====== Problem of the Week ======
~~NOTOC~~
<box 85% round orange|  Problem 2 (due Monday, September 22 )>

Find all functions $F:\mathbb R^2\longrightarrow \mathbb R$ such that

(i) If $ABCD$ is any rectangle on the plane $\mathbb R^2$ then $F(A)+F(C)=F(B)+F(D)$;

(ii) The second order partial derivatives $\displaystyle \frac{\partial^2 F}{\partial x \partial x}$ ,
$\displaystyle \frac{\partial^2 F}{\partial x \partial y}$, $\displaystyle \frac{\partial^2 F}{\partial y \partial x}$, $\displaystyle \frac{\partial^2 F}{\partial y \partial y}$ exist and are continuous on $\mathbb R^2$ (this means that $F$ is of class $C^2$);

(iii) $F(0,0)=0$, $F(1,0)=1=F(0,1)$, $\displaystyle \frac{\partial F}{\partial x}(0,0)=0$.


</box>


===== Overview =====

Every other Monday (starting 08/25/25), we will post a problem to engage our mathematical community in the problem solving activity and to enjoy mathematics outside of the classroom. 
Students (both undergraduate and graduate) are particularly encouraged to participate as there is no better
way to practice math than working on challenging problems. If you have a solution and want to be a part of it, e-mail your solution to Marcin
Mazur ([[mailto:mazur@math.binghamton.edu|mazur@math.binghamton.edu]]) by the due date.  We will post our solutions as well as novel solutions from the participants and record the names of those who've got the most number of solutions throughout each semester. 

When you submit your solutions, please provide a detailed reasoning rather than just an answer. Also, please include some short info about yourself for our records. 

===== Previous Problems and Solutions=====

    * [[pow:Problem1f25|Problem 1]] Solution submitted by Raisha Chowdhury, Gerald Marchesi, Josiah Moltz, and Mathew Wolak.

    * [[pow:Spring 2025]]

    * [[pow:Fall 2024]]

    * [[pow:Spring 2024]]

    * [[pow:Fall 2023]]

    * [[pow:Spring 2023]]

    * [[pow:Fall 2022]]

    * [[pow:Spring 2022]]
    
    * [[pow:Fall 2021]]

    * [[pow:Spring 2021]] 

    * [[pow:Fall 2020]]

    * [[pow:Summer Challenge]]
    
    * [[pow:Spring 2020]]