## Search found 12 matches

Sat Nov 26, 2016 9:31 pm
Forum: Combinatorics
Topic: 'BASIC THINGS OF COMBINATORICS
Replies: 2
Views: 1398

### Re: 'BASIC THINGS OF COMBINATORICS

This videoes are pretty awesome to develop the basics of combinatorics.

Sat Nov 12, 2016 6:10 pm
Forum: Number Theory
Topic: SSC Mock'16 : Problem 1
Replies: 0
Views: 867

### SSC Mock'16 : Problem 1

Well, this might be easy but still an interesting one to me.

Find all finite sets of positive integers with at least two elements such that for any two numbers $a,b$ belonging to the set with $a>b$, the number $\dfrac{b^2}{a-b}$ belongs to the set, too.
Sun Nov 06, 2016 10:37 pm
Forum: Algebra
Topic: Iran TST 2012,Exam2,Day2,P4
Replies: 1
Views: 1969

### Re: Iran TST 2012,Exam2,Day2,P4

My solution Let $a \ge b \ge c > 0$. Then by applying chebyshev, we get, $\sum_{cyc}^{} \dfrac{a\sqrt{a}}{bc} \ge \dfrac{1}{3} \left( \sum_{cyc}^{} \dfrac{a}{bc} \right)\left( \sum_{cyc}^{} \sqrt{a} \right)$ Then it's enough to prove that \begin{align*} \dfrac{1}{3} \left( \sum_{cyc} \dfrac{a}{b...
Sat Nov 05, 2016 11:48 am
Forum: Algebra
Topic: A Geometric Inequality
Replies: 1
Views: 1858

### A Geometric Inequality

Let $a,b,c$ be the sides of $\triangle ABC$ and $x$ be any non-negative real number. Prove that,
$a^x \cos A+ b^x \cos B + c^x \cos C \leq \dfrac{1}{2}(a^x+b^x+c^x) .$
Fri Nov 04, 2016 8:23 pm
Forum: Secondary Level
Topic: An exercise
Replies: 3
Views: 2227

### Re: An exercise

oops! I forgot about the case for n=1. :3
Wed Nov 02, 2016 12:17 am
Forum: Algebra
Topic: Inequality with condition $xy+yz+zx=3xyz$
Replies: 2
Views: 2132

### Re: Inequality with condition $xy+yz+zx=3xyz$

I think 2 is not annoying here as you can simply have by AM-GM,
$\sum_{cyc} (x^2y+\dfrac{1}{y}) \geq 2\sum_{cyc} x$
the desired inequality!!!
Tue Nov 01, 2016 7:51 pm
Topic: IMO 2016 Problem 6
Replies: 1
Views: 2517

### Re: IMO 2016 Problem 6

The main idea of my solution Part (a) : WLOG, we may assume that the line segments are actually chords of a circle and all the chords intersect one another inside the circle. Now we shall label the endpoints one after another like $A_1, A_2,....A_{2n}$. Thus points $A_k$ and $A_{k+n}$ are the end p...
Tue Nov 01, 2016 7:06 pm
Forum: Secondary Level
Topic: An exercise
Replies: 3
Views: 2227

### An exercise

Let $a_1,a_2,......a_n$ are positive real numbers such that $\sum_{i=1}^{n} \dfrac{1}{a_i} = 1$. Prove that,

$\sum_{i=1}^{n} \dfrac{a_i^2}{i} > \dfrac{2n}{n+1}$
Mon Oct 31, 2016 11:44 pm
Forum: Social Lounge
Replies: 53
Views: 39044

My solution The given equation is equivalent to, $p+r= \dfrac{2015-r}{q}$ or, $p+q+r= \dfrac{2015-r}{q} + q$ Now let $a= \dfrac{2015-r}{q}$ and $b=q$. Then we have, $p+q+r=a+b$ with $q=b$, $r=2015-ab$ and $p=a+ab-2015$. So now we've to actually find the minimum and maximum values of $a+b$. As, ...