wins an item; ties split ownership equally.Given valuation matrix
, can we always find bid vectors 
and 
such that given , the second player can not submit a bid strictly better than and vice-versa?
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)
be an acute scalene triangle inscribed in a circle 
.
intersects 
at 
and 
.
is the midpoint of 
.
be the altitude in 
,
intersects 
.
be the midpoint of 
be the reflection of 
with respect to 
and 
are similar.
on the board gets replaced by 
,
is the largest prime divisor of 
groups for an activity. In the afternoon, the math teacher divided the same students into 
groups for another activity. A student is considered special if the group they belonged to in the afternoon is smaller than the group they belonged to in the morning. Find the minimum number of special students that can exist in the school.
be an arbitrary interior point of 
,
,
,
intersect 
,
,
at 
,
,
,
be the midpoint of 
and 
intersect at 
,
intersects 
,
intersects 
.
is tangent to 
has a right angle at 
.
be the point on line 
such that 
and 
.
and 
is the bisector of 
.
and 
is the bisector of 
.
is a parallelogram. Prove that 
and 
are concurrent.
and circumcircle 
.
and 
,
through 
.
and 
are equal.
March 2025
February 2025