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pow:start

Problem 6 (due on Monday, November 18).

For real numbers $a,b,c$ consider the system of equations \[x^2+2yz=a,\ \ y^2+2xz=b,\ \ z^2+2xy=c.\] Prove that this system has at most one solution in real numbers $x,y,z$ such that $x\geq y\geq z$ and $x+y+z\geq 0$. Prove that such a solution exists if and only if $a+b+c\geq 0$ and $b=\min(a,b,c)$. Here $\min(a,b,c)$ denotes the smallest number among $a,b,c$.

Every other Monday (starting 08/27/24), 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 (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.

- Problem 5 Solved by Prof. Vladislav Kargin and Dr. Mathew Wolak.

- Problem 4 Solved by Levi Axelrod and Dr. Mathew Wolak.

- Problem 3 We have not received any solutions.

- Problem 2 Solution submitted by Levi Axelrod and Surajit Rajagopal.

- Problem 1 Solved by Sasha Aksenchuk, Prof. Vladislav Kargin, Josiah Moltz, and Mithun Padinhare Veettil.

pow/start.txt · Last modified: 2024/11/05 10:01 by mazur