## Secondary and Higher Secondary Marathon

For students of class 11-12 (age 16+)
Posts: 244
Joined: Sat May 07, 2011 12:36 pm
Location: B.A.R.D , kotbari , Comilla

### Re: Secondary and Higher Secondary Marathon

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 .
$\frac{1}{0}$

Fahim Shahriar
Posts: 138
Joined: Sun Dec 18, 2011 12:53 pm

### Re: Secondary and Higher Secondary Marathon

$\frac {n^5} {5} + \frac {n^4} {2} + \frac {n^3} {3} - \frac {n} {30}$

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

$= \frac {n*(n+1)*(2n+1)*(3n^2+3n-1)} {30}$ ; 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. ^_^
Name: Fahim Shahriar Shakkhor
Notre Dame College

SANZEED
Posts: 550
Joined: Wed Dec 28, 2011 6:45 pm

### Re: Secondary and Higher Secondary Marathon

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.
$\color{blue}{\textit{To}} \color{red}{\textit{ problems }} \color{blue}{\textit{I am encountering with-}} \color{green}{\textit{AVADA KEDAVRA!}}$

sm.joty
Posts: 327
Joined: Thu Aug 18, 2011 12:42 am
Location: Dhaka

### Re: Secondary and Higher Secondary Marathon

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.
হার জিত চিরদিন থাকবেই
তবুও এগিয়ে যেতে হবে.........
বাধা-বিঘ্ন না পেরিয়ে
বড় হয়েছে কে কবে.........

sm.joty
Posts: 327
Joined: Thu Aug 18, 2011 12:42 am
Location: Dhaka

### Re: Secondary and Higher Secondary Marathon

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
হার জিত চিরদিন থাকবেই
তবুও এগিয়ে যেতে হবে.........
বাধা-বিঘ্ন না পেরিয়ে
বড় হয়েছে কে কবে.........

Posts: 244
Joined: Sat May 07, 2011 12:36 pm
Location: B.A.R.D , kotbari , Comilla

### Re: Secondary and Higher Secondary Marathon

If the Ans is $90$ . Then I'm ready to post my solution ..
joty ভাই , confirmation দেন ।
$\frac{1}{0}$

Fahim Shahriar
Posts: 138
Joined: Sun Dec 18, 2011 12:53 pm

### Re: Secondary and Higher Secondary Marathon

Nadim Ul Abrar wrote:If the Ans is $90$ . Then I'm ready to post my solution ..
joty ভাই , confirmation দেন ।

Yes. It's $90$.
Name: Fahim Shahriar Shakkhor
Notre Dame College

sm.joty
Posts: 327
Joined: Thu Aug 18, 2011 12:42 am
Location: Dhaka

### Re: Secondary and Higher Secondary Marathon

Nadim Ul Abrar wrote:If the Ans is $90$ . Then I'm ready to post my solution ..
joty ভাই , confirmation দেন ।
Ans is $90$
Now post your solution and a new problem
And Shahrier, you also can post your solution
হার জিত চিরদিন থাকবেই
তবুও এগিয়ে যেতে হবে.........
বাধা-বিঘ্ন না পেরিয়ে
বড় হয়েছে কে কবে.........

Posts: 244
Joined: Sat May 07, 2011 12:36 pm
Location: B.A.R.D , kotbari , Comilla

### Re: Secondary and Higher Secondary Marathon

Just find the possible patterns for this six patterns of central $2 \times 2$ array .
চরকি.PNG (7.19 KiB) Viewed 3699 times
Then consider their possible rotations and Count .

এডিট : আমার ২০০ তম পোস্ট ।
$\frac{1}{0}$

Posts: 244
Joined: Sat May 07, 2011 12:36 pm
Location: B.A.R.D , kotbari , Comilla

### Re: Secondary and Higher Secondary Marathon

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 $N$.

$\frac{1}{0}$