Search found 108 matches

by rah4927
Mon Feb 27, 2017 12:36 am
Forum: Combinatorics
Topic: Combi Marathon
Replies: 48
Views: 27834

Re: Combi Marathon

$\text{Problem 10:}$ Consider $2009$ cards, each having one gold side and one black side, lying in parallel on a long table. Initially all cards show their gold sides. Two players, standing by the same long side of the table, play a game with alternating moves. Each move consists of choosing a bloc...
by rah4927
Sat Feb 25, 2017 12:46 am
Forum: Social Lounge
Topic: সেঞ্চুরি ! সেঞ্চুরি !! সেঞ্চুরি !!!
Replies: 10
Views: 7425

Re: সেঞ্চুরি ! সেঞ্চুরি !! সেঞ্চুরি !!!

Congrats on the thousandth post (am posting it here to get my own post-count closer to hundred ;) )
by rah4927
Tue Feb 21, 2017 12:35 am
Forum: Combinatorics
Topic: Combi Marathon
Replies: 48
Views: 27834

Re: Combi Marathon

$\text{Problem } 5$ Let $m, n$ be positive integers with $m > 1$. Anastasia partitions the integers $1, 2, \dots , 2m$ into $m$ pairs. Boris then chooses one integer from each pair and finds the sum of these chosen integers. Prove that Anastasia can select the pairs so that Boris cannot make his sum...
by rah4927
Tue Feb 21, 2017 12:32 am
Forum: Combinatorics
Topic: Combi Marathon
Replies: 48
Views: 27834

Re: Combi Marathon

Problem 4: Each edge of a polyhedron is oriented with an arrow such that at each vertex, there is at least on arrow leaving the vertex and at least one arrow entering the vertex. Does there always exists two faces on the polyhedron such that the edges on each of it's boundary form a directed cycle?...
by rah4927
Mon Feb 20, 2017 10:35 pm
Forum: Combinatorics
Topic: Combi Marathon
Replies: 48
Views: 27834

Re: Combi Marathon

Problem $3$ Let $n$ be a positive integer. At each of the $2n$ points around a circle we place discs with one white side and one black side. We may perform the following move: select a black disc and flip over its two neighbors. Find all initial configurations from which some sequence of moves lead...
by rah4927
Mon Feb 20, 2017 10:17 pm
Forum: Geometry
Topic: A Problem for Dadu
Replies: 2
Views: 1718

Re: A Problem for Dadu

We will show that the four lines are concurrent at their midpoints. Lemma : $AH_AH_BB$ is a parallelogram Proof : Since $AH_A=2\times \text{distance of O to CD}=BH_B$, and $AH_A||BH_B$, we are done. Now it easily follows that the diagonals $AH_B$ and $BH_A$ bisect each other. We can do this for ever...
by rah4927
Mon Feb 20, 2017 2:26 pm
Forum: Combinatorics
Topic: Combi Solution Writing Threadie
Replies: 10
Views: 9460

Re: Combi Solution Writing Threadie

Nice solu Tanmoy. However, the things that you are doing seem magic until we get to the end. Perhaps you should add a brief summary at the beginning. Something like "We will show that every set of four points can contribute at most $2$ to the original sum".
by rah4927
Sun Feb 19, 2017 11:23 am
Forum: Algebra
Topic: Instructive FE (I desperately need that topic list)
Replies: 1
Views: 1200

Re: Instructive FE (I desperately need that topic list)

Some hints. First try to guess the answer. Done? Let's move on. Now there aren't a lot of things you can do in this problem apart from proving $f$ is injective, and even that doesn't yield much. So you can only plug in values. Plug in $n=1$. First assume that $f(1)=1$. Keep reiterating the values to...
by rah4927
Sun Feb 19, 2017 11:16 am
Forum: Algebra
Topic: $2009$ USA TST Inequality
Replies: 1
Views: 1351

$2009$ USA TST Inequality

Prove that for positive real numbers $x$, $y$, $z$, $$x^3(y^2+z^2)^2 + y^3(z^2+x^2)^2+z^3(x^2+y^2)^2 \geq xyz\left[xy(x+y)^2 + yz(y+z)^2 + zx(z+x)^2\right]$$
by rah4927
Sun Feb 19, 2017 11:15 am
Forum: Number Theory
Topic: Numbers expressible as $\sum_{i=1}^{n}(-1)^{a_{i}}2^{b_{i}}$
Replies: 1
Views: 1385

Numbers expressible as $\sum_{i=1}^{n}(-1)^{a_{i}}2^{b_{i}}$

Let $n$ be a positive integer. Find, with proof, the least positive integer $d_{n}$ which cannot be expressed in the form \[\sum_{i=1}^{n}(-1)^{a_{i}}2^{b_{i}},\]
where $a_{i}$ and $b_{i}$ are nonnegative integers for each $i.$