4 votes

Explain the double sum step by step

Two things: You seem to have moved the $j$ sum out of the $i$ sum incorrectly. Where you write $$\sum_{i=0}^n (i+1) \cdot \sum_{j=0}^i (j+1) \text{,}$$ you actually have $$\sum_{i=0}^n \left( (i+1) ...
Eric Towers's user avatar
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3 votes
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$2n$ knights around a table with namecards, is it possible that for every rotation there is exactly one person with a correct namecard?

Let $1,2,..,,2n$ be the numbers of the knights clockwise. Let $a_i$ be the number of places we have to rotate a table counter-clockwise so that $i$-th knight sits in front of his namecard. If $a_i=a_j$...
Aig's user avatar
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3 votes
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In how many ways it is possible to take out balls from the basket , such that will take out at least one from each color?

Take out one black ball, one white ball and one red ball. Now, there are $19$ black balls, $14$ white balls and $17$ red balls left. As order doesn't matter for this particular problem and balls of ...
Julio Puerta's user avatar
3 votes
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Find two coins of weight a, among n coins, where n-2 coins are coins of weight b

Sudeep has given a nice hint in the comments. First observe that for $n=3$, you require 2 moves in the worst case, so $C\le 0$. By "determine" I mean deduce the type of coin. Next, note that ...
D S's user avatar
  • 3,920
3 votes

Number of possible relations with following restrictions | Discrete Mathematics

Well, I think it is better not representing the relation as $n^2$ directed pairs that either exist or not, but as a graph of $\frac{n\left(n-1\right)}{2}$ general edges between 2 elements that can ...
MBobrik's user avatar
  • 131
2 votes

How to prove the existence of this progression? Struggling with a BDMO problem.

HINT: Consider the geometric sequence $a,ar,ar^2,ar^3,\dots,ar^{2022}$. Choose any Pythagorean triplet $(x,y,z)$. Set $r=x/y$ and $a=y^{2022}$. Do you see what the individual terms become? Does it ...
Sahaj's user avatar
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2 votes

How many natural numbers $a\le100$ are there such that $a=[\frac a2]+[\frac a3]+[\frac a5]$, where [.] represents the greatest integer function?

Yes, the answer is $\frac{29}{100}$. There are a few details that need to be taken care of. You are setting $\gamma$ to be one of $\{1,2,\dots,29\}$ and each value of $\gamma$ generates a value for $n$...
Sathvik's user avatar
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2 votes
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How many natural numbers $a\le100$ are there such that $a=[\frac a2]+[\frac a3]+[\frac a5]$, where [.] represents the greatest integer function?

The answer looks correct. But you probably should demonstrate that $a\le 100$ when $\gamma=1,2,…,29$. It is sufficient to show that $n\le 2$. It is true since $$n=\gamma-[\gamma/2]-[\gamma/3]-[\gamma/...
Aig's user avatar
  • 2,740
2 votes
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In how many ways we can arrange $n$ men and $m$ women in a circle such that between every $2$ men we can have at most $m -1$ women?

The total number of ways we can arrange $n$ men and $m$ women is $(n+m-1)!$. Now, the complementary event of what we want is the $m$ women being between two men, that is, all women being together and ...
Julio Puerta's user avatar
1 vote

Finding formula for sequence

The sequence with entries $a(n)$ for $n$ even and $b(n)$ for $n$ odd is $$ \frac{1 + (-1)^n}{2} a(n) + \frac{1 - (-1)^n}{2} b(n) $$
Robert Israel's user avatar
1 vote

$2n$ knights around a table with namecards, is it possible that for every rotation there is exactly one person with a correct namecard?

Another way to present the same argument (I solved it like this when I saw it). Label the cards around the table in clockwise order as $0,1,2 \ldots 2n-1$ and name the knights with their respective ...
D S's user avatar
  • 3,920
1 vote

Maximum independent set of a residual graph

Not much can be said in general: your graph could consist of some number of independent sets of a fixed size, and then a lot of edges between these. For instance, the graph $K_{t,t}$ consisting of two ...
Mathieu Rundström's user avatar
1 vote

Learning effective

The question is somewhat personal, as there is no guaranteed method for raising your grades; however, I could suggest a couple of things that helped with mine, Printing stuff instead of viewing them ...
Mmm's user avatar
  • 31
1 vote

Periods of difference sequences modulo N

I have made significant progress with this, although mainly on the numerical side. The progress is based on two insights. First, assume $N$ is an even integer which is not a prime power. If the first-...
olekirkeby's user avatar
1 vote

How to prove the existence of this progression? Struggling with a BDMO problem.

A different approach All we need is the Egyptian triangle. Or the most common Pythagorean triple. Let us substitute $2023$ with $5$ for the sake of simplicity. Then the following sequence works: $$3^5\...
Aig's user avatar
  • 2,740
1 vote

Exponential generating function of the sequence $1,0,1,0,\dots$

A few different ways to approach this one. For your sequence: \begin{equation} S = 1, 0, 1,0\dots \end{equation} If we let $T_n$ be the $n$-th value of $S$ with $T_0 = 1$, then we can define $S$ via ...
David Galea's user avatar
1 vote

Exponential generating function of the sequence $1,0,1,0,\dots$

HINT: Have you considered $\frac{e^x+e^{-x}}{2}$. ETA: And that is basically all there is. Let $(a_i)_{i=0}^{\infty}$ be a sequence. Then the exponential generating function $f$ for this sequence is ...
Mike's user avatar
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1 vote

Let $a_{n}$ be the sequence defined inductively by $a_{1} = 2$ and $a_{n+1} = 1/2 ( a_n + 2/a_n)$. Prove this sequence is decreasing

Since you already have $$(a_n)^2\ge2$$ Dividing by $2a_n$ since we know it is positive we get $${a_n\over 2}\ge{1\over a_n}\\a_n\ge{a_n\over 2}+{1\over a_n}\\a_n\ge a_{n+1}$$ Which shows it is ...
RandomGuy's user avatar
  • 1,167
1 vote

Proof: If connected graph G has only one cut-vertex, then every longest path contins the cut vertex.

You will not get better by having others give you the answers. So I will just give you some ideas on what to try. First of all, when dealing with "prove or disprove" questions, don't just ...
Jonas Linssen's user avatar
1 vote

Show that ${n^n}^{n}>n(n!)((n!)!)$ where n is a positive integer greater than or equal to $3$.

HINT: Take logarithms to conclude $$n^{n^n} \ge ((n!)!)^4,$$ and note that this gives you what you need. En route, use the easy to see inequality $$M^M \ge 2^M (M!).$$ IF YOU NEED MORE ELABORATION: ...
Mike's user avatar
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1 vote

How many natural numbers $a\le100$ are there such that $a=[\frac a2]+[\frac a3]+[\frac a5]$, where [.] represents the greatest integer function?

Let $a=6q+r$. Then we need $r =\left[\dfrac{r}{2} \right]+\left[\dfrac{r}{3} \right]+\left[\dfrac{q+r}{5} \right]$ When $r=0$, $q$ can be $1,2,3,4$ For $r=1,2,3,4,5$ it is easy to see that $q$ has $5$ ...
Hari Shankar's user avatar
  • 3,558

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