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--- pow:problem2f25 [2025/09/22 18:41] – created mazur
+++ pow:problem2f25 [2025/09/25 02:26] (current) – mazur
@@ Line 1: / Line 1: @@
+<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>
+The problem was solved by Gerald Marchesi, Josiah Moltz, and Mathew Wolak. The only function which
+satisfies the conditions of the problem is $F(x,y)=x^2+y^2$. For detailed solutions and additional
+discussion see the following link {{:pow:2025fproblem2.pdf|Solution}}.