Search found 183 matches

by asif e elahi
Mon Dec 05, 2016 3:06 pm
Forum: Primary Level
Topic: Books and Websites for primary category preparation
Replies: 2
Views: 4378

Re: Books and Websites for primary category preparation

Sadly there aren't much resource for Primary category. You may consider learning some algebra,counting and divisibility. The following links should help 1. AMC 8 problems http://artofproblemsolving.com/wiki/index.php/AMC_8_Problems_and_Solutions 2. AOPS videos http://artofproblemsolving.com/videos 3...
by asif e elahi
Mon Nov 14, 2016 6:26 pm
Forum: Secondary: Solved
Topic: 2015 regional secondary maymenshing question 10
Replies: 1
Views: 2850

Re: 2015 regional secondary maymenshing question 10

Hint
Let $f(i)$ denote the largest odd divisor of $i$ for all $i \in \mathbb{N}$. Prove that $f(i)\neq f(j)$ for $2007\leq i<j\leq 4012$.
by asif e elahi
Fri Nov 04, 2016 12:23 am
Forum: Secondary Level
Topic: An exercise
Replies: 3
Views: 2286

Re: An exercise

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

\[ \sum_{i=1}^{n} \dfrac{a_i^2}{i} > \dfrac{2n}{n+1} \]
Wrong for $n=1$.
Hint
Prove $\exists i$ so that $a_i\geq n$.
Also it can be proved that $L.H.S> n^2$
by asif e elahi
Mon Oct 17, 2016 1:30 am
Forum: Geometry
Topic: Prove parallel
Replies: 2
Views: 1414

Prove parallel

Let $ABCD$ be an isosceles trapezium so that $AD\parallel BC$ and $AB=CD$. Let $\omega$ be the incircle of $\triangle BCD$ and $\Gamma$ be any circles that touches $\overline{AB}$ and $\overline{AC}$ and intersects $BC$ at $U$ and $V$. Let $DU$ and $DV$ meet $\omega$ at $J$ and $K$ respectively. Pro...
by asif e elahi
Fri Oct 14, 2016 3:44 pm
Forum: Geometry
Topic: Tangent Circles
Replies: 4
Views: 2516

Re: Tangent Circles

$Hints:$ Prove a lemma: $W,W'$ are two circles which are reflections of eachother under $BC$. Let $M$ be the midpoint of $BC$. Let $P,Q$ be two lines intersecting $W,W'$ at respectively such that $U,V,W,X$ lies on the same side of $BC$. Then $UVWX$ is cyclic. Then, show that the centres of two resp...
by asif e elahi
Thu Oct 13, 2016 4:10 pm
Forum: Geometry
Topic: Perpendicular lines through the foot of an altitude
Replies: 2
Views: 2402

Re: Perpendicular lines through the foot of an altitude

Let $M$ be the midpoint of $BC$. Applying Cotangent rule in $\triangle AED$, we get \begin{align*} \cot \angle DHE &=\frac{ HD\cdot \cot \angle HAE- AH\cdot \cot \angle HDE}{AD} \\ & = \frac{HD\cdot\frac{AD}{BD}-AH\cdot \cot \angle ODM}{AD}\\ & = \frac{HD}{BD}-\frac{AH\cdot \frac{DM}{OM}}{AD}\\ & = ...
by asif e elahi
Sat Sep 03, 2016 12:54 pm
Forum: International Mathematical Olympiad (IMO)
Topic: IMO Marathon
Replies: 184
Views: 47419

Re: IMO Marathon

Problem $\boxed{49}$ Is it possible to write $\dbinom n 2$ consecutive natural numbers on the edges of $K_n$ such that for every path or cycle of length $3$, if the numbers written on the $3$ edges are $a,b,c$ then $\gcd(a,c)\mid b$ is satisfied? In case of a path, the edge $b$ lies between the edg...
by asif e elahi
Sun Aug 21, 2016 1:09 pm
Forum: Number Theory
Topic: gcd and divisibility
Replies: 5
Views: 1567

Re: gcd and divisibility

Take $864k$ as a multiple of $864$. Now if $1944$ divides $864k$, we must have $\dfrac{864k}{1944}\in \mathbb{N}$. Which implies $9\mid k$. The probability of happening this is $\dfrac{1}{9}$ when we choose $k$ randomly.
by asif e elahi
Wed Aug 17, 2016 2:36 am
Forum: Geometry
Topic: A Problem of Romanian TST
Replies: 10
Views: 3576

Re: A Problem of Romanian TST

tanmoy wrote:Because then $OF \perp O_{1}O_{2}$ and as $O_{1}O_{2} \parallel MN$,so we will be done
Why $O_{1}O_{2} \parallel MN$?
by asif e elahi
Tue Aug 16, 2016 10:21 pm
Forum: Geometry
Topic: A Problem of Romanian TST
Replies: 10
Views: 3576

Re: A Problem of Romanian TST

So,our goal is to show that $F$ also lies on $l$ i.e. if $ME \cap AF=P,NG \cap BF=Q$,we have to prove that $FP \cdot FA=FQ \cdot FB$. That' not true actually. But the problem is very easy from here. Let $\omega=\bigodot (M,MA),\lambda=\bigodot(N,NB)$ Let $\omega\cap AF=A,X$ and $\lambda\cap BF=B,Y$...