-3
votes
0answers
56 views

Which of the following is correct?

Let $$X = \frac{1}{1001} + \frac{1}{1002} + \frac{1}{1003} + \cdots + \frac{1}{3001}$$ Then find the correct option. (A) $X < 1$ (B) $X > \frac{3}{2}$ (C) $1 < X < \frac{3}{2}$ (D) ...
0
votes
1answer
33 views

An unusual inequality

Problem: $(x_i)_{i=1}^n$ is a finite sequence of positive integers. Define $f\big(S\big)=\displaystyle \sum_{i\,\in\, S\,\subseteq\, [n]}x_i$, and suppose $f$ is injective. Prove that: ...
1
vote
1answer
102 views

Proof of $x^y<y^x$ when $e\le y<x$ without calculus

I know that you can prove $e\le y<x$ implies $x^y<y^x$ by the following method: $$x^y<y^x\iff y \log x<x\log y\iff \frac{\log x}x<\frac{\log y}y,$$ and since $\frac d{dx}\frac{\log ...
0
votes
1answer
42 views

Bounds on functions using inequalities?

I'm studying inequalities as part of a course on Numbers, Proofs and Mathematical Induction. There is one type of question that I don't understand, primarily because there's only one example in the ...
4
votes
1answer
52 views

If distinct numbers $a,b,c\in\mathbb N^+$ satisfy $(a+b)(a+c)=(b+c)^2$, prove that $(b-c)^2>8(b+c)$.

If distinct numbers $a,b,c\in\mathbb N^+$ satisfy $$(a+b)(a+c)=(b+c)^2$$prove that $$(b-c)^2>8(b+c).$$ The first thing I did after I saw the problem was turning the inequality into this: ...
3
votes
2answers
58 views

Let $a_1,a_2,\ldots,a_{100}$ be real numbers,each less than one,satisfy $a_1+a_2+\ldots+a_{100} >1$.

Let $a_1,a_2,\ldots,a_{100}$ be real numbers,each less than one,satisfy $$a_1+a_2+\ldots+a_{100}>1$$ Prove the following statements: (i) Let $n_0$ be the smallest integer $n$ such that ...
2
votes
1answer
37 views

Is $\frac{k!!}{j!!(k-j)!!}\leq\frac{k!}{j!(k-j)!}$ for all integers $j$ and $k$, where $0\leq j\leq k$?

For all integers $j$ and $k$, where $0\leq j\leq k$, is the inequality $\frac{k!!}{j!!(k-j)!!}\leq\frac{k!}{j!(k-j)!}$ true? I have a feeling that it is and it would be helpful to me if it is, ...
0
votes
1answer
27 views

Proving that a certain sequence is bounded from above

Let $p_1,p_2,p_3,..$ be the sequence of primes in increasing order ($p_1=2,p_2=3,...$) .Let $x_n$ be given by: ...
0
votes
1answer
34 views

$\frac{2}{\frac{a}{x}+\frac{b}{y}}\leq ax+by$

$\frac{2}{\frac{a}{x}+\frac{b}{y}}\leq ax+by$, where a+b=1 and $a,b,x,y>0$ real numbers. Any hints? part (a) was showing $\frac{2}{\frac{1}{x}+\frac{1}{y}}\leq \sqrt{xy}\leq \frac{x+y}{2}$. To ...
1
vote
1answer
48 views

Let $a_1,a_2,\ldots,a_{100}$ be real numbers,each less than one,satisfy $a_1+a_2+\ldots+a_{100} >1$.Prove the following statements

Let $a_1,a_2,\ldots,a_{100}$ be real numbers,each less than one,satisfy $$a_1+a_2+\ldots+a_{100}>1$$ Prove the following statements: (i) Let $n_0$ be the smallest integer $n$ such that ...
3
votes
2answers
95 views

If $a_1+a_2+\ldots+a_{2000}>a_1a_2\ldots a_{2000}$, prove that at least $1990$ of those numbers are equal to $1$.

If $a_1,a_2,\ldots,a_{2000}\in\mathbb N$ and$$a_1+a_2+\ldots+a_{2000}>a_1a_2\ldots a_{2000}$$ Prove that at least $1990$ of those numbers are equal to $1$. That's an unusual problem for me and I ...
1
vote
1answer
73 views

If $m,n\in \mathbb N$ and $n>m$, prove that $lcm(m,n)+lcm(m+1,n+1)>\frac{2mn}{\sqrt{n-m}}$.

Where $lcm$ is the least common multiple. I've changed it to: $$\frac{mn}{\gcd(m,n)}+\frac{(m+1)(n+1)}{\gcd(m+1,n+1)}>\frac{2mn}{\sqrt{n-m}}$$ Can't see how to continue. Is there a way to ...
1
vote
4answers
146 views

Which is larger :: $y!$ or $x^y$, for numbers $x,y$.

This is a generalization of this question :: Which is larger? $20!$ or $2^{40}$?. No explicit general solution was presented there and I'm just curious :D Thank-you. Edit :: I want a most-general ...
0
votes
1answer
28 views

Prove that there exist two infinite sequences that simultaneously satisfies all these conditions

Prove that there exist two infinite sequences $\langle a_n\rangle_{n\geq 1}$ and $\langle b_n\rangle_{n\geq 1}$ of positive integers such that the following conditions hold simultaneously: $$1 < ...
5
votes
1answer
131 views

How to prove this inequality $\pi(x) > \log x - 1$ involving the prime counting function?

Problem Prove that $\pi(x) > \log x - 1$. Progress Based on a hint and very elementary methods, I got that $$ \prod_{p \leq x} (1-p^{-1})^{-1} \leq \prod_{k=2}^{\pi(x)+1} (1-k^{-1})^{-1}. $$ The ...
5
votes
2answers
249 views

Proving Inequality with the Greatest Integer Function

Show that $$[(m+n)x]+[(m+n)y] \ge [mx+(n-1)y]+[my+(n-1)x]$$ where $m,~n \in \Bbb{N}$ and $0\le x,~y < 1$. I've tried everything for about half a day and still couldn't figure it out. ...
1
vote
1answer
21 views

Interesting continued fraction problem $|r_i-u_0/u_1|\le\frac1{k_ik_{i+1}}$

Let $u_0/u_1$ be a rational number in lowest terms, and write $u_0/u_1=\langle a_0, a_1,...,a_n\rangle$ in standard continued fraction notation. Show that if $0\le i<n$, then ...
4
votes
1answer
86 views

Does every record of the arithmetic derivative of natural numbers occur at a practical number?

Consider the arithmetic derivative of natural numbers, as defined here. By this definition, for every integer $n>1$, with canonical prime factorization ...
1
vote
1answer
67 views

Determine whether $\sigma(n)<e^\gamma n \omega(n)$ for all $n$ not of the form $2^x$

Determine whether $\sigma(n)<e^\gamma n \omega(n)$ for all $n$ not of the form $2^x$. In words (to define the symbols), the sum of the divisors of $n$ is less than the product of Euler's number to ...
1
vote
0answers
36 views

Ineqality regarding LCM of $1, 2, \ldots, n$

While going through F. Beukers proof of irrationality of $\zeta(3)$ I found the inequality $d_{n} < 3^{n}$ for all sufficiently large values of $n$ where $d_{n}$ denotes the LCM of all the numbers ...
1
vote
2answers
240 views

Proving that if $2a + 3b \ge 12m + 1$, then $a \ge 3m + 1$ or $b \ge 2m + 1$ [duplicate]

Let $a$, $b$, $m$ be integers. Prove that if $2a + 3b \ge 12m + 1$, then $a \ge 3m + 1$ or $b \ge 2m + 1$. I need help proving this. I am not sure what to do. Thank you for all of the edits. ...
2
votes
2answers
48 views

Prove modular inequalities $ab + ac\le a(b+ac)$ and $(a+b)(a+c)\ge a+b(a+c)$

How to prove $$(a\cdot b)+(a\cdot c)\le a\cdot\big(b+(a\cdot c)\big)$$ and $$(a+b)\cdot(a+c)\ge a+\big(b\cdot(a+c)\big)\;?$$ I have tried this. Using distributive property, I think we can get ...
2
votes
3answers
82 views

How to prove $(1+x)^n\geq 1+nx+\frac{n(n-1)}{2}x^2$ for all $x\geq 0$ and $n\geq 1$?

I've got most of the inductive work done but I'm stuck near the very end. I'm not so great with using induction when inequalities are involved, so I have no idea how to get what I need... ...
0
votes
0answers
42 views

What proportion of the positive integers satisfy $I(n^2) < (1 + \frac{1}{n})I(n)$, where $I(x)$ is the abundancy index of $x$?

Let $\sigma(x)$ denote the classical sum-of-divisors function, and let $$I(x) = \frac{\sigma(x)}{x}$$ be the abundancy index of the positive integer $x$. My question is this: What proportion of ...
0
votes
1answer
85 views

What proportion of the positive integers satisfy $I(n) < \frac{2n}{n + 1} \leq I(n^2)$ < 2?

Let $$I(x) = \frac{\sigma(x)}{x}$$ be the abundancy index of the positive integer $x$. Note that $\sigma(x)$ is the classical sum-of-divisors function. For example, $$\sigma(12) = 1 + 2 + 3 + 4 + ...
0
votes
0answers
97 views

Can an odd perfect number be divisible by 101?

Preamble - This question is an offshoot from the following earlier questions here at MSE: Can an odd perfect number be divisible by 825? Can an odd perfect number be divisible by 165? Odd perfect ...
1
vote
2answers
154 views

Does the following inequality hold if and only if $N$ is an odd deficient number?

Let $N \in \mathbb{N}$. (That is, let $N$ be a positive integer.) This is in reference to two of my earlier questions here at MSE: Does the following inequality hold true, in general? Does this ...
3
votes
0answers
162 views

A question about odd perfect numbers

Edit [in response to a comment from anon]: Hereinafter, $N$ is a positive integer, $\sigma(N)$ is the sum-of-divisors of $N$, $\omega(N)$ is the number of distinct prime factors of $N$, and ...
1
vote
2answers
142 views

Does this inequality hold true, in general?

Let $$N = \prod_{i=1}^{\omega(N)}{{p_i}^{\alpha_i}}$$ be the prime factorization of the positive integer $N$. Does the following inequality hold true in general? ...
2
votes
1answer
132 views

Floor Inequalities

Proving the integrality of an fractions of factorials can be done through De Polignac formula for the exponent of factorials, reducing the question to an floored inequality. Some of those inequalities ...
6
votes
4answers
279 views

Greatest integer $n$ where $n \lt (\sqrt5 +\sqrt7)^6$

I'm really not sure how to do this. I factored out a power of $3$ and squared so that I have $2^3 (6+\sqrt{35})^3 \gt n$ , and I know that if I can prove that $12^3-1 \le (6+\sqrt{35})^3 \lt 12^3$ I ...
1
vote
1answer
81 views

Proving the existence of integer solutions to linear inequalities

Let $b_k\in\mathbb{Z}^n$ for $1\le k\le m$ for some $m,n$. I wish to prove the existence of two vectors $x,y\in\mathbb{Z}^n$ such that for all $k$, $b_k\cdot x\ne 0$ and $b_k\cdot y\ne 0$ with ...
4
votes
1answer
95 views

Finding an upper bound on a fraction

$0<\varepsilon <1$. If $n_k$ and $a_k$ are positive integers for which $$n_{k+1}=a_{k+1}n_k+n_{k-1}$$ Let $L\in\mathbb{N}.$ If $L>a_k \ge 3$, what's the smallest upper bound I can place on ...
2
votes
1answer
62 views

Does this inequality have any solutions for composite $n \in \mathbb{N}$?

Does this inequality have any solutions for composite $n \in \mathbb{N}$? $$\sqrt{2} < \frac{\sigma_1(n^2)}{n^2} < \frac{4n^2}{(n + 1)^2}$$ Note that $\sigma_1$ is the sum-of-divisors ...
3
votes
1answer
97 views

Does this inequality have any solutions in $\mathbb{N}$?

Does this (number-theoretic) inequality have any solutions $x \in \mathbb{N}$? $$\frac{\sigma_1(x)}{x} < \frac{2x}{x + 1}$$ Notice that we necessarily have $x > 1$.
1
vote
2answers
80 views

Find the greatest integer $N$ such that…

Find the greatest integer $N$ such that $N<\dfrac{1}{\sqrt{33+\sqrt{128}}+\sqrt{2}-8}$. The way I did it is this: first, I rewrote the biggest square root as $\sqrt{1+2*16+8\sqrt{2}}$. Then I ...
2
votes
1answer
58 views

Prove that $n^2 \leq \frac{c^n+c^{-n}-2}{c+c^{-1}-2}$.

Let $c \not= 1$ be a real positive number, and let $n$ be a positive integer. Prove that $$n^2 \leq \frac{c^n+c^{-n}-2}{c+c^{-1}-2}.$$ My initial thought was to try and induct on $n$, but the ...
2
votes
1answer
136 views

Average of divisors of n.

Let n be a natural number and let $f(n)=\frac{\sigma(n)}{d(n)}$ be the arithmetical average of n's divisors. Either prove or give a counterexample that for all natural numbers like n, which are not ...
1
vote
1answer
89 views

Finding integers to satisfy two inequalities.

Let $a,b,c $ be integers. We want to prove that there exists some integers $r,u,s,t$ such that $ru-st=1$ and $$|2art+b(ru+ts)+2csu|\le |ar^2+brs+cs^2|\le |at^2+btu+cu^2|$$ This problem is from: ...
9
votes
4answers
117 views

Prove That $x=y=z$

If $x, y,z \in \mathbb{R}$, and if $$ \left ( \frac{x}{y} \right )^2+\left ( \frac{y}{z} \right )^2+\left ( \frac{z}{x} \right )^2=\left ( \frac{x}{y} \right )+\left ( \frac{y}{z} \right )+\left ( ...
6
votes
5answers
162 views

$n!>n^m$ for $n\ge?$

I want to find a natural number $N$ in terms of $m(\in\mathbb N)$, such that $$n!>n^m \;, \forall n \ge N$$ Also, (how) can we prove that $n!-n^m$ is an increasing sequence for $n\ge N$? I was ...
2
votes
1answer
80 views

Calculate or bound infimum

Let $a_1, \ldots, a_n \in\mathbb R$ and nonnegative let $b\geq1$ and $c\in [0,1]$. Calculate or bound from above $$ \inf \left\{d>0: \sum_{i=1}^n \ln ...
6
votes
1answer
85 views

How to prove $\sum_{i=1}^{n-1}\frac{1}{\operatorname{lcm}(a_i,a_{i+1})}\lt1$ where $a_i\in\mathbb N$ and $a_i\lt a_{i+1}$?

Let $a_1,a_2,\ldots ,a_n\in\mathbb N$ and $a_1\lt a_2\lt\cdots\lt a_n$. Then how to prove $$\sum_{i=1}^{n-1}\frac{1}{\operatorname{lcm}(a_i,a_{i+1})}\lt1$$ Thanks in advance
19
votes
8answers
809 views

Comparing $2013!$ and $1007^{2013}$

I have to compare the following two numbers: $$2013! \text{ and } 1007^{2013}$$ where $n! = 1 \times 2 \times \cdots \times (n-1) \times n$. I tried in different ways to group the $1 \times 2 ...
2
votes
0answers
53 views

Lower bound on diophantine system of inequalities with all but one non-linear constraint

I have a system of $n+1$ diophantine inequalities, in the following form: $$f_{1}(x_1, x_2, \dots, x_n) \geq 0$$ $$f_{2}(x_1, x_2, \dots, x_n) \geq 0$$ $$\vdots$$ $$f_{m}(x_1, x_2, \dots, x_n) ...
1
vote
1answer
286 views

How to justify $\phi(n) \ge \sqrt{n}$

If $\phi(n)$ is the Euler-totient function, how can I show that $\phi(n) \ge \sqrt{n}$?
4
votes
1answer
201 views

$x^2+y^2=z^2(1+xy)$ prove $z=\min \{x;y;z\}$ (with $x,y,z \in \mathbb{Z^+}$)

$x,y,z \in \mathbb{Z^+}$ such that $x^2+y^2=z^2(1+xy)$. Prove $z=\min \{x;y;z\}$ $$x^2+y^2=z^2(1+xy) \iff xy = \frac{x^2+y^2} {z^2} - 1$$. Assum $z>y \implies xy < x^2/z^2$, we have $xy \in Z ...
1
vote
3answers
46 views

inequality with one number and a sum of numbers

Let $x_1, \ldots, x_n $ be non-zero real number, such that $\sqrt{x_1^2 + \cdots + x_n^2}=1$. Show that for any $i = 1, \ldots, n$, $$|x_i| \leq \sqrt{\frac{x_1^2 + \cdots + x_n^2}{n}}= ...
2
votes
2answers
48 views

Proof of a comparison inequality

I'm working on a problem that's been giving me some difficulty. I will list it below and show what work I've done so far: If a, b, c, and d are all greater than zero and $\frac{a}{b} < ...
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.