## BDMO Secondary National 2021 #3

Discussion on Bangladesh Mathematical Olympiad (BdMO) National
Mehrab4226
Posts: 211
Joined: Sat Jan 11, 2020 1:38 pm

### BDMO Secondary National 2021 #3

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
$25\{r\} + [r]=125$
The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
-Henri Poincaré

Pro_GRMR
Posts: 46
Joined: Wed Feb 03, 2021 1:58 pm

### Re: BDMO Secondary National 2021 #3

Mehrab4226 wrote:
Sat Apr 10, 2021 12:42 pm
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
$25\{r\} + [r]=125$
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 $r = 101.04, 102.08, \dots, 124.96, 125$
Summing the partially arithmetic series we get the answer of $\frac{101.04+124.96}{2} \times 24 + 125 = \boxed{2837}$
"When you change the way you look at things, the things you look at change." - Max Planck

Pritom12345
Posts: 2
Joined: Sat Apr 03, 2021 7:41 pm

### 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?

gwimmy(abid)
Posts: 9
Joined: Tue Apr 06, 2021 11:23 am

### Re: BDMO Secondary National 2021 #3

Pritom12345 wrote:
Sun Apr 11, 2021 9:21 pm
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?
It is considered in the solution
Umm....the healer needs healing...

Pro_GRMR
Posts: 46
Joined: Wed Feb 03, 2021 1:58 pm

### 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.
"When you change the way you look at things, the things you look at change." - Max Planck

Lelu
Posts: 1
Joined: Fri Apr 16, 2021 12:02 pm