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Re: MPMS Problem Solving Marathon

PROBLEM 3:
Find all *odd* integers $n$ for which $4n^2-6n+45$ is a perfect square.

PROBLEM 4:
Find all positive integers $m$ and $n$ such that $7^m+11^n$ is a perfect square.
by Phlembac Adib Hasan
Fri Jun 02, 2017 1:59 pm
 
Forum: News / Announcements
Topic: MPMS Problem Solving Marathon
Replies: 8
Views: 464

MPMS Problem Solving Marathon

This is a general problem solving marathon for members of Mymensingh Parallel Math School (MPMS). However, feel free to participate, even if you are not a member. PROBLEM 1: $p$ is a prime number of the form $4k+1$. Prove that there exists an integer $a$ so that $a^2+1$ is divisible by $p$. PROBLEM ...
by Phlembac Adib Hasan
Wed May 24, 2017 12:15 am
 
Forum: News / Announcements
Topic: MPMS Problem Solving Marathon
Replies: 8
Views: 464

Points contained in a bounded area

$n$ points lie on a plane so that the triangle formed by any three of them has an area of at most $1\;\text{unit}^2$. Prove that all the points are contained in a triangle with area of at most $4\;\text{unit}^2$.
by Phlembac Adib Hasan
Sun Dec 04, 2016 9:33 pm
 
Forum: Secondary Level
Topic: Points contained in a bounded area
Replies: 1
Views: 231

Two triangles and three collinear points

We are given triangles $ABC$ and $DEF$ such that $D\in BC, E\in CA, F\in AB$, $AD\perp EF, BE\perp FD, CF\perp DE$. Let the circumcenter of $DEF$ be $O$, and let the circumcircle of $DEF$ intersect $BC,CA,AB$ again at $R,S,T$ respectively. Prove that the perpendiculars to $BC,CA,AB$ through $D,E,F$ ...
by Phlembac Adib Hasan
Mon Nov 07, 2016 10:19 pm
 
Forum: Geometry
Topic: Two triangles and three collinear points
Replies: 1
Views: 223

Bulldozer on the coordinate plane

On the coordinate plane, there are finitely many walls, (= disjoint line segments) none of which are parallel to either axis. A bulldozer starts at an arbitrary point and moves in the $+x$ direction. Every time it hits a wall, it turns at a right angle to its path, away from the wall, and continues ...
by Phlembac Adib Hasan
Mon Nov 07, 2016 10:17 pm
 
Forum: Combinatorics
Topic: Bulldozer on the coordinate plane
Replies: 0
Views: 126

Functional divisibility

$ k$ is a given natural number. Find all functions $ f: \mathbb{N}\rightarrow\mathbb{N}$ such that for each $ m,n\in\mathbb{N}$ the following holds: \[ f(m)+f(n)\mid (m+n)^k\]
by Phlembac Adib Hasan
Mon Nov 07, 2016 10:12 pm
 
Forum: Number Theory
Topic: Functional divisibility
Replies: 2
Views: 286

Dominoes in a chessboard

A $100\times 100$ chessboard is cut into dominoes ($1\times 2$ rectangles). Two persons play the following game: At each turn, a player glues together two adjacent cells (which were formerly separated by a cut-edge). A player loses if, after his turn, the $100\times 100$ chessboard becomes connected...
by Phlembac Adib Hasan
Mon Nov 07, 2016 10:10 pm
 
Forum: Combinatorics
Topic: Dominoes in a chessboard
Replies: 0
Views: 139

Re: Programming Question

To write a desktop app, you need to use a framework. I would recommend Qt, a very popular framework. Visit its website: https://www.qt.io/
by Phlembac Adib Hasan
Mon Nov 07, 2016 12:32 pm
 
Forum: Computer Science
Topic: Programming Question
Replies: 6
Views: 1638

Largest convex polygon in an array

Draw a $2004\times 2004$ array of points. What is the largest integer $n$ for which it is possible to draw a convex $n$-gon whose vertices are chosen from the points in the array?
by Phlembac Adib Hasan
Mon Nov 07, 2016 11:59 am
 
Forum: Combinatorics
Topic: Largest convex polygon in an array
Replies: 0
Views: 151

Finitely many 'good' numbers

For a positive integer $n$, denote by $\tau (n)$ the number of its positive divisors. For a positive integer $n$, if $\tau(m) < \tau(n)$ for all positive integers $m<n$, we call $n$ a good number. Prove that for any positive integer $k$, there are only finitely many good numbers not divisible by $k$.
by Phlembac Adib Hasan
Mon Nov 07, 2016 11:55 am
 
Forum: Number Theory
Topic: Finitely many 'good' numbers
Replies: 0
Views: 166
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