2
votes
3answers
80 views

Maximize $\sqrt{2x + 13} + \sqrt[3]{3y+5} + \sqrt[4]{8z+12}$

Given three non-negative (as pointed out by Calvin Lin) real numbers $x+y+z = 3$, find the maximum value of $\sqrt{2x + 13} + \sqrt[3]{3y+5} + \sqrt[4]{8z+12}$. (Source : Singapore Math Olympiad ...
0
votes
2answers
31 views

sequence $a_n = \lceil \sqrt{2}n \rceil $

I was trying to prove $\lceil \sqrt{2}n \rceil + \lceil \sqrt{2}m \rceil \geq \lceil \sqrt{2}(n+m) \rceil$ where $m,n\in \mathbb{z}$ Direct proof I tried but could not figure out. I tried fixing m ...
2
votes
3answers
123 views

Finding the values of $A,B,C,D,E,F,G,H,J$

Given that the letters $A,B,C,D,E,F,G,H,J$ represents a distinct number from $1$ to $9$ each and $$\frac{A}{J}\left((B+C)^{D-E} - F^{GH}\right) = 10$$ $$C = B + 1$$ $$H = G + 3$$ find (edit: ...
4
votes
2answers
82 views

Finding the value of $(bc-ad)(ac-bd)(ab-cd)$

Let $a,b,c,d$ be $4$ distinct non-zero integers such that $a+b+c+d = 0$. It is know that the number $$M = (bc - ad)(ac - bd)(ab-cd)$$ lies strictly between $96100$ and $98000$. Determine the value ...
5
votes
1answer
125 views

How prove this number theory inequality $\left(\dfrac{1}{N}\sum_{n=1}^{N}(\omega{(n)})^k\right)^{\frac{1}{k}}\le k+\sum_{q\le N}\frac{1}{q}$

show that: for any positive numbers $k$ and $N$, have $$\left(\dfrac{1}{N}\sum_{n=1}^{N}(\omega{(n)})^k\right)^{\frac{1}{k}}\le k+\sum_{q\le N}\dfrac{1}{q}$$ where $\displaystyle\sum_{q\le N}$ is ...
18
votes
4answers
291 views

Find the smallest k such that $n^k > \sum_{i=0}^{n-1} i^k$

Let $n \in \mathbb{N}$. Is it possible to find the smallest $k \in \mathbb{N}$ such that $$n^k > \sum_{i=1}^{n-1} i^k \ ?$$ It's easy to prove that such $k$ exist because: $$n^k > 1^k + 2^k ...
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 ...
2
votes
0answers
87 views

Sum of inverse of pairwise square roots [closed]

Prove that $$\sum_{i=1}^{50} \frac{1}{\sqrt{2i-1}+\sqrt{2i}} \gt \frac 92$$
0
votes
0answers
43 views

Given a set of inequalities, use what algorithm to minimize one variable?

I have the following set of inequalities: $$C_1 \le y-T_1x_1 \lt C_1+D_1$$ $$C_2 \le y-T_2x_2 \lt C_2+D_2$$ $$C_3 \le y-T_3x_3 \lt C_3+D_3$$ where $x_1, x_2, x_3$ and $y$ are non-negative integer ...
7
votes
1answer
104 views

Inequality with four positive integers looking for upper bound

Umm. This comes from Diophantine quartic equation in four variables and will finish the most important part if it can be done. Four positive integers $w,x,y,z.$ One equation and two inequalities $$ ...
11
votes
1answer
115 views

A claim that a is a square

We have integers $a,b,c,d$ such that $a<b\le c<d$ and $ad=bc$ and $\sqrt{d}-\sqrt{a}\le 1$.Show that $a$ is a perfect square.This question is pretty unbelievable for me.anyway I don't know if I ...
0
votes
0answers
24 views

Exponential Diophantine Inequality

how would one go about solving inequality of the form $|a2^n-b2^k|>1 $ for $a,b \in R$ and $n,k \in Z$. Assume that $|a|>|b|$. Any help will be appreciated. Thank you
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 ...
-1
votes
1answer
83 views
13
votes
2answers
197 views

Prove $\log_5{30}<\log_8{81}$

It's easy to prove this by calculator or computer, and I wonder can we prove that $$\log_5{30}<\log_8{81}\tag 1$$ by pencil and paper ? Thanks in advance ! Edit: $(1)$ can be written as ...
0
votes
1answer
36 views

Equality question

Hi I'm a bit confused with this? $\frac{1}{x} < 0 \iff x\frac{1}{x} < x\times 0 =0 \iff 1 < 0$ This was another question that I saw which was $\frac{1}{x} < 0$ but when I multiplied by ...
3
votes
1answer
98 views

Lower bound on a number theoretic function

Let $n$ be a positive odd integer, let $$n_j = \Bigl\{\frac{n}{2^{j+1}}\Bigr\}\,,$$ where $\{x\}$ denotes the fractional part of $x$, and finally let $k = \lceil \log_2 n\rceil$. Consider the ...
4
votes
1answer
111 views

Generalization of an inequality $0\lt e^6-{\pi}^4-{\pi}^5\lt 0.00002$

Question : Is the following true? For any $n\in\mathbb N$, there exists a triple $(k,l,m)\ (k,l,m\in\mathbb N)$ such that $$0\lt e^k-{\pi}^l-{\pi}^m\lt{10}^{-n}.$$ Motivation : A friend ...
0
votes
0answers
30 views

Lower-Upper bounds on the cardinality of a bounded set

Let $S$ be a finite set which is a subset of $\{(\alpha ,\beta ):\alpha , \beta \in \mathbb{Z}, \alpha\geq 0, \beta \geq 0\}$ and $ T(x,y)=\sum_{(\alpha ,\beta ) \in S} h_{\alpha, \beta} ...
1
vote
1answer
132 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
1answer
169 views

Regarding Chebyshev's theta function

It is known that $x\sim \theta \left ( x \right )$, where $$\theta \left ( x \right )= \sum_{p\leqslant x}\log p.$$ For all values of x for which it has been calculated, $x> \theta \left ( x ...
3
votes
2answers
97 views

How prove this inequality $|n\sqrt{2009}-m|>\dfrac{1}{kn}$

if $k>\sqrt{2009}+\sqrt{2010}$,show that for any positive integer numbers $m,n$, have $$|n\sqrt{2009}-m|>\dfrac{1}{kn}$$ My try: $$\Longleftrightarrow(n\sqrt{2009}-m)^2k^2n^2>1$$ ...
6
votes
2answers
176 views

How find this$\frac{1}{{{p}_{1}}}+\frac{1}{{{p}_{2}}}+…+\frac{1}{{{p}_{n}}}<10$

Let ${{p}_{1}},{{p}_{2}},...,{{p}_{n}}$ be the prime numbers less than ${{2}^{100}}$. Prove that $$\frac{1}{{{p}_{1}}}+\frac{1}{{{p}_{2}}}+...+\frac{1}{{{p}_{n}}}<10$$ This problem is from this ...
0
votes
1answer
55 views

Relation of $e$ to other numbers…

I found the following result, When i was working on my calculator . $$x^y < y^x \quad ,x < y \quad \text{ for } x,y<e$$ $$x^y > y^x \quad ,x < y \quad \text{ for } x,y>e$$ I can't ...
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 + ...
11
votes
1answer
214 views

How to prove that $\sqrt{2}+\sqrt{3}>\pi$

Does someone know a other proofs (using properties of $\pi$) of following inequality: $$\sqrt{2}+\sqrt{3}>\pi$$ First proof: the area of ​​regular 48-gon circumscribed to the unit circle is ...
8
votes
1answer
178 views

Prove that $\dfrac{\pi}{\phi^2}<\dfrac{6}5 $

How to prove that$$\frac{\pi}{\phi^2}<\frac{6}5 $$ from the definition of $\pi$ and $\phi$? ($\pi=4\int_{0}^1\sqrt{1-t^2}dt,\phi=\dfrac{\sqrt5+1}2.$)
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
1answer
137 views

How prove this $|\{n\sqrt{3}\}-\{n\sqrt{2}\}|>\frac{1}{20n^3}$

Prove that $$|\{n\sqrt{3}\}-\{n\sqrt{2}\}|>\dfrac{1}{20n^3}$$ let $t=\{n\sqrt{2}\}-\{n\sqrt{3}\}$ and $k=[n\sqrt{3}]-[n\sqrt{2}]$ then we have ...
0
votes
0answers
59 views

How prove this $|t-2\sqrt{2}n|\le2\sqrt{2}n+\frac{1}{20}$

let $$t=\{n\sqrt{2}\}-\{n\sqrt{3}\},n\in N^{+}$$ show that $$|t-2\sqrt{2}n|\le2\sqrt{2}n+\dfrac{1}{20}$$ where $\{x\}=x-\lfloor{x}\rfloor$ my idea: let ...
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? ...
3
votes
0answers
76 views

$\pi^4+\pi^5 < e^6 $ [duplicate]

Any idea about this inequality: $$\large \pi^4+\pi^5<e^6$$ Any hints would be appreciated.
6
votes
2answers
167 views

If a sequence of natural numbers satisfies $\gcd(a_{i+1},a_{i})>a_{i-1}$, then $a_{n}>2^n$

Given a sequence $\{a_{n}\}$ in $\mathbb{N}$ such that $\gcd(a_{i+1},a_{i})>a_{i-1},$ for any $i\ge 2$, show that $a_{n}>2^{n-1}$. Thank you everyone, my friend asked me about this problem, ...
0
votes
0answers
88 views

estimate $\sum_{x<p\le x+y} \log{p}/p$

In his paper the prime number theorem via the large sieve, A. Hildebrand made use of the following inequality $$\sum_{x<p\le x+y} \frac{\log{p}}{p} \le (2+o(1))\log{\frac{x+y}{x}}$$ where $x\ge y$ ...
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 ...
-2
votes
1answer
94 views

Comparing $\gamma^e$ and $e^\gamma$ [duplicate]

How can I calculate without calculator or something like this the values of $\gamma^e$ and $e^\gamma$ in order to compare them? ($\gamma$ the Euler-Mascheroni constant) Note: the shape of this ...
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 $x > 1$.
2
votes
1answer
81 views

The smallest positive integer $n$ satisfying a given condition

Given any positive integer $g$, what is the smallest positive integer $n$ such that $$\left\lceil \dfrac{(n-3)(n-4)}{12}\right\rceil>g.$$$\lceil x\rceil$ is a ceiling function of $x$.
20
votes
2answers
300 views

How prove that:$\varphi(2)+\varphi(3)+\varphi(4)+\cdots+\varphi(n)\ge\frac{n(n-1)}{4}+1$

Prove that for $n\ge 3$, $$\varphi(2)+\varphi(3)+\varphi(4)+\cdots+\varphi(n)\ge\dfrac{n(n-1)}{4}+1$$ where $\varphi$ is the Euler's totient function I think we must use this ...
1
vote
1answer
48 views

Checking if a solution exists across two inequalities

If I have $ay < x < by$ $cx < y < dx$ With $a,b,c,d$ as known (they are real-valued, can be positive or negative or 0) and $x,y$ unknown, is there a methodical way to see if a solution ...
4
votes
0answers
88 views

how prove $\phi(n)\ge \frac{n}{6\log \log (n)} $ $\forall n\ge5 $

How to prove$\forall n\ge5 $ $$\phi(n)\ge \frac{n}{6\log \log (n)} $$ $\phi$ is Euler function Thanks in advance
4
votes
1answer
63 views

Trying to understand Theorem 2.27 in a recent paper on the Chebyshev function

In February 2013, Sadegh Nazardonyavi and Semyon Yakubovich posted on arxiv: Sharper estimates for Chebyshev's functions $\vartheta$ and $\psi$. I have a question about Theorem 2.27 on page 22. My ...
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 ...
1
vote
1answer
79 views

Solution to a functional equation

Let $n,i$ be positive integers and $C$ a strictly positive real value. Consider the equation for $f$ : $$1*\ln(f(n)) = C * \sum_{3<i* \ln(i) < \sqrt{n}} \left(\ln[f( i* \ln(i) )-1)] - \ln[(f( i* ...
0
votes
0answers
78 views

Using Stirling's Formula to approximate a difference of logarithms of factorials in the same way as Jitsuro Nagura.

In Jitsuro Nagura's classic proof of a prime existing between $x$ and $\frac{6x}{5}$, he uses Stirling's formula to show that: $$T\left(x\right) - T\left(\frac{x}{2}\right) - ...