Search found 1015 matches
- Mon Nov 07, 2016 10:10 pm
- Forum: Combinatorics
- Topic: Dominoes in a chessboard
- Replies: 0
- Views: 2096
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...
- Mon Nov 07, 2016 12:32 pm
- Forum: Computer Science
- Topic: Programming Question
- Replies: 6
- Views: 13771
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/
- Mon Nov 07, 2016 11:59 am
- Forum: Combinatorics
- Topic: Largest convex polygon in an array
- Replies: 0
- Views: 2073
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?
- Mon Nov 07, 2016 11:55 am
- Forum: Number Theory
- Topic: Finitely many 'good' numbers
- Replies: 0
- Views: 2094
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 $k$.
- Mon Nov 07, 2016 11:48 am
- Forum: Combinatorics
- Topic: USAMO 2005/4 (Stable table)
- Replies: 1
- Views: 2590
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...
- Mon Nov 07, 2016 11:32 am
- Forum: Algebra
- Topic: A Geometric Inequality
- Replies: 1
- Views: 3469
- Sat Oct 15, 2016 2:10 pm
- Forum: Algebra
- Topic: Sum of the entries
- Replies: 1
- Views: 3399
Re: Sum of the entries
($1000^{th}$ post, yay) Solution: Enumerate the rows and columns from the top-left corner. Then the entry in $i$-th row and $j$-th column is $\dfrac {1}{i+j-1}$. Suppose the set of selected $n$ entries is $E$. Now using Cauchy-Schwartz's Inequality, \[ \begin{split} \sum_{(i,j)\in E} \frac{1}{i+j-1}...
- Wed Oct 12, 2016 10:43 am
- Forum: Combinatorics
- Topic: Existence of a subset satisfying specific conditions
- Replies: 0
- Views: 2876
Existence of a subset satisfying specific conditions
Let $A$ be a finite set of positive integers. Prove that $\exists B\subseteq A$ satisfying the following conditions: i) if $b_1,b_2\in B$ are distinct, then neither $b_1$ and $b_2$ nor $b_1+1$ and $b_2+1$ are multiples of each other, and ii) for any $a\in A$, we can find a $b\in B$ such that either ...
- Mon Oct 10, 2016 10:22 am
- Forum: Junior Level
- Topic: Finding value
- Replies: 1
- Views: 3568
Re: Finding value
$\dfrac{x+1}{x}=2\Longrightarrow x+1=2x\Longrightarrow x=1$
So $\dfrac{x^4+1}{x^4}=\dfrac{1+1}{1}=2$
So $\dfrac{x^4+1}{x^4}=\dfrac{1+1}{1}=2$
- Mon Oct 10, 2016 9:59 am
- Forum: Geometry
- Topic: Perpendicular lines through the foot of an altitude
- Replies: 2
- Views: 4157
Perpendicular lines through the foot of an altitude
In an acute triangle $ABC$ with circumcenter $O$ and orthocenter $H$, let $D$ be the foot of the altitude from $A$. The line perpendicular to $OD$ at $D$ meets segment $AB$ at $E$. Prove $\angle DHE=\angle ABC$
Rules: trig bashing is allowed, but not encouraged. Look for synthetic solutions.
Rules: trig bashing is allowed, but not encouraged. Look for synthetic solutions.