## Search found 1015 matches

- Sun Jan 20, 2019 7:54 am
- Forum: Site Support
- Topic: How to use LaTeX
- Replies:
**59** - Views:
**588525**

### Re: How to use LaTeX

@samiul_samin, thank you! They have been updated in the first post.

- Mon Jan 07, 2019 8:01 pm
- Forum: News / Announcements
- Topic: MPMS Problem Solving Marathon
- Replies:
**11** - Views:
**8194**

### 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...

- Mon Jan 07, 2019 7:42 pm
- Forum: News / Announcements
- Topic: Spam
- Replies:
**1** - Views:
**3204**

### Re: Spam

Thank you for bringing this to our attention. It has been taken care of. :)

- Wed Feb 14, 2018 4:59 am
- Forum: Algebra
- Topic: lowest Value
- Replies:
**2** - Views:
**4207**

### Re: lowest Value

**Hint:**

1.

- Fri Jun 02, 2017 1:59 pm
- Forum: News / Announcements
- Topic: MPMS Problem Solving Marathon
- Replies:
**11** - Views:
**8194**

### 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.

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.

- Wed May 24, 2017 12:15 am
- Forum: News / Announcements
- Topic: MPMS Problem Solving Marathon
- Replies:
**11** - Views:
**8194**

### 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 ...

- Sun Dec 04, 2016 9:33 pm
- Forum: Secondary Level
- Topic: Points contained in a bounded area
- Replies:
**1** - Views:
**1723**

### 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 $4\;\text{unit}^2$.

- Mon Nov 07, 2016 10:19 pm
- Forum: Geometry
- Topic: Two triangles and three collinear points
- Replies:
**1** - Views:
**1728**

### 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 $D,E,F$ ...

- Mon Nov 07, 2016 10:17 pm
- Forum: Combinatorics
- Topic: Bulldozer on the coordinate plane
- Replies:
**0** - Views:
**1477**

### 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 ...

- Mon Nov 07, 2016 10:12 pm
- Forum: Number Theory
- Topic: Functional divisibility
- Replies:
**2** - Views:
**1838**

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