মনে করো, \(r\) একটা ধনাত্মক বাস্তব সংখ্যা। \([r]\) দিয়ে আমরা \(r\)-এর পূর্ণসাংখ্যিক অংশ বোঝাই আর \(\{r\}\) দিয়ে আমরা \(r\)-এর ভগ্নাংশিক অংশটা বোঝাই। যেমন যদি \(r=32.86\) হয়, তাহলে \(\{r\}=0.86\) এবং \([r]=32\)। এমন সব ধনাত্মক সংখ্যা \(r\)-এর যোগফল কত যদি \(25\{r\}+[r]=125\) হয়?
Let $r$ be a positive real number. Denote $[r]$ by the integer part of $r$ and by $\{r\}$ the fractional part of $r$. For example, if $r=32.86$, then $\{r\}=0.86$ and $[r]=32$ . What is the sum of all positive numbers $r$ satisfying
\[ 25\{r\} + [r]=125\]
BDMO Secondary National 2021 #3
- Mehrab4226
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Re: BDMO Secondary National 2021 #3
As $\{r\}$ is a fraction,Mehrab4226 wrote: ↑Sat Apr 10, 2021 12:42 pmLet $r$ be a positive real number. Denote $[r]$ by the integer part of $r$ and by $\{r\}$ the fractional part of $r$. For example, if $r=32.86$, then $\{r\}=0.86$ and $[r]=32$ . What is the sum of all positive numbers $r$ satisfying
\[ 25\{r\} + [r]=125\]
$0 \leq \{r\} < 1$
$\Longrightarrow 0 \leq 25\{r\} < 25$
Also Notice as $25\{r\} = 125 - [r]$,
$25\{r\}$ is an integer.
By taking different integer values of $25\{r\}$ from 0 to 24 we see that $r = 101.04, 102.08, \dots, 124.96, 125$
Summing the partially arithmetic series we get the answer of $\frac{101.04+124.96}{2} \times 24 + 125 = \boxed{2837}$
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Re: BDMO Secondary National 2021 #3
Isn't 125 a real number?
I mean it fulfills the condition:: [125] + {0} = 125.0 and there no condition like {r} != 0
so, why won't we add that number?
I mean it fulfills the condition:: [125] + {0} = 125.0 and there no condition like {r} != 0
so, why won't we add that number?
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Re: BDMO Secondary National 2021 #3
It is considered in the solutionPritom12345 wrote: ↑Sun Apr 11, 2021 9:21 pmIsn't 125 a real number?
I mean it fulfills the condition:: [125] + {0} = 125.0 and there no condition like {r} != 0
so, why won't we add that number?
Umm....the healer needs healing...
Re: BDMO Secondary National 2021 #3
There were some mistakes. The forum doesn't give me edit access anymore.
This is the correct version
As $\{r\}$ is a fraction,
$0 \leq \{r\} < 1$
$\Longrightarrow 0 \leq 25\{r\} < 25$
Also Notice as $25\{r\} = 125 - [r]$,
$25\{r\}$ is an integer.
By taking different integer values of $25\{r\}$ from 0 to 24 we see that $r = 101.96, 102.92, \dots, 124.04, 125$
Summing the partially arithmetic series we get the answer of $\frac{101.96+124.04}{2} \times 24 + 125 = \boxed{2837}$
The answer is the same but Notice the decimal value of $r$ was wrong.
This is the correct version
As $\{r\}$ is a fraction,
$0 \leq \{r\} < 1$
$\Longrightarrow 0 \leq 25\{r\} < 25$
Also Notice as $25\{r\} = 125 - [r]$,
$25\{r\}$ is an integer.
By taking different integer values of $25\{r\}$ from 0 to 24 we see that $r = 101.96, 102.92, \dots, 124.04, 125$
Summing the partially arithmetic series we get the answer of $\frac{101.96+124.04}{2} \times 24 + 125 = \boxed{2837}$
The answer is the same but Notice the decimal value of $r$ was wrong.
"When you change the way you look at things, the things you look at change." - Max Planck