$x^{a} × x^{b} = x^{a+b}$
If $x^{a} = m,x^{b} = n$. Then,
$m + n = mn$
$n = m(n-1)$
Then $n-1 = 1 \rightarrow n=2$
It follows that $m=2$
So only solution $(x,a,b)=(2,1,1)$
Search found 9 matches
- Thu May 20, 2021 12:15 pm
- Forum: Junior Level
- Topic: Cute NT ^-^
- Replies: 1
- Views: 7374
- Tue Apr 20, 2021 4:18 am
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National 2021 Junior Problem 4
- Replies: 4
- Views: 6042
Re: BdMO National 2021 Junior Problem 4
By using Pythagorean theorem, $PC^2+PB^2= BC^2$ And $PC^2+(PB+8)^2= (BC+12)^2$ $or, PC^2+PB^2+16PB+64=BC^2+24BC+144$ Now when we subtract these two equations we get, $PC^2+PB^2+16PB+64-PC^2-PB^2=BC^2+24BC+144-BC^2$ $or, 16PB+64=24BC+144$ $or, 16PB-24BC=80$ From this we can say that, $16PB>24BC$ $or...
- Sat Apr 17, 2021 8:13 pm
- Forum: Junior Level
- Topic: RMO-2010/3
- Replies: 9
- Views: 10693
Re: RMO-2010/3
My solution : divisibility of 4 : if the last two digit is divisible by 4. Then the whole number is divisible by 4 Divisibility by 8 : if the last two digit is divisible by 4 but not by 8 and the 3rd digit is odd. Or if the last digit is divisible by 4 and 8 and the 3rd digit is even . Then the num...
- Sat Apr 17, 2021 10:37 am
- Forum: Junior Level
- Topic: RMO-2010/3
- Replies: 9
- Views: 10693
Re: RMO-2010/3
My solution : divisibility of 4 : if the last two digit is divisible by 4. Then the whole number is divisible by 4 Divisibility by 8 : if the last two digit is divisible by 4 but not by 8 and the 3rd digit is odd. Or if the last digit is divisible by 4 and 8 and the 3rd digit is even . Then the numb...
- Sat Apr 17, 2021 8:13 am
- Forum: Higher Secondary Level
- Topic: Pigeonhole Problem
- Replies: 14
- Views: 13096
Re: Pigeonhole Problem
My solution: Before going to the main point , we will first find the maximum amount of number we can choose from 1 to 55 such that triangle inequality doesnt fully work on any one of them Some points 1) for the maximality of the amount , we will choose the number and make it as smallest as possible ...
- Thu Apr 15, 2021 9:21 am
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National 2021 Primary Problem 6
- Replies: 3
- Views: 45267
- Thu Apr 15, 2021 9:18 am
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National 2021 Primary Category Problem 7
- Replies: 2
- Views: 44063
- Thu Apr 15, 2021 6:04 am
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National 2021 Junior Problem 2
- Replies: 3
- Views: 5862
Re: BdMO National 2021 Junior Problem 2
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 ...
- Mon Apr 05, 2021 4:40 pm
- Forum: Junior Level
- Topic: শেষ ৩টি অঙ্ক কত?
- Replies: 12
- Views: 23667
Re: শেষ ৩টি অঙ্ক কত?
Try solving this ;) $1×2×3×...×2018$ has $N$ trailing zeroes. What is the last $3$ digits of $N$? $1×2×3×...×2018$ এর শেষে $N$টি শূন্য আছে। $N$ এর শেষ তিন অংক বের কর। Note:- Trailing zeroes means number are zeroes at last of any number . Example :- 10 has 1 trailing zeroes. 2000100 has 2 trailing ze...