2
votes
0answers
59 views

All those unit fractions add to 1?

Consider $$S(n)=\{x \mid x=(a_1 ,a_2,a_3 \cdots a_n) \text{ where } \sum_{r=1}^{n}\frac{1}{a_r} =1 \}$$ Now let $|S(n)|$ denote the cardinaly (order) of set $S(n)$. Thus: $S(1)= \{(1)\} \implies ...
0
votes
1answer
48 views

How to solve a inequality with fractions and roots in denominator and numerator

The inequality is like that: $$ \sqrt{\frac{3x+1}{2}}>1 $$ I have no idea how should i begin with it.
0
votes
0answers
70 views

If I have $\lfloor\frac{E}{K}\rfloor =\lfloor \frac{E}{K + m}\rfloor$, what is the upper limit of 'm' in terms of 'E' and 'k'

Given that E, K, m > 0, then is there a way to find out value of m in terms of E and ...
2
votes
2answers
85 views

The floor of a product of fractions

Evaluate: $ \displaystyle \Bigg \lfloor \prod_{n=0}^{248} \frac{33+8n}{29+8n} \Bigg \rfloor= \Bigg \lfloor \frac{33}{29} \times \frac{41}{37} \times \frac{49}{45} \times\ ...\ \times ...
0
votes
0answers
66 views

difficult inequality to prove

I need help proving this inequality is correct for a homework assignment: $$\displaystyle \left(\frac{13}{4}\right)^{n} \leq ...
5
votes
1answer
131 views

$\frac{\prod_{i=1}^n (1+x_i)-1}{\prod_{i=1}^n (1+x_i/\delta)-1} \stackrel{\text{?}}{\le} \frac{(1+x_n)^n-1}{(1+x_n/\delta)^n-1} $ .

Let $x_1 \le x_2 \le \cdots \le x_n$. Let $\delta>1$ be some positive real numbers. I assume that $0\le x_i <1$, for $i=1,\ldots,n$ and $x_n >0$. Does the following expression hold? $$ ...
1
vote
1answer
133 views

$\sum_{k_1+k_2+\cdots+k_N=n,\ k_i\ge0\in\mathbb Z}\frac1{\prod_{j=1}^{N}\{(N-1)k_j+1\}}\le 1$ is true for any $n,N\in\mathbb N$?

Is the following true for any $n,N\in\mathbb N$? $$\sum_{k_1+k_2+\cdots+k_N=n,\ k_i\ge0\in\mathbb Z}\frac1{\prod_{j=1}^{N}\{(N-1)k_j+1\}}\le 1$$ Motivation : I've known the $N=3$ case. ...
0
votes
0answers
12 views

If $d/dx_t ({\dot y}_t/y_t) > 0$ and $dy_t/dx_t < 0$ what can I then say about the sign of $d{\dot y}_t/dx_t$?

Assume that the rate of change in $y_t$ over time is ${{{{\dot y}_t}} \over {{y_t}}} = {x_t}$, where $x_t >0$. The derivative of this expression with respect to $x_t$ will be positive (well, it ...
5
votes
2answers
129 views

REVISTED$^2$: Fraction Existence Proof

Question 1: I'm asked to prove that there exists an $n\in\mathbb{N}$ such that $$\frac{1}{n+1}\leq\frac{a}{b}<\frac{1}{n},$$ where $0<\frac{a}{b}<1$. Here $\frac{a}{b}$ is a fraction in ...
2
votes
2answers
244 views

How to prove the fraction identity without using calculator

How to prove without calculator that $$ \frac{1}{1001} + \frac{1}{3001} > \frac{1}{1000}$$
0
votes
1answer
80 views

summation of fractions and inequalities

I am trying to prove that $\sum_{i=1}^{n}\frac{1}{a_i}\leqslant 2$, where all $a_i$ are less than 1000, and all $a_i$ have a lowest common multiple greater than 1000. This is what I have done so far: ...
7
votes
5answers
367 views

Solving inequality $\frac{2x}{x-2}>1$

I'm trying to solve $$\frac{2x}{x-2}>1$$ but I can't seem to get the correct answer. I'm doing something wrong but I don't know what; that is why I'm asking. This is what I've got: ...
1
vote
1answer
79 views

Looking for hints of this inequality

I think the following two inequalities are true. However, the proof may not be easy. Does anyone have any hints? Thank you very much! Fix $a>1$. there exists two constants $K_1$ and $K_2$, such ...
1
vote
2answers
85 views

Why is nothing being done to right side numerator?

From what I understand is that you have to multiply both sides by $2$, so that on the left side 2 cancel out and you are left with 3(t-7) but why does right side turn into 2t-12? So now you have: ...
17
votes
7answers
968 views

Simple proof that $8\left(\frac{9}{10}\right)^8 > 1$

This question is motivated by a step in the proof given here. $\begin{align*} 8^{n+1}-1&\gt 8(8^n-1)\gt 8n^8\\ &=(n+1)^8\left(8\left(\frac{n}{n+1}\right)^8\right)\\ &\geq ...