Search found 46 matches
- Fri Apr 23, 2021 6:59 pm
- Forum: National Math Camp
- Topic: BdMO 2021 National Camp Discussion Thread
- Replies: 6
- Views: 4128
Re: Example
Finding the magic number is tough for newbies like me ): This is how I did it: We see that $1$ obviously works. Then show that the perfect square must be odd and so, $n$ will have to be even, If $n \geq 2$. If not then $2^n+12^n+2011^n\not\equiv1$(mod $8$) And then we do the same thing as Dustan di...
- Fri Apr 23, 2021 4:43 pm
- Forum: National Math Camp
- Topic: BdMO 2021 National Camp Discussion Thread
- Replies: 6
- Views: 4128
Example
This is an example.
I'm sharing a hint for problem 1 of IMO Prep Pset-1.
Problem-
Find, with proof, all positive integers $n$ for which $2^n+12^n+2011^n$ is a perfect square.
Hint-
I'm sharing a hint for problem 1 of IMO Prep Pset-1.
Problem-
Find, with proof, all positive integers $n$ for which $2^n+12^n+2011^n$ is a perfect square.
Hint-
- Fri Apr 23, 2021 4:27 pm
- Forum: National Math Camp
- Topic: BdMO 2021 National Camp Discussion Thread
- Replies: 6
- Views: 4128
BdMO 2021 National Camp Discussion Thread
Hey, Campers! This thread's purpose is:- 1. To post our solutions to the problem sets. (Using Latex is easier this way) 2. To check each other's solutions and learn from them. 3. To share hints about the problems. Good Practices to follow:- 1. Use [hide][/hide] to hide your hint/solution. 2. Describ...
- Mon Apr 19, 2021 7:07 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National 2021 Secondary Problem 12
- Replies: 3
- Views: 4888
Re: BdMO National 2021 Secondary Problem 12
Without restrictions the answer would be $\binom{10}{5}^2=63504$. We now try to find the number of ways such that Gamakichi and Gamatatsu have at least one common point in their path. To do so, we make a bijection: Bijection: Consider a path of the two toads where there is at least one common point...
- Fri Apr 16, 2021 11:01 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BDMO Secondary National 2021 #3
- Replies: 5
- Views: 7878
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 ...
- Thu Apr 15, 2021 4:00 pm
- Forum: Social Lounge
- Topic: Share your AoPS id name
- Replies: 8
- Views: 37601
Re: Share your AoPS id name
Mine is the same as here- Pro_GRMR
- Tue Apr 13, 2021 8:48 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National 2021 Junior Problem 1
- Replies: 1
- Views: 5989
Re: BdMO National 2021 Junior Problem 1
মিতা একটা সারিতে \(2024\)-টা ধনাত্মক পূর্ণসংখ্যা এমনভাবে লিখেছে যাতে পরপর যেকোনো চারটা সংখ্যার গুণফল \(2100\) হয়। সারির প্রথম সংখ্যাটা \(7\), \(1011\)-তম সংখ্যাটা \(5\), \(2014\)-তম সংখ্যাটা \(20\)। সারির সবার শেষের সংখ্যাটা কত? Mita wrote $2024$ positive integers in a row such that the product of...
- Tue Apr 13, 2021 8:29 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National 2021 Secondary Problem 9
- Replies: 6
- Views: 7917
Re: BdMO National 2021 Secondary Problem 9
Choose 1 to 10 numbers 1,2,3 and 4,5,6 and 7,8,9 and 10 (1,2,3 and 4,5,6 and 7,8,9 every 3 of them are enemies with each other) 1 isn't enemy with 4, 2 isn't enemy with 5, 3 isn't enemy with 6, (all 6 here) There are 7,8,9 and 10 left From them you can take any 1 number with 10 (all 2 here) 2+6=8 C...
- Sun Apr 11, 2021 10:48 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National 2021 Higher Secondary Problem 4
- Replies: 1
- Views: 2162
Re: BdMO National 2021 Higher Secondary Problem 4
$P(x)$ is a polynomial in $x$ with non-negative integer coefficients. If $P(1)=5$ and $P(P(1))=177$, what is the sum of all possible values of $P(10)$? Let $P(x)= a_0+a_1x+a_2x^2+ \dots+a_nx^n$ where $a_n \geq 1$. Notice, $P(1)= a_0+a_1\cdot1+a_2\cdot1^2+ \dots+a_n\cdot1^n = a_0+a_1+a_2+ \dots+a_n=...
- Sun Apr 11, 2021 10:24 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National 2021 Higher Secondary Problem 5
- Replies: 1
- Views: 2031
Re: BdMO National 2021 Higher Secondary Problem 5
How many ways can you roll three $20$-sided dice such that the sum of the three rolls is exactly $42$? Here the order of the rolls matters. (Note that a $20$-sided die is very much like a regular six-sided die other than the fact that it has $20$ faces instead of the regular six.) Re-stating the qu...