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Re: Secondary and Higher Secondary Marathon

Posted: Sun Nov 18, 2012 5:27 pm
by Nadim Ul Abrar
Prove that if $n$ is a natural number,
$\displaystyle \frac {n^5}{5}+\frac {n^4}{2}+\frac {n^3}{3}−\frac {n}{30}$
is always an integer.

Source : An NT textbook .

Re: Secondary and Higher Secondary Marathon

Posted: Mon Nov 19, 2012 12:52 am
by Fahim Shahriar
$\frac {n^5} {5} + \frac {n^4} {2} + \frac {n^3} {3} - \frac {n} {30}$

$= \frac {6n^5+15n^4+10n^3-n} {30}$

$= \frac {n*(n+1)*(2n+1)*(3n^2+3n-1)} {30}$ ; it will be a integer if the numerator is divisible by $30$.

Surely one of $n$ and $(n+1)$ is divisible by $2$.
One of $n$, $(n+1)$ and $(2n+1)$ is divisible by $3$.

When $n≡1,3 (mod 5)$; $(3n^2+3n-1)$ is divisible by $5$.
When $n≡2 (mod 5)$; $(2n+1)$ is divisible by $5$.
When $n≡4 (mod 5)$; $(n+1)$ is divisible by $5$.

So it is always an integer. ^_^

Re: Secondary and Higher Secondary Marathon

Posted: Mon Nov 19, 2012 11:49 am
by SANZEED
Problem $\boxed {11}$:
Prove that the sum $1^k+2^k+3^k+...+n^k$ where $n$ is an arbitrary integer and $k$ is odd, is divisible by $1+2+3+...+n$.

Source: The USSR Olympiad Problem Book-The Divisibility of Integers.

Re: Secondary and Higher Secondary Marathon

Posted: Wed Nov 28, 2012 8:00 pm
by sm.joty
we know
$1+2+3+...+n = \frac{n(n+1)}{2}$
Let,
$S=1^k+2^k+3^k+...+n^k$

We know that for any odd $n$
$a+b|a^n+b^n$
so $1+(n-1)|1^k+(n-1)^{k}$
$2+(n-2)|2^k+(n-2)^{k}$
.
.
.
so all terms of this sequence is divisible by $n$
Now,
$1+(n+1)-1|1^k+n^k$
$2+(n+1)-2|2^k+(n-1)^k$
.
.
.
so $S$ is also divisible by $n+1$
we conclude that
$n(n+1)|S$
so,
$\frac{2S}{n(n+1)}=c$
where $c$ is a natural number.
so
$\frac{S}{n(n+1)/2}=c$
so
S is divisible by $\frac{n(n+1)}{2}$
so are done.

Re: Secondary and Higher Secondary Marathon

Posted: Wed Nov 28, 2012 8:09 pm
by sm.joty
Problem $\boxed {12}$
How many different $4 \times 4$ arrays whose entries are all $1$'s and $-1$'s have the property that the sum of entries in each row is $0$ and sum of entries in each column is $0$

Source: AIME 1997

Re: Secondary and Higher Secondary Marathon

Posted: Sun Dec 02, 2012 11:04 pm
by Nadim Ul Abrar
If the Ans is $90$ . Then I'm ready to post my solution .. :?
joty ভাই , confirmation দেন ।

Re: Secondary and Higher Secondary Marathon

Posted: Mon Dec 03, 2012 10:48 am
by Fahim Shahriar
Nadim Ul Abrar wrote:If the Ans is $90$ . Then I'm ready to post my solution .. :?
joty ভাই , confirmation দেন ।


Yes. It's $90$.

Re: Secondary and Higher Secondary Marathon

Posted: Mon Dec 03, 2012 1:55 pm
by sm.joty
Nadim Ul Abrar wrote:If the Ans is $90$ . Then I'm ready to post my solution .. :?
joty ভাই , confirmation দেন ।
Good job, Nadim. :D
Ans is $90$
Now post your solution and a new problem :)
And Shahrier, you also can post your solution

Re: Secondary and Higher Secondary Marathon

Posted: Mon Dec 03, 2012 5:28 pm
by Nadim Ul Abrar
Just find the possible patterns for this six patterns of central $2 \times 2$ array .
চরকি.PNG
চরকি.PNG (7.19KiB)Viewed 19109 times
Then consider their possible rotations and Count .





এডিট : আমার ২০০ তম পোস্ট । :shock: :D :lol: :? :x :( :cry: :mrgreen:

Re: Secondary and Higher Secondary Marathon

Posted: Mon Dec 03, 2012 5:35 pm
by Nadim Ul Abrar
P $13$

A number ($ \geq2$), is called product-perfect if it is equal to the product of all of its proper divisors. For example, $6=1×2×3$, hence $6$ is product-perfect. How many product-perfect numbers are there below $50$?

Note: A proper divisor of a number $N$ is a positive integer less than $N$ that divides $N$.

Source : Facebook .