# inequality's questions - English 1answer

16.433 inequality questions.

### 7 Minimizing a symmetric sum of fractions without calculus

Calculus tools render the minimization of $$\frac{(1-p)^2}{p} + \frac{p^2}{1-p}$$ on the interval $(0,1/2]$ to be a trivial task. But given how much symmetry there is in this expression, I was curious ...

### 1 Is $\|u v\|_{L^2} \lesssim \|u\|_{L^\infty}\|v\|_{L^2}$ true because of Holder's inequality?

Let $f: S \to \mathbb{R}$ be a Lebesgue integrable function on a measure space $(S,\Sigma,\mu)$, then we may have \begin{align} \|f\|_{L^1} &:= \int_S |f| d\mu \\ \|f\|_{L^2} &:= \left(\...

### 17 Proof for triangle inequality for vectors

Generally,the length of the sum of two vectors is not equal to the sum of their lengths. To see this consider the vectors $u$ and $v$ as shown below. By considering $u$ and $v$ as two sides of a ...

### Question from a Moscow summer camp

Prove that, given $a,b,c > 0$ and $n$ a positive integer, $$\frac{a^n}{b+c}+\frac{b^n}{a+c}+\frac{c^n}{a+b} \geq \frac{a^{n-1}+b^{n-1}+c^{n-1}}{2}\ .$$ I've tried every rathole for hours on this ...

### 4 If $xyz=1$, prove $\frac{1}{y(x+y)}+\frac{1}{z(y+z)}+\frac{1}{x(z+x)} \geqslant \frac{3}{2}$

$x,y,z$ are positive real numbers such that $$xyz=1$$ Prove that $\dfrac{1}{y(x+y)}+\dfrac{1}{z(y+z)}+\dfrac{1}{x(z+x)} \geqslant \dfrac{3}{2}$. I have no idea how to solve this problem. I've tried ...

### 6 If $abc=1$ then $\sum\limits_{cyc}^{}{\frac{1}{b(a+b)}}\ge \frac{3}{2}$

If $abc=1$ for positive $a,b,c$, then $\sum\limits_{cyc}^{}{\dfrac{1}{b(a+b)}}\ge \dfrac{3}{2}$ I have tried the following,in decreasing order of success: 1)AM-GM:$a+b+c\ge 3$ and $ab+bc+ca\ge 3$ 2)...

### -3 Triangle Inequality (IMO 1991)

1 answers, 34 views inequality triangle
enter image description here Solved 2nd part but how to solve first part?

### Sphere-Sphere Collision

1 answers, 19 views linear-algebra inequality
A large, hollow sphere of internal radius R contains a smaller solid sphere of radius r. Write an inequality which would indicate that the small sphere has impacted the interior surface of the ...

### Proving that $-(2n+1/n+1) \leq 0$ for all n a natural number.

I was just wondering if someone can help me with a real basic proof. Prove that $-\frac{2n+1}{n+1} \leq 0 \forall n \in \mathbb N$. Is it just enough to show that $-\frac{2n+1}{n+1} > 0$ cannot ...

### 2 A polynomial inequality in three variables

I have a polynomial $P(A,B,C)$ where $A,B,C \in \mathbb{R}$ and $A>0,B>0$. $$P(A,B,C)=C^2+A^2-2CA+4AB-2B+C-A$$ When will $P(A,B,C)>0$ $P(A,B,C)=0$ $P(A,B,C)<0$ I have tried factorizing ...

### 1 Is there any relation between $\lfloor n/k\rfloor$ and $\lfloor n/(k+1)\rfloor$?

2 answers, 47 views inequality floor-function
Given two integers $n$ and $k$ such that $n\geq k+1$. Can we find any relation between $\left\lfloor\dfrac{n}{k}\right\rfloor$ and $\left\lfloor \dfrac{n}{k+1}\right\rfloor$? At first, I thought ...

### Classify the following function as increasing or decreasing

Let $f(x)=ax^3+bx^2+cx+d$, where $a,b,c,d$ are real is an increasing function. Given that $3b^2<c^2$. Define $g(x)= af’(x) + bf’’(x) + c^2$. Also , define $p(x) = \int_x^m g(t)dt$. Classify $p(x)$ ...

### 1 Solving the logarithmic inequality $(\log_3 x)^2 \lt \log_9( x^4)$

3 answers, 33 views inequality logarithms
The inequality for x is: $(\log_3 x)^2 \lt \log_9( x^4)$ I plotted the graphs to find as answer: $1\lt x \lt 9$ My approach: In my attempt to solve it, I used the logarithm properties to equal the ...

### 1 Show that $\sum_{k=1}^{n/2} \dbinom{n}{k}\alpha^{k(n-k)} \rightarrow 0$ for $\alpha <1$

1 answers, 37 views algebra-precalculus limits inequality