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: ...
2
votes
2answers
63 views

Solving inequalities with absolute values

This is the question: $$ \left| \frac{x+2}{3(x-1)} \right| \leq \frac{2}{3} $$ And this is my working out, first I squared both the numerator and denominator, then solved it as if it was a normal ...
0
votes
3answers
61 views

Proving this inequality

I am having trouble with proving an inequality. Assume we have two positive real numbers $a$ and $b$ such that $a+b=1$ and numbers $x > 0$ and $y > 0$. Prove: $$\frac{2}{\frac{a}{x} + ...
9
votes
1answer
134 views

Inequality $\frac{a + \sqrt{ab} + \sqrt[3]{abc}}{3} \leq \sqrt[3]{a \cdot \frac{a+b}{2} \cdot \frac{a+b+c}{3}}.$

Someone can to help me with a hint in the following problem: Show that for any $a,b,c>0$, $$\frac{a + \sqrt{ab} + \sqrt[3]{abc}}{3} \leq \sqrt[3]{a \cdot \frac{a+b}{2} \cdot ...
1
vote
2answers
353 views

Proving Holder's inequality using Jensen's inequality

Let $p$ and $q$ be positive reals such that $\frac{1}{p}+\frac{1}{q} = 1$, so that $p,q$ in $(1,\infty)$. For $\vec a$ and $\vec b \in \mathbb{R}^2$ prove that $|\vec a \cdot \vec b | \leq ||\vec ...
7
votes
3answers
171 views

If $x_1, \ldots, x_6$ are positive real numbers that add up to $2$. Show that:

If $x_1,x_2,x_3,x_4,x_5$ and $x_6$ are positive real numbers that add up to $2$, then: $$2^{12} \leq \left(1+\dfrac{1}{x_1}\right) ...
0
votes
1answer
115 views

Prove using Jensen's Inequality

Let $\alpha_1, \alpha_2, . . . , \alpha_n$ be the interior angles of a convex (but not necessarily regular) n-gon. Prove, that for all integers $n\geq3$: $$\cos \alpha_1 + \cos \alpha_2 + \cdots + ...
3
votes
0answers
56 views

A challenging non homogenous fractional inequality.

The following problem is a challenging generalization of several difficult inequalities, where none of the usual methods used in inequalities seems to work. I would like to know if someone has a ...
1
vote
2answers
134 views

Graphing - Absolute Value and Circle

The diagram Shows The Graphs of $y = |x + 2|$ and $y = \sqrt{4 - x^2}$ Write down the solution for $\sqrt{4 - x^2}$ is equal to or less than $y = |x + 2|$.
14
votes
3answers
357 views

How to prove this innocent inequality?

I need to show that $$ (a-d)^2+(b-c)^2\geq 1.6 $$ if $a^2+4b^2=4$ and $cd=4$. I was told that there exist nice and sweet elementary solution.
5
votes
1answer
111 views

Maximizing an unusual function (Putnam 1996)

“Fish," he said, "I love you and respect you very much. But I will kill you dead before this day ends.” -- Ernest Hemingway, The Old Man and the Sea I have, with varying degrees of concentration, ...
9
votes
2answers
148 views

Proving that $\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\dots+\frac{1}{\sqrt{100}}<20$

How do I prove that: $$\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\dots+\frac{1}{\sqrt{100}}<20$$ Do I use induction?
1
vote
2answers
71 views

Prove $|A| + |B| \ge |A \cup B|$ if $A$ and $B$ are finite sets

I have a question like this: Prove $|A| + |B| \ge |A \cup B|$ if $A$ and $B$ are finite sets. Here is the solution but I don't agree with it: Let $A = \{a_1, \dots, a_n\}$ and $B = \{b_1, \dots, ...
4
votes
4answers
263 views

Proof of $n^2 \leq 2^n$.

I am trying to prove that $n^2 \leq 2^n$ for all natural $n$ with $n \ne 3$. My steps are: induction base case: $n=0:$ $0² \leq 2⁰$ which is okay. inductive step: $n \rightarrow n+1:$ ...
10
votes
5answers
495 views

Prove $|a+b|+|a-b| \geq |a|+|b|$

I am fighting with this proof-writing problem for a while. The statement says $$|a+b|+|a-b| \geq |a|+|b|.$$ I know the triangle inequality which says$$|a+b| \leq |a|+|b|.$$ How can I use this ...
1
vote
1answer
61 views

inequality fraction

I saw this problem in an Australian maths olympiad: $6/10 < a/b < 10/15$ The problem asked for the lowest possible value of $b\in \mathbb{Z}$. I tried manipulating but couldnt derive one of ...
1
vote
2answers
92 views

Solving Equation Difficulty (analytically or by plotting)

I found this interesting problem in a programming forum. It would be great to get some help.. Solve for $K$, $K-1 \leq 27\cdot\log_{10}(9(K-1)) $ We plotted it to find that $K\leq77$. Can you solve ...
1
vote
1answer
660 views

Solving modular inequalities/constraint solving

A few of my current programming problems boil down to solving inequalities over modular domains and possibility could benifit from knowledge of efficient maths/algorithms rather than brute force ...