All Questions
133
questions
1
vote
1
answer
86
views
Number repositioning
You are given $2^n$ numbers, in one step you move the numbers in odd positions to the beginning of the list and the numbers in even positions to the end of the list, keeping the initial order among ...
- 258
3
votes
2
answers
282
views
How many 2x2 matrices with module 27 inputs exist with the condition being invertible?
I've been checking the following discrete mathematics exercise:
How many 2x2 matrices with module 27 inputs exist with the condition being invertible?
Checking many information in the internet I ...
- 875
0
votes
2
answers
204
views
Show that $\binom {2n}n \leq(2n)^{\pi (2n)}$ where $\pi(2n) $ is number of prime number less than $2n$
Show that $$\binom {2n}n \leq(2n)^{\pi (2n)}$$
where, as usual, $\pi(2n) $ is number of prime number less than $2n$.
I was solving basic techniques of combinatorial theory by Daniel Cohen.
I was ...
- 103
6
votes
1
answer
213
views
Periodic sequences resulting from a summation over the Thue–Morse sequence
Let $s_2(n)$ denote the sum of digits of $n$ in base-2 (OEIS sequence A000120), and $t_n=(-1)^{s_2(n)}$. Note that $t_n$ is the signed Thue–Morse sequence (OEIS sequence A106400), satisfying the ...
- 47.3k
8
votes
5
answers
312
views
Show that $ \sum_{k=0}^{n} \binom{2n+1}{2k} 3^k $ is divisible by $2^n$
I want to prove that
$$
\sum_{k=0}^{n} \binom{2n+1}{2k} 3^k = \sum_{k=0}^{2n} \binom{2n}{k} 3^{\lceil k/2 \rceil}
$$
is divisible by $2^n$.
I tried induction the following way
\begin{align*}
\...
- 18.2k
1
vote
2
answers
69
views
what's the minimum slices does it needs to slice a cube into $1100$ equal parts? [closed]
I have tried using prime factorization which is $2^2 \times 5^2 \times 11$ and found out that I'll need $37$ slices. but in the question paper which is a multiple choice question there's no such ...
- 79
2
votes
2
answers
140
views
Problem on Principle of Inclusion-Exclusion
How many integers $1, 2,....., 11000$ are invertible modulo $880$?
$880$ can be rewritten as $2^4\cdot5\cdot11$.
So I am supposed to find the number of integers in this range that have $2$, $5$ or $...
- 2,042
0
votes
3
answers
2k
views
Difference between surjections, injections and bijections
I am a little confused by the definitions of these different types of functions:
I think the definition of a surjection is pretty clear in that each element of x is mapped to some value of y.
But I'...
- 2,042
1
vote
1
answer
79
views
I have this home work problem i cannot proved this sum over product of fibonacci sequence i used definition of fibonacci numbers and geometric sums
how to prove this fibonacci identity?
$\sum_{k=0}^{n-3} F_k F_{n-k-3}$ = $(1/5)[ (n-3) L_{n-3} - F_{n-3}]$
I used generating functions and geometric sum and how i could get the result.?
- 53
2
votes
2
answers
588
views
How many ordered quadruples $(w, x, y, z)$ of non-negative integers are there such that $wxyz = 288$? Why?
I found the prime factorization
$$288 = 2^5 \cdot 3^2$$
and then I tried to set up some algebraic equations, but got stuck. How would we proceed so that we get the answer? Especially when we are ...
- 2,042
5
votes
2
answers
656
views
Construct $4 \times 4$ magic square with fixed "1"
The method I have found to generate $4\times 4$ magic squares gives me a result in which the number "1" is at of the corners of the square. How can we extend this to a method to generate a magic ...
- 2,760
2
votes
1
answer
371
views
Proof of an integer partitions inequality
I came across an interesting problem the other day.
Let $P_n$ be the number of partitions of a positive integer $n$. For instance $P_4$ = $5$, as there are five ways of partitioning $4$:
$4$
$3+1$
$...
- 233
2
votes
3
answers
689
views
Integer Partitions asymptotic behaviour
Let $ P(n) $ be the number of partitions of number $n$.
Prove that $ P(n)$, grows faster than any polynomial from $n$.
I am looking for an elementary (rather bijective) proof of the fact.
- 571
1
vote
1
answer
73
views
Pairing $2n$ real numbers
Let $\{l_1,l_2,\dots,l_{2n}\}$ be a set of real numbers.
I need to divide those numbers into -$n$- pairs such that the sum of their multiplications (of each pair) will be as maximum as possible.
I ...
- 63
2
votes
2
answers
265
views
A consequence of the Fundamental Theorem of Arithmetic.
The book said that:"If $S={1,2,3,.....,200}$, then for each $x \in S$, we may write $x=2^{k}y$, with $k \geq 0$, and gcd(2,y)=1." and the book added that this result follows from the Fundamental ...
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