Search found 230 matches
- Wed Apr 21, 2021 12:28 pm
- Forum: National Math Camp
- Topic: National Camp Exam 2018 P9
- Replies: 1
- Views: 2396
Re: National Camp Exam 2018 P9
The equation \begin{align*} 9x³-3x²-3x-1 & =0 \end{align*} has a real root of the form $\frac{√3a+√3b+1}{c}$ where a, b, c are positive integers. Find a + b + c. I think you didn't type the question correctly. Short solution: Given, $9x^3-3x^2-3x-1=0$ Or,$10x^3-x^3-3x^2-3x-1=0$ Or,$10x^3-(x+1)^3=0 ...
- Sat Apr 17, 2021 9:06 pm
- Forum: Junior Level
- Topic: RMO-2010/3
- Replies: 9
- Views: 12966
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 10:42 am
- Forum: Junior Level
- Topic: RMO-2010/3
- Replies: 9
- Views: 12966
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...
- Thu Apr 15, 2021 10:50 pm
- Forum: Number Theory
- Topic: EGMO 2021 P1
- Replies: 3
- Views: 6929
Re: EGMO 2021 P1
At first note, $2021 \equiv -1\text{ (Mod 3)}$ $\therefore 2021^{2021} \equiv -1 \text{ (Mod 3)}$ So the number should change into $-1$ mod 3. So there are 3 cases In $\text{mod 3}$ $m \equiv 0$ $2m+1 \equiv 1$ $3m \equiv 0$ $m \equiv 1$ $2m+1 \equiv 0$ $3m \equiv 0$ $m \equiv -1$ $2m+1 \equiv -1$ ...
- Wed Apr 14, 2021 9:37 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National 2021 Secondary Problem 9
- Replies: 6
- Views: 8783
Re: BdMO National 2021 Secondary Problem 9
$\def\lf{\lfloor}$ $\def\rf{\rfloor}$ Before we start to calculate how many pokemon Cynthia can catch, we need to make some ideas clear. If we have a graph(not multigraph) with n vertices such that each vertex having degree $= 2$, the number of vertices we can choose such that no $2$ vertices share...
- Wed Apr 14, 2021 1:53 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National 2021 Higher Secondary Problem 8
- Replies: 5
- Views: 4533
Re: BdMO National 2021 Higher Secondary Problem 8
We claim there are only 2 values of n for which Shakur has a winning strategy. But first, Note that if Shakur starts with any number say $n$ and $n+3$ is not a square number then Tiham can choose $2$ and then Shakur has to choose $1$ and lose the game. Now our Test subjects are narrowed down to $31...
- Wed Apr 14, 2021 2:01 am
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National 2021 Higher Secondary Problem 8
- Replies: 5
- Views: 4533
Re: BdMO National 2021 Higher Secondary Problem 8
We claim there are only 2 values of n for which Shakur has a winning strategy. But first, Note that if Shakur starts with any number say $n$ and $n+3$ is not a square number then Tiham can choose $2$ and then Shakur has to choose $1$ and lose the game. Now our Test subjects are narrowed down to $31...
- Sun Apr 11, 2021 10:00 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National 2021 Higher Secondary Problem 8
- Replies: 5
- Views: 4533
Re: BdMO National 2021 Higher Secondary Problem 8
Shakur and Tiham are playing a game. Initially, Shakur picks a positive integer not greater than $1000$. Then Tiham picks a positive integer strictly smaller than that. Then they keep on doing this taking turns to pick progressively smaller and smaller positive integers until someone picks $1$. Aft...
- Sun Apr 11, 2021 9:30 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO National 2021 Higher Secondary Problem 1
- Replies: 2
- Views: 5538
Re: BdMO National 2021 Higher Secondary Problem 1
For a positive integer $n$, let $A(n)$ be equal to the remainder when $n$ is divided by $11$ and let $T(n)=A(1)+A(2)+A(3)+\dots+A(n)$. Find the value of $A(T(2021))$. $A(1)+A(2)+\cdots +A(11)=1+2+\cdots 10+0=55$ This will repeat for every multiple of 11, like, $A(12)+A(13)+\cdots +A(22)=55$ Now, $2...
- Sun Apr 11, 2021 8:03 pm
- Forum: Physics
- Topic: why so sun
- Replies: 8
- Views: 20992
Re: why so sun
An interesting thing about the sun and nuclear energy on mars is, The Mars rover sent by NASA in 2003-2004 was actually solar-powered. Which used solar panels for collecting energy. But unfortunately, martian dust used to cover those solar panels and resulting in the reduction of energy for the rove...