BdMO National 2021 Junior Problem 2

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Anindya Biswas
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BdMO National 2021 Junior Problem 2

Unread post by Anindya Biswas » Mon Apr 12, 2021 11:35 am

নিচের সমীকরণটার সব সমাধানের যোগফল বের করো।
\[5^{2r+1}+5^2=5^r+5^{r+3}\]

Find the sum of all solutions of the equation, \[5^{2r+1}+5^2=5^r+5^{r+3}\]
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
John von Neumann

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gwimmy(abid)
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Re: BdMO National 2021 Junior Problem 2

Unread post by gwimmy(abid) » Wed Apr 14, 2021 5:00 pm

$\begin{align*} &5^{2r+1}+5^2 = 5^r+5^{r+3} \\
\Longrightarrow & 5 \cdot (5^r)^2 + 25 = 5^r + 125 \cdot 5^r\\
\Longrightarrow & 5 \cdot (5^r)^2 + 25 = 126\cdot 5^r \\
\Longrightarrow & 5 \cdot (5^r)^2 - 126\cdot 5^r + 25 = 0 \\
\Longrightarrow & 5 \cdot (5^r)^2 -125\cdot 5^r -5^r + 25 = 0\\
\Longrightarrow & 5 \cdot (5^r) (5^r - 25) - 1 (5^r -25)=0\\
\Longrightarrow & (5^r - 25)(5^r - 1) =0 \\
\Longrightarrow & 5^r = 25\text{ or }5^r = \frac{1}{5}\\
\Longrightarrow & r = \boxed{2} \text{ or } r = \boxed{-1}
\end{align*}$
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Marzuq
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Re: BdMO National 2021 Junior Problem 2

Unread post by Marzuq » Thu Apr 15, 2021 6:04 am

My solution
It can be easily observable that an equation where both side are sum of two number with same base but different power is not possible unless the base is $-1, 0, 1, 2$

So we will consider the powers are same.
In the equation
$5^{2r+1} + 5^{2} = 5^{r} + 5^{r+3}$
We will split the equation into two case:
Case 1: $2r+1 = r+3$ and $ 2=r$
This two equation can be true at the same time and here $r = 2$

Case 2: $2r + 1 = r$ and $2 = r+3$
This two equation can also be true at the same time and here $r = -1$

So the solution of $r = (2,-1)$
And their sum $= (2) + (-1) = \boxed{1}$
And if there is any error , then inform me. A solution without using algebra :mrgreen:

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gwimmy(abid)
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Re: BdMO National 2021 Junior Problem 2

Unread post by gwimmy(abid) » Fri Apr 16, 2021 4:05 pm

Marzuq wrote:
Thu Apr 15, 2021 6:04 am
My solution
It can be easily observable that an equation where both side are sum of two number with same base but different power is not possible unless the base is $-1, 0, 1, 2$

So we will consider the powers are same.
In the equation
$5^{2r+1} + 5^{2} = 5^{r} + 5^{r+3}$
We will split the equation into two case:
Case 1: $2r+1 = r+3$ and $ 2=r$
This two equation can be true at the same time and here $r = 2$

Case 2: $2r + 1 = r$ and $2 = r+3$
This two equation can also be true at the same time and here $r = -1$

So the solution of $r = (2,-1)$
And their sum $= (2) + (-1) = \boxed{1}$
And if there is any error , then inform me. A solution without using algebra :mrgreen:
i like the solution!
Umm....the healer needs healing...

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