3
votes
1answer
292 views
+500
Conjecture regarding trapping rational numbers in some special intervals
Conjecture:
Let $b\in\mathbb{N}_{\geq3}$ and $\{x_i\}$ be a collection of $b−2$ rational numbers greater than $1$. Does there always exist a natural number $a$ such that for all $i$ there exists some ...
0
votes
0answers
52 views
Inequality with natural numbers.
Let $d_1 \geq d_2 \geq \dots \geq d_n$, $n\geq 3 $ natural numbers.
Prove that
$$ \prod_{i=1}^n(d_i + 1 ) \leq \binom{ \sum_{i=1}^n d_i -d_n +n +2 }{n} - \sum_{j=1}^n \binom{ (\sum_{i=1}^n d_i) - d_j ...
0
votes
0answers
113 views
proving inequality for combinatorial sum
If somone can prove the following for every $d\leq r$ (for $d=0,1$ its easy, see below, the case d=r may be also simple, I didn't find something helpful)
$$\frac{(d!)^2}{2^{n-2d}}\sum_{k=0}^{n}{n ...
4
votes
1answer
32 views
Combinatorics question about addition
I noticed the following happening and I wonder if it can be proved:
Assume $x_1, \ldots, x_n$ are positive integers and $h$ is their least common multiple. Now assume
$$a_1x_1 + \cdots a_nx_n > ...
1
vote
1answer
47 views
Counting problem - How many times an inequality holds?
Let $k>2$ be a natural number and let $b$ be a non-negative real number.
Now, for each $n$, $m \in \{ 1, 2, ... k \}$, consider the following inequalities:
$$ mb < k - n $$
We have $k^2$ ...
2
votes
0answers
130 views
Find a tight upper bound of the following expectation.
I am stuck in finding a tight upper bound (as tight as possible) of the following expectation $$E\left [ (1-a\cdot b^{X})^{m} \right ]$$ where $X\sim B(n-1,p)$ is a binomial random variable.In ...
0
votes
0answers
51 views
proving an inequality by induction
Not sure how to proceed. I'm trying to prove that the following inequality is true. I know that $t_2 = 6$ and $t_3=17$ from the problem statement. The base case is obvious.
$t_{r+1} \leq (r+1) (t_r ...
1
vote
0answers
160 views
how to solve the following expectation? closed-form expression or approximation
Suppose there is a binomial random variable $X\sim B(n-1,p)$,how to solve the following expectation $$E[(1- b^{X})^{m}]$$ where $b\in (0,1]$ and $m\in \mathbb{N} $ are all constants.I have tried my ...
7
votes
1answer
201 views
On Magnitudes of Sums of Roots of Unity and a Simple Trigonometric Inequality
The Problem
Let $r,q,m$ be positive integers such that $4 \leq r$ and $1<m,q\leq r/2$. Is it the case that
$$\left | \sum_{k=0}^{q-1} \zeta^{km}\right | < \left |\sum_{k=0}^{q-1} ...
1
vote
3answers
102 views
bound for the product of numbers
Let $n \in N$. Fix $m \in [-n,n]$. I am curious, how to bound from above the following expression
$$
(n-m)^{\frac{n-m}{2}+1}(n+m)^{\frac{n+m+1}{2}}\leq \quad ?
$$
Thank you.
2
votes
0answers
83 views
Inequality with binomial coefficient
Let $n$ be a natural number, $m\in [-n, n]$. Let $p=0,\ldots, \frac{n+m}{2}$.
Show, that for all $p$,
$$
{n \choose \left[{\frac{n+m}{2}}\right]}\geq \frac{2^{n+1/2}}{\sqrt{n-p/2}}.
$$
Thank you for ...
11
votes
2answers
234 views
Find the number of pairs $(m, n)$ of positive integers such that $\frac{ m}{n+1} < \sqrt{2} < \frac{m+1}{n}$
Find the number of pairs $(m, n)$ of positive integers such that $\frac{ m}{n+1} < \sqrt{2} < \frac{m+1}{n}$ Constraint: $m$ and $n$ are both less than or equal to 1000
I toiled over this ...
4
votes
1answer
70 views
inequality with numbers--when its true?
Help me please to understand when the inequality true.
Let $n<N,$ where $n, N$ are natural numbers.
For which $n$ and $N$ the following is true
$$
n^{2n+1}\leq N^{N+1}?
$$
Thank you.
1
vote
0answers
171 views
Does this hold?
Strayed on the following question. Assume that $x_{1}$,$\ldots$, $x_{d}\ge0$ with $x_{1}+\ldots+x_{d}=1$
and $y_{1},\ldots,y_{d}\in\mathbb{R}$. Does
$$
\min_{1\le i\ne j\le ...
-2
votes
1answer
71 views
Count the number of integer solution to $\sum_{i=1}^ {4}{a_i\times b_i} \geq 8 $? [closed]
Count the number of integer solution to $\sum_{i=1}^ {4}{a_i\times b_i} \geq 8 $ such that
condition 1: $1 \leq a_i \leq 7$
condition 2: $1 \leq b_i \leq 4$
condition 3: $\sum_{i=1}^{4} {a_i} = ...
0
votes
1answer
89 views
Count the number of integer solution to $\sum_{i=1}^ {2}{a_i\times b_i} \geq 6 $ [closed]
Count the number of integer solution to $\sum_{i=1}^ {2}{a_i\times b_i} \geq 6 $ such that
condition 1: $1 \leq a_i \leq 7$
condition 2: $1 \leq b_i \leq 4$
condition 3: $\sum_{i=1}^{2} {a_i} = 8$
...
0
votes
2answers
80 views
Counting $2010^2$-tuples
Here's a problem i invented myself, but i'm not sure about my solution. I'll show it later, so people can enjoy trying to find one:
Consider the function $f:\mathbb{R}^2\to\{1,2,...,2012\}$ that ...
3
votes
1answer
102 views
An inequality involving Stirling numbers of the second kind
The task is to prove the following inequality:
$\begin{Bmatrix} mn\\ n \end{Bmatrix} \geqslant \frac{(mn)!}{(m!)^nn!}$ , where $m, n \in \mathbb{N_+}$
and to determine when the equality ...
4
votes
1answer
153 views
Inequality involving sums of fractions of products of binomial coefficients
Let $n\in\mathbb{N}$.
For $0\le l\le n$ consider
\begin{equation}
b_l:=4^{-l} \sum_{j=0}^l \frac{\binom{2 l}{2 j} \binom{n}{j}^2}{\binom{2 n}{2 j}}\text{.}
\end{equation}
Do you know a technique how ...
2
votes
3answers
209 views
Two inequalities with binomial coefficients
I have two inequalities that I can't prove:
$\displaystyle{n\choose i+k}\le {n\choose i}{n-i\choose k}$
$\displaystyle{n\choose k} \le \frac{n^n}{k^k(n-k)^{n-k}}$
What is the best way to prove ...
3
votes
3answers
398 views
Count the number of solutions of the inequality $x + y + z \leq N$
Problem
Given $A, B, C $ and $N$. How many integer solutions are there of the following inequality:
$$x + y + z \leq N$$
where $0 \leq x \leq A, 0 \leq y \leq B, 0 \leq z \leq C$?
When $A + ...
1
vote
3answers
147 views
Combinatorial inequality $\binom{n}{j}\leqslant 2^n$
I was trying to prove (or to find a counterexample) of the following inequality:
$$\binom{n}{j}\leqslant 2^n$$
As I coudn't find a proof/counterexample, I tested some numbers and could see it ...
1
vote
1answer
104 views
Comparing two binomial coefficient sums
Let $j \in N, n\in N, n>1, q\geq 2$.
I would like to show that
$$
\sum_{j=\frac{n}{\ln n}}^{\sqrt n/2}(2j-n)^q{n \choose j}<\sum_{j=\sqrt n/2}^{{\frac n2}}(2j-n)^q{n \choose j}
$$
Any help would ...
2
votes
2answers
142 views
A combinatorial inequality
How can I prove
$$
\log \binom nk \leq k \left(1 +\log\frac{n}{k}\right)
$$
where $\binom\cdot\cdot$ stands for combination.
I tried to use stirling approximation but I couldn't get the inequality.
3
votes
1answer
232 views
Equidistant Points in the Plane problem by Paul Erdos
This problem was actually posed by Erdos in Geombinatorics (Oct. 1994) and goes as follows:
Let $f(n)$ be the largest integer for which there are $n$ distinct points $x_1,x_2,\dots,x_n$ in the plane ...
-1
votes
2answers
122 views
how to prove an integer inequality
Given a positive integer $n$, is there a simple way to see that
$$ (n+3)^{n+2}(2n+5)^{n+2}(n+1)^{n+2}(2n+1)^n \ge (n+2)^{2n+4}(2n+3)^{2n+2} $$
Any hint is welcome.
0
votes
0answers
208 views
Inequality with Stirling's numbers
I supect that for all $n>k>0$:
$k^2\left\{ \begin{array}{c}n\\k\end{array} \right\}^2 +2k\left\{ \begin{array}{c}n\\k\end{array} \right\}\left\{ \begin{array}{c}n\\k-1\end{array} ...
6
votes
4answers
230 views
How many sequence of integers ($j_1 , j_2 , . . . , j_k$) are there such that $0 ≤ j_1 ≤ j_2 ≤ . . . ≤ j_k ≤ n$?
I need to solve the problem,
How many sequence of integers ($j_1 , j_2 , . . . , j_k$) are there
such that $0 ≤ j_1 ≤ j_2 ≤ . . . ≤ j_k ≤ n$?
I've been given a hint,
(Hint: Reduce the ...
0
votes
1answer
31 views
If we define the expression $P(x)=x^2$ and an expression $Q(x) = |4x|$, then for how many integer values is $P(x) -Q(x)$ a positive quantity?
If we define the expression $P(x)=x^2$ and an expression $Q(x) = |4x|$, then for how many integer values is $P(x) -Q(x)$ a positive quantity?
$ a)2 \quad\quad\quad b) 4 \quad\quad c) 6 ...
2
votes
1answer
83 views
Bounding the number of integer solutions of the following inequality
Let $r\geq 1$ be a real number, $-1\leq x\leq 1$ a real number and $y>2$ a real number. We consider this data to be fixed.
How can I obtain an upper bound on the number of $(a,b,c,d)\in ...
4
votes
1answer
79 views
Is any of the sets a subset of a union of other sets?
I have eleven sets, all of them are subsets of $X:=\{(a,b,c,d)\in[-1,1]^4: a\le b,\text{ and } c\le d\}$:
$$\begin{align*}
A_1&:=\{(a,b,c,d)\in X: b\ge 0,\ c\le a+b+d\}\\
...
12
votes
1answer
489 views
Combinatorial proof of arithmetic geometric mean inequality
It is a well known fact that for positive reals $x_1, x_2, \dots, x_n$, their arithmetic mean is no less than their geometric mean:
$$ \frac{x_1 + x_2 + \dots + x_n}{n} \ge \sqrt[n]{x_1 x_2 \dots ...
11
votes
1answer
171 views
Is there a better bound than $(n^2)^{n^3}$ for the order of the commutator subgroup of a group whose center has index $n?$
Let $G$ be a group and $[G:Z(G)]=n<\infty,$ where $Z(G)$ is the center of $G$. A theorem says that in this case
$$o([G,G])\leq(n^2)^{n^3}, $$
where $o(\cdot)$ denotes order and $[G,G]=G'$ is ...
6
votes
4answers
288 views
Inequality with central binomial coefficients
For every even positive number $N$ we have:
$$ {2N \choose N } < 2^N {N \choose N/2 } < 2 {2N \choose N } $$
(Furthermore, $\frac{2^N {N \choose N/2 }}{{2N \choose N }} \to \sqrt{2} $ for ...
1
vote
3answers
373 views
27
votes
3answers
843 views
A combinatorial proof of $n^n(n+2)^{n+1}>(n+1)^{2n+1}$?
The statement is, of course, simply that the sequence $\left(1+\frac{1}{n}\right)^n$ is increasing.
Since the numbers $n^m$ have quite natural combinatorial interpretations, it makes me wonder if a ...
2
votes
1answer
147 views
Justifying a pair of inequalities involving the exponential function
I'm reading Fan Chung's Spectral Graph Theory. There's a pair of inequalities I don't know how to justify. Chung doesn't attempt to explain them, so maybe they're very obvious. Example 1.19 on page ...
17
votes
3answers
397 views
An Inequality Involving Bell Numbers: $B_n^2 \leq B_{n-1}B_{n+1}$
The following inequality came up while trying to resolve a conjecture about a certain class of partitions (the context is not particularly enlightening):
$$
B_n^2 \leq B_{n-1}B_{n+1}
$$
for $n \geq ...
9
votes
4answers
189 views
Bounding ${(2d-1)n-1\choose n-1}$
Claim: ${3n-1\choose n-1}\le 6.25^n$.
Why?
Can the proof be extended to
obtain a bound on ${(2d-1)n-1\choose
n-1}$, with the bound being $f(d)^n$
for some function $f$?
(These numbers ...
3
votes
2answers
130 views
Calculate max/min of $x_1 x_2+y_1 y_2+z_1 z_2+w_1 w_2$
What is a good way to calculate max/min of
$$x_1 x_2+y_1 y_2+z_1 z_2+w_1 w_2$$
where $x_1+y_1+z_1+w_1=a$ and $x_2+y_2+z_2+w_2=b$ and $x, y, z, w, a, b \in \mathbb{N} \cup \{0 \}$, and please explain ...
