## Search found 550 matches

Thu Sep 25, 2014 11:14 pm
Forum: Algebra
Topic: Another Tricky FE
Replies: 3
Views: 1963

Solution: Let $f(x)=x^{3}+g(x) \forall x\in \mathbb{R}$. Now the given statement can be written as $(x-y)(x+y)^{3}+(x-y)g(x+y)-(x+y)(x-y)^{3}-(x+y)g(x-y)=4xy(x^{2}-y^{2})$. Simplification yields $(x-y)g(x+y)=(x+y)g(x-y)$. Let us denote the last statement by $Q(x,y)$. Then $Q(x+1,x)\Rightarrow g(2x+1... Mon Sep 15, 2014 10:07 pm Forum: Number Theory Topic: Infinintely many squares of a form Replies: 7 Views: 3436 ### Re: Infinintely many squares of a form Does there exist infinite many pairs$(m,n)$of positive integers such that$(a^2+b^2+3ab)$is a perfect square? What if we are asked to find all ordered$(a,b)$such that$(a^2+b^2+3ab)$is a perfect square? Tue Aug 26, 2014 11:04 pm Forum: Number Theory Topic: F.E (Canada-2002) Replies: 2 Views: 1368 ### Re: F.E (Canada-2002) Let us denote the statement$mf(n)+nf(m)=(m+n)f(m^{2}+n^{2})$by$P(m,n)$. Then,$P(0,n)\Rightarrow nf(0)=nf(n^{2})$. So$f(n^{2})=f(0) \forall n\in\mathbb{N}_{0}$. Now let$n_{0}$be any non-negative integer. Let us take$m_{0}$such that$(m_{0},n_{0})=1$. Then clearly$(n_{0},m_{0}^{2}+n_{0})=1$.... Sun Aug 24, 2014 12:09 am Forum: International Mathematical Olympiad (IMO) Topic: IMO 2014 - Day 1 Problem 3 Replies: 2 Views: 2168 ### Re: IMO 2014 - Day 1 Problem 3 In my solution the reflections of$C$wrt$B,D$are$C_1,C_2$respectively. I also proved that$C_1$is the isogonal conjugate of$T$wrt$\triangle ASH$in the same way Asif did. Thus it is easy to see that$\angle TSH=\angle C_1SB$. Now,$\angle AHT=\angle C_1HS=\angle C_1CS=\angle BC_1S=90^{\circ...
Mon Feb 10, 2014 6:10 pm
Forum: Junior Level
Topic: Last goes first
Replies: 3
Views: 2053

### Re: Last goes first

@Fatin:
It is easier to solve the equation, $10a+2=3\times (200000+a)$.
Thu Feb 06, 2014 6:09 pm
Forum: Combinatorics
Topic: rotating a colored square
Replies: 5
Views: 3294

### Re: rotating a colored square

asif e elahi wrote:How many times is the square rotated ?
the question said $90^{\circ}$.
Tue Feb 04, 2014 6:21 pm
Topic: BdMO National Secondary 2011/10
Replies: 7
Views: 3760

### Re: BdMO National Secondary 2011/10

I have done this: Clearly we have to find $n$ such that $10^{n}+1$ is not square free, and we know $10^{11}+1$ is one (Thanks to Masum vai and LTE). Now $\displaystyle\frac{100000000001}{11^{2}}=826446281$. Let us take $a=82644628100=826446281\times 10^{2}$. Then the repetition of $a$,an $11$ digit ...
Sun Feb 02, 2014 9:57 pm
Topic: BdMO National 2013: Higher Secondary 4
Replies: 5
Views: 3644

### Re: BdMO National 2013: Higher Secondary 4

I got $\frac{11}{6}$. Is it the answer?
Wed Jan 29, 2014 7:45 pm
Topic: BdMO National 2013: Higher Secondary 4
Replies: 5
Views: 3644

### Re: BdMO National 2013: Higher Secondary 4

Is it enough to find the successor term of $\frac{14}{17}$ in fairy sequence $f_{17}$?
Sun Jan 26, 2014 6:58 pm
(I am posting this via a friend of mine) Let us colour the points which are chosen. Now call a point "Row happy" if it is the only coloured point in its row. Similarly define "Column happy" points. Now by the pigeon-hole principal, at least one of the rows contain more than $1$ point. So the number ...