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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: 155

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: 165

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: 93

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: 184

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: 101

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: 1372

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: 104

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: 104

USAMO 2005/4 (Stable table)

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...
by Phlembac Adib Hasan
Mon Nov 07, 2016 11:48 am
 
Forum: Combinatorics
Topic: USAMO 2005/4 (Stable table)
Replies: 1
Views: 156

Re: A Geometric Inequality

Hint:
Chebyshev’s Inequality
by Phlembac Adib Hasan
Mon Nov 07, 2016 11:32 am
 
Forum: Algebra
Topic: A Geometric Inequality
Replies: 1
Views: 144
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