## Search found 192 matches

- 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 $\...

- 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.

- 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?

- 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 $N|m-a, N|n-b$.

- 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"

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

- 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...

- 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 $2$.

- 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.

- 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...

- 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 $S$.