Search found 107 matches
Re: Tricky FE
This part is a text book problem.
Re: Tricky FE
$f(n)=f(1)+ \frac {1}{2.3} +\frac {1}{3.4} + \frac {1}{4.5}+....+\frac{1}{(n+1)(n+2)}= f(1)+\frac{1}{2}-\frac{1}{3}+\frac {1}{3}-\frac{1}{4}+\frac{1}{4}+....+\frac{1}{n+1}-\frac{1}{n+2}$=$ f(1)+\frac{1}{2}-\frac {1}{n+2}$. And $f(1)$ is determined by our choice. P.S: I don't see any 'tricky' part. 8-)
- Wed Sep 24, 2014 10:13 am
- Forum: Secondary Level
- Topic: Exponential Diophantine Eq
- Replies: 3
- Views: 3895
Exponential Diophantine Eq
Solve the following equation in natural numbers:
$3^m-7^n=2$.
Source: AOPS
$3^m-7^n=2$.
Source: AOPS
Re: Tricky FE
Fix $f(1)$ any rational number. Then all other $f(n)$ are uniquely determined by the given relation and all of them must also be rational. Thus there are infinitely many solutions.
- Tue Sep 23, 2014 12:27 pm
- Forum: National Math Olympiad (BdMO)
- Topic: Junior 2009/5
- Replies: 3
- Views: 3971
Re: Junior 2009/5
If the base (as in fig 1) is parallel to the y axis, it is easily seen that the largest area is achieved when we move the base to the end of the line and the base is largest. In such situations we can get atmost $\frac {1}{2}n^2$. Now consider that the base isn't parallel to the axis. By flipping, w...
- Tue Sep 23, 2014 12:56 am
- Forum: Algebra
- Topic: Functional Equation( Japan final round 2008)
- Replies: 6
- Views: 5125
- Tue Sep 23, 2014 12:55 am
- Forum: Algebra
- Topic: Functional Equation( Japan final round 2008)
- Replies: 6
- Views: 5125
Functional Equation( Japan final round 2008)
Find all functions $ f : R\rightarrow R$ such that for any $x,y$, the relation $f(x+y)f(f(x)-y)=xf(x)-yf(y)$ satisfies.
- Sat Sep 20, 2014 9:52 pm
- Forum: Higher Secondary Level
- Topic: Functional Equation (Canada 1969)
- Replies: 10
- Views: 8303
Re: Functional Equation (Canada 1969)
Hints for two different solutions:
- Sat Sep 20, 2014 8:46 pm
- Forum: Higher Secondary Level
- Topic: Functional Equation (Canada 1969)
- Replies: 10
- Views: 8303
Functional Equation (Canada 1969)
Find all functions $f:N \rightarrow N$ such that $f(2)=2$ and for any two $m,n ; f(mn)=f(m)f(n)$ and $f(n+1)>f(n)$.
- Sat Sep 20, 2014 10:26 am
- Forum: Number Theory
- Topic: IMO Shortlist 1991
- Replies: 5
- Views: 50877
Re: IMO Shortlist 1991
$1990^{{1991}^{1992}}+1992^{{1991}^{1990}}=[{1990}^{{1991}^2}]^{{1991}^{1990}}+1992^{{1991}^{1990}}$
Now apply LTE twice.
Now apply LTE twice.