Search found 192 matches

by Anindya Biswas
Thu May 06, 2021 5:03 pm
Forum: National Math Camp
Topic: Problem - 01 - National Math Camp 2021 Number Theory Exam - "GCD, Coprime, Divisibility"
Replies: 1
Views: 265

Re: Problem - 01 - National Math Camp 2021 Number Theory Exam - "GCD, Coprime, Divisibility"

Let $g=\text{gcd}(a,b)$ $\therefore\exists x,y\in\mathbb{Z}$ such that $\text{gcd}(x,y)=1$ and $a=gx, b=gy$. By Bézout's identity , $\exists k_1,k_2\in\mathbb{Z}$ such that $k_2x-k_1y=1$. Claim : Choosing $m=a+Nk_1,n=b+Nk_2$ satisfies the necessary condition. Proof : It's sufficient to prove that $\...
by Anindya Biswas
Thu May 06, 2021 4:32 pm
Forum: National Math Camp
Topic: Problem - 03 - National Math Camp 2021 Number Theory Exam - "Infinitely many prime divisors"
Replies: 0
Views: 156

Problem - 03 - National Math Camp 2021 Number Theory Exam - "Infinitely many prime divisors"

Let $P(x)$ be a nonzero integer polynomial, that is, the coefficients are all integers. We call a prime $q$ "interesting" if there exists some natural number $n$ for which $q|2^n+P(n)$. Prove that there exist infinitely many “interesting” primes.
by Anindya Biswas
Thu May 06, 2021 4:27 pm
Forum: National Math Camp
Topic: Problem - 02 - National Math Camp 2021 Number Theory Exam - "Group theory"
Replies: 0
Views: 163

Problem - 02 - National Math Camp 2021 Number Theory Exam - "Group theory"

Let $p$ be a prime number. We call a subset $S$ of $\{1,2,\cdots,p-1\}$ "good" if it satisfies the property that for every $x,y\in S, xy\text{ mod }{p}$ is also in $S$. How many "good" sets are there?
by Anindya Biswas
Thu May 06, 2021 4:19 pm
Forum: National Math Camp
Topic: Problem - 01 - National Math Camp 2021 Number Theory Exam - "GCD, Coprime, Divisibility"
Replies: 1
Views: 265

Problem - 01 - National Math Camp 2021 Number Theory Exam - "GCD, Coprime, Divisibility"

Let $N$ be a nonzero integer. Given any $a,b$ such that $\text{gcd}(a,b,N)=1$. Prove that you can find $m,n$ such that $\text{gcd}(m,n)=1$ and $N|m-a, N|n-b$.
by Anindya Biswas
Tue May 04, 2021 11:22 pm
Forum: National Math Camp
Topic: Problem - 05 - National Math Camp 2021 Combinatorics Test - "Checkers on a board"
Replies: 3
Views: 301

Re: Problem - 05 - National Math Camp 2021 Combinatorics Test - "Checkers on a board"

Mehrab4226 wrote:
Tue May 04, 2021 10:11 pm
No one?? :'(
We have to show if condition 1,2 satisfies, then the conclusion is, but not necessarily the 2nd condition must be true if the inequality is true
by Anindya Biswas
Fri Apr 30, 2021 5:40 pm
Forum: National Math Camp
Topic: Problem - 05 - National Math Camp 2021 Combinatorics Test - "Checkers on a board"
Replies: 3
Views: 301

Problem - 05 - National Math Camp 2021 Combinatorics Test - "Checkers on a board"

We place some checkers on an $n\times n$ checkerboard so that they follow the conditions : Every square that does not contain a checker shares a side with one that does; Given any pair of squares that contain checkers, we can find a sequence of squares occupied by checkers that start and end with th...
by Anindya Biswas
Fri Apr 30, 2021 5:33 pm
Forum: National Math Camp
Topic: Problem - 04 - National Math Camp 2021 Combinatorics Test - "Alternating Parity"
Replies: 2
Views: 256

Problem - 04 - National Math Camp 2021 Combinatorics Test - "Alternating Parity"

Let $n\geq1$ be an integer. A non-empty set is called “good” if the arithmetic mean of its elements is an integer. Let $T_n$ be the number of good subsets of $\{1,2,3,\cdots,n\}$. Prove that for all integers $n$, $T_n$ and $n$ leave the same remainder when divided by $2$.
by Anindya Biswas
Fri Apr 30, 2021 5:25 pm
Forum: National Math Camp
Topic: Problem - 03 - National Math Camp 2021 Combinatorics Test - "All you see is isosceles"
Replies: 1
Views: 248

Problem - 03 - National Math Camp 2021 Combinatorics Test - "All you see is isosceles"

Show that for all positive integers $n\geq8$, you can cut any quadrilateral into $n$ isosceles triangles.
by Anindya Biswas
Fri Apr 30, 2021 5:22 pm
Forum: National Math Camp
Topic: Problem - 02 - National Math Camp 2021 Combinatorics Test - "The result is invariant"
Replies: 1
Views: 233

Problem - 02 - National Math Camp 2021 Combinatorics Test - "The result is invariant"

There are $2021$ stones in a pile. At each step, Lazim chooses a pile with at least two stones, splits it into two piles, and multiplies the sizes of the resulting two piles. He keeps doing this until there are $2021$ piles each containing exactly one stone. Finally, he adds up all the products he o...
by Anindya Biswas
Fri Apr 30, 2021 5:14 pm
Forum: National Math Camp
Topic: Problem - 01 - National Math Camp 2021 Combinatorics Test - "Points not maintaining social distance"
Replies: 1
Views: 221

Problem - 01 - National Math Camp 2021 Combinatorics Test - "Points not maintaining social distance"

You have a set $S$ of $19$ points in the plane such that given any three points in $S$, there exist two of them whose distance from each other is less than $1$. Prove that there exists a circle of radius $1$ that encloses at least $10$ points of $S$.