Search found 1015 matches

by Phlembac Adib Hasan
Sun Jan 20, 2019 7:54 am
Forum: Site Support
Topic: How to use LaTeX
Replies: 59
Views: 221752

Re: How to use LaTeX

@samiul_samin, thank you! They have been updated in the first post.
by Phlembac Adib Hasan
Mon Jan 07, 2019 8:01 pm
Forum: News / Announcements
Topic: MPMS Problem Solving Marathon
Replies: 11
Views: 1846

Re: MPMS Problem Solving Marathon

Problem 5 A Hydra has $2019$ heads and is immune to damage from conventional weapons. However, with one blow of a magical sword, Hercules can cut off its $9, 10, 11$ or $12$ heads. In each of these cases, $5, 18, 7$ and $0$ heads grow on its shoulder. The Hydra will die only if all the heads are cu...
by Phlembac Adib Hasan
Mon Jan 07, 2019 7:42 pm
Forum: News / Announcements
Topic: Spam
Replies: 1
Views: 484

Re: Spam

Thank you for bringing this to our attention. It has been taken care of. :)
by Phlembac Adib Hasan
Wed Feb 14, 2018 4:59 am
Forum: Algebra
Topic: lowest Value
Replies: 2
Views: 372

Re: lowest Value

Hint:
1.
$abc=1$. Now apply AM-GM.
2.
#2 directly follows from AM-GM. For #1, can you find a clever way to apply AM-GM?
by Phlembac Adib Hasan
Fri Jun 02, 2017 1:59 pm
Forum: News / Announcements
Topic: MPMS Problem Solving Marathon
Replies: 11
Views: 1846

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
Wed May 24, 2017 12:15 am
Forum: News / Announcements
Topic: MPMS Problem Solving Marathon
Replies: 11
Views: 1846

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
Sun Dec 04, 2016 9:33 pm
Forum: Secondary Level
Topic: Points contained in a bounded area
Replies: 1
Views: 605

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
Mon Nov 07, 2016 10:19 pm
Forum: Geometry
Topic: Two triangles and three collinear points
Replies: 1
Views: 530

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 10:17 pm
Forum: Combinatorics
Topic: Bulldozer on the coordinate plane
Replies: 0
Views: 362

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 10:12 pm
Forum: Number Theory
Topic: Functional divisibility
Replies: 2
Views: 622

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: \[ f(m)+f(n)\mid (m+n)^k\]