pow:start
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pow:start [2025/11/18 23:27] mazur |
pow:start [2026/03/25 15:53] (current) mazur |
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| ====== Problem of the Week ====== | ====== Problem of the Week ====== | ||
| ~~NOTOC~~ | ~~NOTOC~~ | ||
| - | <box 85% round orange| Problem 7 (due Monday, December 1 )> | + | <box 85% round orange| Problem 4 (due Monday, March 30 ) > |
| - | + | ||
| - | Consider an $m\times n$ rectangle divided into $mn$ unit squares. Let $T$ be the set of all vertices of the unit squares. At each point of $T$ we draw a short arrow (say of length $1/2$) pointing up, down, left, or right | + | |
| - | with the condition that no arrow sticks outside the rectangle. Prove that regardless of how the arrows are chosen, | + | |
| - | there always must exist two vertices of the same unit square at which the arrows point in opposite directions. | + | |
| + | Let $n>0$ be an odd integer. Prove that there exists a set $S=\{A_1, \ldots, A_{2n}\}$ of $2n$ distinct points in the plane which are not collinear and such that if $i+j\neq 2n+1$ then the line $A_iA_j$ contains a third point from $S$. | ||
| </box> | </box> | ||
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| ===== Overview ===== | ===== 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. | + | Every other Monday (starting 01/26/26), 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 | 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 | 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 | ||
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| ===== Previous Problems and Solutions===== | ===== Previous Problems and Solutions===== | ||
| - | * [[pow:Problem6f25|Problem 6]] Solution submitted by Gerald Marchesi and Mathew Wolak. | + | * [[pow:Problem3s26|Problem 3]] No solutions were submitted. |
| - | + | ||
| - | * [[pow:Problem5f25|Problem 5]] Solution submitted by Ashton Keith (Purdue University), Gerald Marchesi, and Alif Miah. | + | |
| - | + | ||
| - | * [[pow:Problem4f25|Problem 4]] Solution submitted by Alif Miah,Takeru Sueyoshi (from Anglo-Chinese Junior College based in Singapore), and Mathew Wolak. | + | |
| - | * [[pow:Problem3f25|Problem 3]] No solutions were submitted. | + | * [[pow:Problem2s26|Problem 2]] Solved by Prof. Emmett Wyman. |
| - | * [[pow:Problem2f25|Problem 2]] Solved by Gerald Marchesi, Josiah Moltz, and Mathew Wolak. | + | * [[pow:Problem1s26|Problem 1]] No solutions were submitted. |
| - | * [[pow:Problem1f25|Problem 1]] Solution submitted by Raisha Chowdhury, Gerald Marchesi, Josiah Moltz, and Mathew Wolak. | + | * [[pow:Fall 2025]] |
| * [[pow:Spring 2025]] | * [[pow:Spring 2025]] | ||
pow/start.1763526427.txt · Last modified: 2025/11/18 23:27 by mazur