Dhaka Secondary 2009/2
Forum rules
Please don't post problems (by starting a topic) in the "Secondary: Solved" forum. This forum is only for showcasing the problems for the convenience of the users. You can post the problems in the main Divisional Math Olympiad forum. Later we shall move that topic with proper formatting, and post in the resource section.
Please don't post problems (by starting a topic) in the "Secondary: Solved" forum. This forum is only for showcasing the problems for the convenience of the users. You can post the problems in the main Divisional Math Olympiad forum. Later we shall move that topic with proper formatting, and post in the resource section.
If $(DBC)^2=BCABC$ find the value of $D$?
Re: Dhaka Secondary 2009/2
D = 2 . Because DBC = 276
- leonardo shawon
- Posts:169
- Joined:Sat Jan 01, 2011 4:59 pm
- Location:Dhaka
Re: Dhaka Secondary 2009/2
what is DBC? Where did it come?
Ibtehaz Shawon
BRAC University.
long way to go .....
BRAC University.
long way to go .....
-
- Posts:7
- Joined:Wed Jan 26, 2011 9:36 am
- Location:Sylhet
- Contact:
Re: Dhaka Secondary 2009/2
FroM (DBC)^2
Re: Dhaka Secondary 2009/2
Sudip: Please post complete solution (or at least an outline on how to solve it). Posting just the answer is strongly discouraged as we can not learn anything from a number, can we?
"Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin
Please install LaTeX fonts in your PC for better looking equations,
learn how to write equations, and don't forget to read Forum Guide and Rules.
Please install LaTeX fonts in your PC for better looking equations,
learn how to write equations, and don't forget to read Forum Guide and Rules.
Re: Dhaka Secondary 2009/2
Moon vai I am posting a soultion,theoritical though,ar pura puri observation er upor prove kora,(observation gular nirghat prove ase but ami janina)so viewers please look for a better solution -
ami amar koyekta observation mention kori( shobari egula jana so skip korleo chole):-
1.last digit 5 hole shetar square(je kono power) e last digit alltime 5 hoy and second last digit hoy 2
2. last digit 6 hole shetar khetreo last digit (1) follow kore but second last digit e ei sequence ta paoa jay
a) second last digit jodi 0 hoy tahole square er por 2nd last digit hobe 3(ofcourese)
b)last two digit 16 hole square er por 56 hoy
c)last two digit 26 hole 76 paoa jai
d)last two digit 36 hole 96 paoa jai ... blah blah blah
point hoilo last two digit jodi (0,1,2,3,4)6 type hoi tahole 2nd last digit hoy 3,5,7,9,1.....(2)
same happenns when(5,6,7,8,9)type hoy tahole loop akare again 3,5,7,9,1 ashe(3)...(program e deksi,really ashe, but kan ashe thikmoton akhono prove kore dekhate pari nai) eita apnara try koren pls....(ei kothagular aktao ami exma hall e liktam na)
$COME$ $TO$ $THE$ $SOLUTION$-
DBC three digit but BCABC two digit , and $(300)^2=90000, 32\times300=9600 + 256= 9856$, kajei number ta 316 xceed korte parbe na.so D(max)=3
metioned above, $(DBC)^2=BCABC$ er mane hoilo c should be either 5 or 6...
case 1
if $C=5$ then
$B=2$ hobe so the equation will be like $(D25)^2= 25A25, 25^2=625$, so $A=6$
Akhon D= 1 hoile B er value 1,2,3 er moddhe, B=2 kajei D er value 1 hobe. Akhon D= 1
hole $(125)^2$ er small part - $10000+2.100.25= 10000+5000=15000+$ ja ashe tai.....
So D= 1 hole B= 2 hobe na, so c=5 hobe na
Case 2-
if c=6 then
b= 1,3,5,7,9 er kisu akta... (2) and (3) explore kore pai
b= 7 hole $BC^2$ er second last digit 7 orfe last 2 digit $76$, kajei $BCABC= 76A76$
akhon D=1 hole B= 1,2,3 , but b=7, D er value 2 hole B er value $[2^2,3^2)=4,5,6,7,8$
So $D= 2$ hole or $DBC =276$ holei all condition fulfil hoy....
SO the result is $D=2$..........
ami amar koyekta observation mention kori( shobari egula jana so skip korleo chole):-
1.last digit 5 hole shetar square(je kono power) e last digit alltime 5 hoy and second last digit hoy 2
2. last digit 6 hole shetar khetreo last digit (1) follow kore but second last digit e ei sequence ta paoa jay
a) second last digit jodi 0 hoy tahole square er por 2nd last digit hobe 3(ofcourese)
b)last two digit 16 hole square er por 56 hoy
c)last two digit 26 hole 76 paoa jai
d)last two digit 36 hole 96 paoa jai ... blah blah blah
point hoilo last two digit jodi (0,1,2,3,4)6 type hoi tahole 2nd last digit hoy 3,5,7,9,1.....(2)
same happenns when(5,6,7,8,9)type hoy tahole loop akare again 3,5,7,9,1 ashe(3)...(program e deksi,really ashe, but kan ashe thikmoton akhono prove kore dekhate pari nai) eita apnara try koren pls....(ei kothagular aktao ami exma hall e liktam na)
$COME$ $TO$ $THE$ $SOLUTION$-
DBC three digit but BCABC two digit , and $(300)^2=90000, 32\times300=9600 + 256= 9856$, kajei number ta 316 xceed korte parbe na.so D(max)=3
metioned above, $(DBC)^2=BCABC$ er mane hoilo c should be either 5 or 6...
case 1
if $C=5$ then
$B=2$ hobe so the equation will be like $(D25)^2= 25A25, 25^2=625$, so $A=6$
Akhon D= 1 hoile B er value 1,2,3 er moddhe, B=2 kajei D er value 1 hobe. Akhon D= 1
hole $(125)^2$ er small part - $10000+2.100.25= 10000+5000=15000+$ ja ashe tai.....
So D= 1 hole B= 2 hobe na, so c=5 hobe na
Case 2-
if c=6 then
b= 1,3,5,7,9 er kisu akta... (2) and (3) explore kore pai
b= 7 hole $BC^2$ er second last digit 7 orfe last 2 digit $76$, kajei $BCABC= 76A76$
akhon D=1 hole B= 1,2,3 , but b=7, D er value 2 hole B er value $[2^2,3^2)=4,5,6,7,8$
So $D= 2$ hole or $DBC =276$ holei all condition fulfil hoy....
SO the result is $D=2$..........
- nafistiham
- Posts:829
- Joined:Mon Oct 17, 2011 3:56 pm
- Location:24.758613,90.400161
- Contact:
Re: Dhaka Secondary 2009/2
A little information.
if $N=.........25$ then, $N^n=.........25$
and, if $N=..........76$ then $N^n=.........76$
these are the only two numbers of two digits like this.
from 'আমি তপু' by Muhammad Zafar Iqbal
if $N=.........25$ then, $N^n=.........25$
and, if $N=..........76$ then $N^n=.........76$
these are the only two numbers of two digits like this.
from 'আমি তপু' by Muhammad Zafar Iqbal
\[\sum_{k=0}^{n-1}e^{\frac{2 \pi i k}{n}}=0\]
Using $L^AT_EX$ and following the rules of the forum are very easy but really important, too.Please co-operate.
Using $L^AT_EX$ and following the rules of the forum are very easy but really important, too.Please co-operate.