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

- Sun Dec 04, 2016 9:33 pm
- Forum: Secondary Level
- Topic: Points contained in a bounded area
- Replies:
**1** - Views:
**127**

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 10:19 pm
- Forum: Geometry
- Topic: Two triangles and three collinear points
- Replies:
**1** - Views:
**144**

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 10:17 pm
- Forum: Combinatorics
- Topic: Bulldozer on the coordinate plane
- Replies:
**0** - Views:
**73**

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

- Mon Nov 07, 2016 10:12 pm
- Forum: Number Theory
- Topic: Functional divisibility
- Replies:
**2** - Views:
**131**

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...

- Mon Nov 07, 2016 10:10 pm
- Forum: Combinatorics
- Topic: Dominoes in a chessboard
- Replies:
**0** - Views:
**78**

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/

- Mon Nov 07, 2016 12:32 pm
- Forum: Computer Science
- Topic: Programming Question
- Replies:
**6** - Views:
**1258**

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?

- Mon Nov 07, 2016 11:59 am
- Forum: Combinatorics
- Topic: Largest convex polygon in an array
- Replies:
**0** - Views:
**79**

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 $k$.

- Mon Nov 07, 2016 11:55 am
- Forum: Number Theory
- Topic: Finitely many 'good' numbers
- Replies:
**0** - Views:
**78**

Legs $L_1, L_2, L_3, L_4$ of a square table each have length $n$, where $n$ is a positive integer. For how many ordered 4-tuples $(k_1, k_2, k_3, k_4)$ of non-negative integers can we cut a piece of length $k_i$ from the end of leg $L_i \; (i=1,2,3,4)$ and still have a stable table? (The table is st...

- Mon Nov 07, 2016 11:48 am
- Forum: Combinatorics
- Topic: USAMO 2005/4 (Stable table)
- Replies:
**1** - Views:
**121**